text:
A global community of change makers and entrepreneurs transforming the future of healthcare
What is SoPE?
The Society of Physician Entrepreneurs (SoPE) is a global biomedical and healthcare innovation network with membership open to all healthcare professionals, entrepreneurs and industry stakeholders. SoPE was founded in 2011 by three physicians who shared a common vision – to empower physicians and other professionals to innovate healthcare through entrepreneurship. SoPE continues to gain momentum across the United States, Turkey, China and the United Arab Emirates, as well as with industry partners who directly support our members' efforts. Members include doctors, nurses, dentists, healthcare entrepreneurs, investors, engineers, academics, IT professionals and industry representatives. What brings them together? Our community members share a common focus – working at the crossroads of innovation and commercialization to make a difference in healthcare. SoPE provides a platform for members to accelerate and catalyze innovation, as well as a separate supporting 501(c)(3) tax-exempt Foundation for scholarship and philanthropic endeavors to advance healthcare.
What is SoPE's Purpose?
To provide a global community-based platform for biomedical and healthcare entrepreneurs to connect and collaborate. Our vision, mission and activities are ultimately about championing healthcare innovation from the inside out – better treatments, improved access, at lower costs.
Not just for physicians, SoPE supports healthcare professionals and individuals from all backgrounds to achieve their entrepreneurial goals. Making a better world through improved healthcare starts with a good idea – it's our goal to facilitate the transformation of good ideas into great innovations and our community is growing.
SoPE's peer-to-peer network empowers entrepreneurs by providing the education, connections, experience and funding opportunities necessary to commercialize ideas, inventions and discoveries. SoPE accelerates healthcare innovation by encouraging investment and support for new therapies and approaches, including drugs, devices, diagnostic tools, vaccines, digital health products, or business process innovations. SoPE was created to assist you in your journey from the science of innovation to the business of transforming healthcare.
What is the Real Problem that SoPE is trying to solve?
The long pipeline from idea to innovation is fraught with obstacles to commercialization. SoPE was established to fill the gap between medical professionals with great ideas and industrial partners with the resources and knowledge to make things happen.
SoPE recognizes that private practice, academic and community-based medical professionals are an extremely valuable source of healthcare-related innovations. Yet, healthcare professions often require a commitment that makes focusing on other projects or the notion of a "portfolio career" seem difficult or even impossible.
Healthcare professionals may also have limited hands-on knowledge or bandwidth to start and run a profitable business focused on developing a new healthcare product or idea. SoPE is the best place to gain knowledge from experienced and professional healthcare entrepreneurs while establishing influential contacts for support at all stages of commercialization.
How is SoPE Different?
Founded by skilled physicians who recognize that the best ideas often come from thinking outside the box, SoPE is unique in its make-up, leadership and mission. SoPE is the only organization founded and led by physicians with the specific purpose of providing support to entrepreneurial healthcare professionals as well as all stakeholders in the life science and healthcare communities.
What is the SoPE Solution?
SoPE operates in the very early stages of innovation commercialization to encourage, educate, and connect individuals through access to a global network of healthcare professionals, entrepreneurs and industry partners.
Have an idea but aren't sure how to go about making a prototype, protecting intellectual property or getting it into the hands of healthcare providers? SoPE gives new ideas a chance by empowering entrepreneurs with the tools and knowledge to take the first steps to commercialization.
Our resources allow members to explore strategies for all phases of commercialization including development, production and marketing. SoPE's community of experts can provide guidance, information and even connections to secure funding for your innovative healthcare idea.
Because improving healthcare is your passion and your business. Because a community is defined by the shared values of its members. Because your innovation has the potential to impact the way healthcare is delivered. Come join the conversation and see all that SoPE has to offer:
Supportive Community of Experienced Healthcare Entrepreneurs
SoPE connects you to like-minded entrepreneurs, support specialists and industry leaders who catalyze the innovation commercialization process. SoPE is your premier resource for information, support, advice and network connections.
Access to Global Industry Partners
Every inspiration needs proper guidance and resources to reach its full potential. Our industry partners possess expertise across a diverse range of healthcare and life science sectors and are always in search of new and innovative collaborations.
SoPE's ever expanding knowledge-base includes educational materials, news articles on cutting-edge technologies, information on available funding sources, guidance for investors and many other resources to help you on your path to innovation. SoPE members also receive significant discounts to major healthcare and life science conferences and training events throughout the year.
Gain access to online, local and national gatherings happening year-round. Our events bring together entrepreneurs, investors and industry leaders to facilitate interaction and foster new partnerships.
For our nominal annual membership fee, you will gain exclusive access to network contacts, member resources, expertise, events and training that will provide immeasurable value to you and your entrepreneurial endeavors.
DR. ARLEN MEYERS
president & co-founder
DR. JEFFREY HAUSFELD
chairman & co-founder
DR. STEVEN LEVINE
director & co-founder
DR. RANIA NASIS
DR. SUDHEN DESAI
"Our mission is to help our members get their ideas to patients, and we do that by providing education, resources and networks…what we do is connect dots, so our members can create things."
Dr. Meyers is a retired professor of otolaryngology, dentistry, and engineering at the University of Colorado Anschutz Medical campus and President and CEO of SoPE. He has founded several medical device companies and is also the cofounder of a medical tourism company. His primary research centers around biomedical and health innovation and entrepreneurship and life science technology commercialization. He consults for and speaks to companies, governments, colleges and universities around the world who need expertise and contacts in the areas of bio entrepreneurship, bioscience, healthcare, healthcare IT, medical tourism — nationally and internationally, new product development, product design, and financing new ventures. Dr. Meyers is a Fulbright Scholar and a former Harvard-Macy Fellow. A devoted and prolific member of the SoPE community, Dr. Meyers also leads the Colorado Chapter of SoPE.
"The Society of Physician Entrepreneurs stands at the crossroads of innovation and commercialization in healthcare and the life sciences."
With over two decades of active practice in otolaryngology and a list of successful healthcare and real estate ventures, Dr. Hausfeld is a consummate physician entrepreneur. His diverse expertise spans both medicine and business, allowing practical insights into bioentrepreneurship and innovation attainable only through years of experience. In addition to leading the National Capital Chapter of SoPE, he is tasked with being the SoPE Global Chapter Coordinator, establishing the guidelines and framework to set up new SoPE Chapters throughout the world. Dr. Hausfeld's commitment to SoPE derives from his dedication to improving healthcare delivery and his belief in the platform's ability to connect similar forward-thinking individuals.
"SoPE is really a misnomer – it's a Society of Patient-Centered Entrepreneurs that includes physicians, dentists, nurses, chiropractors, researchers, engineers, lawyers, accountants, investors, etc."
A practicing physician specializing in otolaryngology and allergy, Dr. Levine has a Connecticut private practice for more than twenty-five years and is Assistant Clinical Professor of Surgery at Yale University. In addition to pursuing business interests in the delivery of auditory healthcare diagnostic products, Dr. Levine has a special interest in the booming market of healthcare information technology and possesses expertise developing HIPAA-compliant tools and software. Dr. Levine also leads the Tristate New York Chapter of SoPE with focus on its integration into the New York bio-entrepreneurship community.
RANIA NASIS, MD, MBA
Rania is a seasoned executive who helps build early stage healthcare startups. She is founder and CEO of Starlings, an advisory firm that works with early stage healthcare startups on strategy, operations and finding product-market fit. She is also a Director of the Society of Physician Entrepreneurs (SoPE), a non-profit global biomedical and healthcare innovation network, co-leads the New York Tristate SoPE Chapter, and previously founded & launched the Los Angeles SoPE Chapter.
Her experience spans the healthcare, technology and finance sectors. She was formerly Chief Operating Officer of Element3 Health, a healthcare technology company in the senior technology space. Prior to that, she was Managing Director at Poliwogg Holdings, a healthcare focused investment firm. Dr. Nasis was also Managing Director at Regenerative Medicine Strategy Group, a biotech management consulting firm, and General Manager at California Stem Cell, Inc., a cellular therapeutics company (acquired by NeoStem, Inc.), where she guided daily operations and oversaw development of novel cell therapy products and research tools. She was also the co-founder and CEO of Wanzo, a digital health startup targeting childhood obesity through gamification of activity via a social network.
She received her medical degree from New Jersey Medical School, Masters of Business Administration from the Paul Merage School of Business at University of California at Irvine and Bachelor of Science degree from Drexel University. She has published multiple business articles and has made scientific contributions to peer-reviewed journals and textbooks. She is committed to building communities that foster healthcare entrepreneurship and supporting startups that use technology to improve health, well-being and the quality of care.
Dr. Desai earned his Doctorate of Medicine with Distinction in Research and Alpha Omega Alpha honors from Albany Medical College, after completing the six-year BS/MD program in conjunction with Rensselaer Polytechnic Institute. During his latter years of medical school, he was a scholar in the Clinical Research Training Program at the NCI/NIH, a clinical fellowship geared towards the development of translational researchers. He then went on to residency at Stanford University (General Surgery) and UCSF (view site), followed by fellowship in Vascular and Interventional Radiology at Northwestern University. He has practiced Interventional and Diagnostic Radiology for over eight years. He consults with multiple early-stage life sciences companies, venture capital firms and business schools. He has been appointed to multiple committees for the Society of Interventional Radiology and has lectured at the SIR annual meetings. As leader of the Houston chapter of the Society of Physician Entrepreneurs (SoPE), Dr. Desai works to engage physicians interested in innovation and idea development, as well as to provide mechanisms and insights to assist ideas seeing the light of day.


A sermon preached at St. Andrew's On-the-Sound Episcopal Church in Wilmington NC on October 20, 2013. The texts were Jeremiah 31:27-34 and 2 Timothy 3:14-4:5.
For thousands of years, human beings have been telling stories. From the obvious fact that we have inherited a Bible filled with stories that are (approximately) three thousand years old, to the even more ancient tales the Neanderthals painted with dirt and grime on the walls of their cavernous homes, humans have always been story-telling creatures. At their best, stories have a way of capturing not just our intellectual interest, but they grab us at our emotional core, tugging on the strings of our hearts, inviting us in to feel what the characters feel, to journey with them, to arrive at the end somehow changed or transformed.
Though, not all stories are about the journey. Think about how often stories get told at family gatherings. Even stories filled with inconsequential details take on a greater significance. Telling stories is a way to remember the past so that the present and the future maintain a sense of connection, of continuity. Speaking of family gatherings, one of my favorite things to happen when stories get told is how one story gets repeated time and time again, and each time a new detail emerges, or if you're like me, a new detail gets exaggerated. But no matter how tired the story is or how exaggerated it becomes, it's the telling of the story that matters because it's a reminder that we are caught up in something bigger than just the present moment and the problems it brings.
Some stories, however, dwell too much on the darkness of this present age. The never-ceasing news cycle replays horror stories of crime, tragedy, and violence. Wars and rumors of wars persist. Speaking of darkness, we've entered into a season where ghosts, ghouls, and goblins adorn our porches and legends of violence, betrayal, murder and mayhem cause a great fear to settle upon our hearts and minds. So the doors get locked twice, and the porch light is left on, and we shudder at the sound of anything that goes bump in the night. I'm fascinated by the Halloween season because I marvel at the stories we choose to tell, and the films we choose to watch…how we choose to dwell in the darkness if only but for a moment.
Dwelling in darkness certainly isn't just a modern phenomenon. The Hebrew Scriptures contain a great many stories about the people of Israel seemingly mired by darkness and destruction. Second only to their enslavement by Egypt, the Babylonian exile stands as a time of great darkness and gloom for the people of Israel. Their land stripped from them, their wealth and riches snatched by Babylonian hands, and their people divided and sent into exile. And leading up to this exile and captivity, the prophet Jeremiah had been warning God's people of a coming calamity, but they wouldn't heed his warnings. And so the Babylonians arrived and conquered Jerusalem. Though Israel's story had been a narrative of God's chosen people living into prosperity, they chose another narrative. They turned to false gods and forsook the ways of their forbearers. They chose a story of rebellion, and this choice ultimately cost them their identity.
Imagine how Jeremiah's words must have felt to those who were living in the midst of destruction. Any sense of abandonment by God would have been washed away with a flood of joy. Jeremiah chose to tell another story in the face of deep darkness and gloom. His story was one of repentance and restoration, of a new age where God would be known more intimately than ever before, and all the sins of the past would be wiped clean. Jeremiah chose to tell a story of hope.
During this season of stewardship, what kind of story will you and I choose to tell with our lives? We've been given an opportunity to think critically about how our money is spent and how we fill the hours of our days. And when all is said and done, we realize that stewardship is about much more than money or time. It's about a life lived in service to this grand story of grace and redemption. Jeremiah chose to tell a story of hope to a broken and beaten down people. The Apostle Paul admonished Timothy to "continue in what he has learned" and to "proclaim the message and be persistent whether the time is favourable or unfavourable." Last Sunday, Richard reminded us as a congregation filled with God's children to be thankful for all the blessings we have even in the midst of the troubles that may work against us. So, what kind of story will you tell? Will you live your life in such a way that the story of God's grace, redemption, and salvation will shine brightly? Will you be thankful for your blessings and seek to tell the story of God's peaceful kingdom with the sharing of those blessings? Will you help St. Andrew's to become a parish ever more faithful to the story of grace and transformation we have received, where the Gospel is preached and righteousness flourishes? What kind of story do you want your life to tell?


text:
// Code generated by client-gen. DO NOT EDIT.

package fake

import (
	rest "k8s.io/client-go/rest"
	testing "k8s.io/client-go/testing"
	v1 "knative.dev/net-certmanager/pkg/client/certmanager/clientset/versioned/typed/certmanager/v1"
)

type FakeCertmanagerV1 struct {
	*testing.Fake
}

func (c *FakeCertmanagerV1) Certificates(namespace string) v1.CertificateInterface {
	return &FakeCertificates{c, namespace}
}

func (c *FakeCertmanagerV1) CertificateRequests(namespace string) v1.CertificateRequestInterface {
	return &FakeCertificateRequests{c, namespace}
}

func (c *FakeCertmanagerV1) ClusterIssuers() v1.ClusterIssuerInterface {
	return &FakeClusterIssuers{c}
}

func (c *FakeCertmanagerV1) Issuers(namespace string) v1.IssuerInterface {
	return &FakeIssuers{c, namespace}
}

// RESTClient returns a RESTClient that is used to communicate
// with API server by this client implementation.
func (c *FakeCertmanagerV1) RESTClient() rest.Interface {
	var ret *rest.RESTClient
	return ret
}

redpajama_set_name: RedPajamaGithub


All dealings including all quotations and any other order placed following such quotation are subject to the following conditions of sale in which Berties Direct Ltd is referred to as "the Company"
Prices quoted by the Company are current but we reserve the right to make adjustments if necessary due to currency fluctuations, raw material or factory increases. Unless otherwise stated, all prices are exclusive of any applicable value added tax, for which the customer shall be additionally liable to the Company.
All calculations of equivalent pricing and price per unit are approximate.
Payment must be received for the whole of the price of the goods you order, and any applicable charges for carriage, before your order can be accepted unless we have agreed otherwise in advance in writing.
No payment shall be deemed to have been received, until the Company has received cleared funds. Unless otherwise agreed payment is required prior to delivery of the goods.
For so long as any amount to be paid for goods remains owing to The Company, goods will remain with The Company and will not pass to The Customer until the full payment has been received, subsequent orders and deliveries will be refused until such payment is received.
Delivery periods and dates are given in good faith, but are not the subject of any warranty or condition, and time shall not be of the essence of the contract in these respects. No liability will attach to the Company if delivery periods or dates are not met for any reason whatsoever.
Please advise us of any non delivery within 10 days and confirm in writing to enable us to claim under the carriers terms. Any shortage or damage on delivery must be reported to us immediately and confirmed in writing within 3 days.
Please call us on 01939 234232 to obtain details on how to return your product.
Until such time as the property in and legal title to the Goods passes to the Customer, the Customer shall hold the Goods as the Company's fiduciary agent and bailee and shall keep the Goods separate from those of the Customer and third parties and properly stored, protected, insured and identified as the Company's property. Until that time, the customer shall be entitled to resell or use the Goods in ordinary course of its business, but shall account to the Company for the proceeds of sales of the Goods, including insurance proceeds, and shall keep all such proceeds separate from any moneys of the customer and of third parties.
The Customer shall not be entitled to pledge or charge, by way of security for any indebtedness, any of the goods which remain the property of the Company but, if the Customer does so, all moneys owing by the Customer to the Company shall (without prejudice to any other right or remedy of the reseller) forthwith become due and payable.
The Company shall be entitled to maintain an action for the price of the goods notwithstanding that title in them has not passed to the customer.


Alicia GallegosPfc. Travis Collins of Greeley kisses his wife Amy Collins during a homecoming celebration for Bravo Company of the 244th Engineering Battalion at the City Park in Fort Collins.
FORT COLLINS — The most difficult part about returning home from the war in Iraq for Windsor resident Dan Klema is finding his place in the fabric of life he left behind.
even when it comes to simple things, such as what's for dinner.
"He says to me, 'For a year I walked into a place where they put food on a plate and I ate it," Jewell-Klema said.
The Klemas attended a celebration Saturday, held in Fort Collins City Park in honor of Bravo Company, 244th Engineering Battalion, which returned home a week ago. Although Klema was stationed with the Bravo Company in Iraq, he was actually in a different company and returned home separately last week.
American flags waved in the spring breeze as more than 200 family members and friends came to honor the soldiers homecoming.
Sen. Wayne Allard and Colorado State University head football coach Sonny Lubick spoke at the event.
Klema said he has never been more proud to be an American than he is today.
"We have it made in the shade compared to some people," he said.
Jewell-Klema said it was the little things she missed most about her husband while he was gone.
"It was not having someone to talk to at night, or not having someone to help pick up the girls from soccer," she said.
Although, Jewell-Klema said she is overjoyed that her family is complete once more, she worries about her husband's adjustment to civilian life.
"The hard thing for Dan is going back to the work force," she said.
Pfc. Travis Collins of the Bravo Company agrees that being back in the country has its obstacles.
Collins was married a month before he was shipped to Iraq, and his wife Amy said they never got the quality time that most newlyweds have.
"I worried we wouldn't be the same people anymore," Amy Collins said.
Klema said he is certain the adjustment back to family life won't take very long.
In fact, Alex Klema, 9, brought her Dad to school for show and tell last week, and Klema was an immediate hit with the children.
Klema-Jewell said it is her husband's selfless attitude that makes him a strong soldier and loving father.
Despite getting used to changes that come with being gone for a year, Collins summed up the emotion of most returning soldiers as he hugged his wife: "It feels great to be back here with my family," he said.


text:
namespace extensions {

namespace {

const char kExtensionAPIPerContextKey[] = "extension_api_binding";

using APISignature = std::vector<std::unique_ptr<ArgumentSpec>>;

// Returns the expected APISignature for a dictionary describing an API method.
std::unique_ptr<APISignature> GetAPISignature(
    const base::DictionaryValue& dict) {
  std::unique_ptr<APISignature> signature = base::MakeUnique<APISignature>();
  const base::ListValue* params = nullptr;
  CHECK(dict.GetList("parameters", &params));
  signature->reserve(params->GetSize());
  for (const auto& value : *params) {
    const base::DictionaryValue* param = nullptr;
    CHECK(value->GetAsDictionary(&param));
    std::string name;
    CHECK(param->GetString("name", &name));
    signature->push_back(base::MakeUnique<ArgumentSpec>(*param));
  }
  return signature;
}

// Attempts to match an argument from |arguments| to the given |spec|.
// If the next argmument does not match and |spec| is optional, a null
// base::Value is returned.
// If the argument matches, |arguments| is advanced and the converted value is
// returned.
// If the argument does not match and it is not optional, returns null and
// populates error.
std::unique_ptr<base::Value> ParseArgument(
    const ArgumentSpec& spec,
    v8::Local<v8::Context> context,
    gin::Arguments* arguments,
    const ArgumentSpec::RefMap& type_refs,
    std::string* error) {
  v8::Local<v8::Value> value = arguments->PeekNext();
  if (value.IsEmpty() || value->IsNull() || value->IsUndefined()) {
    if (!spec.optional()) {
      *error = "Missing required argument: " + spec.name();
      return nullptr;
    }
    // This is safe to call even if |arguments| is at the end (which can happen
    // if n optional arguments are omitted at the end of the signature).
    arguments->Skip();
    return base::Value::CreateNullValue();
  }

  std::unique_ptr<base::Value> result =
      spec.ConvertArgument(context, value, type_refs, error);
  if (!result) {
    if (!spec.optional()) {
      *error = "Missing required argument: " + spec.name();
      return nullptr;
    }
    return base::Value::CreateNullValue();
  }

  arguments->Skip();
  return result;
}

// Parses the callback from |arguments| according to |callback_spec|. Since the
// callback isn't converted into a base::Value, this is different from
// ParseArgument() above.
bool ParseCallback(gin::Arguments* arguments,
                   const ArgumentSpec& callback_spec,
                   std::string* error,
                   v8::Local<v8::Function>* callback_out) {
  v8::Local<v8::Value> value = arguments->PeekNext();
  if (value.IsEmpty() || value->IsNull() || value->IsUndefined()) {
    if (!callback_spec.optional()) {
      *error = "Missing required argument: " + callback_spec.name();
      return false;
    }
    arguments->Skip();
    return true;
  }

  if (!value->IsFunction()) {
    *error = "Argument is wrong type: " + callback_spec.name();
    return false;
  }

  *callback_out = value.As<v8::Function>();
  arguments->Skip();
  return true;
}

// Parses |args| against |signature| and populates error with any errors.
std::unique_ptr<base::ListValue> ParseArguments(
    const APISignature* signature,
    gin::Arguments* arguments,
    const ArgumentSpec::RefMap& type_refs,
    v8::Local<v8::Function>* callback,
    std::string* error) {
  auto results = base::MakeUnique<base::ListValue>();

  // TODO(devlin): This is how extension APIs have always determined if a
  // function has a callback, but it seems a little silly. In the long run (once
  // signatures are generated), it probably makes sense to indicate this
  // differently.
  bool signature_has_callback =
      !signature->empty() &&
      signature->back()->type() == ArgumentType::FUNCTION;

  v8::Local<v8::Context> context = arguments->isolate()->GetCurrentContext();

  size_t end_size =
      signature_has_callback ? signature->size() - 1 : signature->size();
  for (size_t i = 0; i < end_size; ++i) {
    std::unique_ptr<base::Value> parsed =
        ParseArgument(*signature->at(i), context, arguments, type_refs, error);
    if (!parsed)
      return nullptr;
    results->Append(std::move(parsed));
  }

  v8::Local<v8::Function> callback_value;
  if (signature_has_callback &&
      !ParseCallback(arguments, *signature->back(), error, &callback_value)) {
    return nullptr;
  }

  if (!arguments->PeekNext().IsEmpty())
    return nullptr;  // Extra arguments aren't allowed.

  *callback = callback_value;
  return results;
}

struct APIPerContextData : public base::SupportsUserData::Data {
  std::vector<std::unique_ptr<APIBinding::HandlerCallback>> context_callbacks;
};

void CallbackHelper(const v8::FunctionCallbackInfo<v8::Value>& info) {
  gin::Arguments args(info);

  // If the current context (the in which this function was created) has been
  // disposed, the per-context data has been deleted. Since it was the owner of
  // the callback, we can no longer access that object.
  v8::Local<v8::Context> context = args.isolate()->GetCurrentContext();
  gin::PerContextData* per_context_data = gin::PerContextData::From(context);
  if (!per_context_data)
    return;

  v8::Local<v8::External> external;
  CHECK(args.GetData(&external));
  auto* callback = static_cast<APIBinding::HandlerCallback*>(external->Value());

#if DCHECK_IS_ON()
  // If there is per-context data, then it should own the callback that is about
  // to be called. Double-check this in debug builds.
  APIPerContextData* data = static_cast<APIPerContextData*>(
      per_context_data->GetUserData(kExtensionAPIPerContextKey));
  DCHECK(data);
  DCHECK(std::any_of(
      data->context_callbacks.begin(), data->context_callbacks.end(),
      [callback](
          const std::unique_ptr<APIBinding::HandlerCallback>& live_callback) {
        return live_callback.get() == callback;
      }));
#endif  // DCHECK_IS_ON()

  callback->Run(&args);
}

}  // namespace

APIBinding::APIBinding(const std::string& api_name,
                       const base::ListValue& function_definitions,
                       const base::ListValue* type_definitions,
                       const base::ListValue* event_definitions,
                       const APIMethodCallback& callback,
                       ArgumentSpec::RefMap* type_refs)
    : api_name_(api_name),
      method_callback_(callback),
      type_refs_(type_refs),
      weak_factory_(this) {
  DCHECK(!method_callback_.is_null());
  for (const auto& func : function_definitions) {
    const base::DictionaryValue* func_dict = nullptr;
    CHECK(func->GetAsDictionary(&func_dict));
    std::string name;
    CHECK(func_dict->GetString("name", &name));
    std::unique_ptr<APISignature> spec = GetAPISignature(*func_dict);
    signatures_[name] = std::move(spec);
  }
  if (type_definitions) {
    for (const auto& type : *type_definitions) {
      const base::DictionaryValue* type_dict = nullptr;
      CHECK(type->GetAsDictionary(&type_dict));
      std::string id;
      CHECK(type_dict->GetString("id", &id));
      DCHECK(type_refs->find(id) == type_refs->end());
      // TODO(devlin): refs are sometimes preceeded by the API namespace; we
      // might need to take that into account.
      (*type_refs)[id] = base::MakeUnique<ArgumentSpec>(*type_dict);
    }
  }
  if (event_definitions) {
    event_names_.reserve(event_definitions->GetSize());
    for (const auto& event : *event_definitions) {
      const base::DictionaryValue* event_dict = nullptr;
      CHECK(event->GetAsDictionary(&event_dict));
      std::string name;
      CHECK(event_dict->GetString("name", &name));
      event_names_.push_back(std::move(name));
    }
  }
}

APIBinding::~APIBinding() {}

v8::Local<v8::Object> APIBinding::CreateInstance(
    v8::Local<v8::Context> context,
    v8::Isolate* isolate,
    APIEventHandler* event_handler,
    const AvailabilityCallback& is_available) {
  // TODO(devlin): APIs may change depending on which features are available,
  // but we should be able to cache the unconditional methods on an object
  // template, create the object, and then add any conditional methods. Ideally,
  // this information should be available on the generated API specification.
  v8::Local<v8::Object> object = v8::Object::New(isolate);
  gin::PerContextData* per_context_data = gin::PerContextData::From(context);
  DCHECK(per_context_data);
  APIPerContextData* data = static_cast<APIPerContextData*>(
      per_context_data->GetUserData(kExtensionAPIPerContextKey));
  if (!data) {
    auto api_data = base::MakeUnique<APIPerContextData>();
    data = api_data.get();
    per_context_data->SetUserData(kExtensionAPIPerContextKey,
                                  api_data.release());
  }
  for (const auto& sig : signatures_) {
    std::string full_method_name =
        base::StringPrintf("%s.%s", api_name_.c_str(), sig.first.c_str());

    if (!is_available.Run(full_method_name))
      continue;

    auto handler_callback = base::MakeUnique<HandlerCallback>(
        base::Bind(&APIBinding::HandleCall, weak_factory_.GetWeakPtr(),
                   full_method_name, sig.second.get()));
    // TODO(devlin): We should be able to cache these in a function template.
    v8::MaybeLocal<v8::Function> maybe_function =
        v8::Function::New(context, &CallbackHelper,
                          v8::External::New(isolate, handler_callback.get()),
                          0, v8::ConstructorBehavior::kThrow);
    data->context_callbacks.push_back(std::move(handler_callback));
    v8::Maybe<bool> success = object->CreateDataProperty(
        context, gin::StringToSymbol(isolate, sig.first),
        maybe_function.ToLocalChecked());
    DCHECK(success.IsJust());
    DCHECK(success.FromJust());
  }

  for (const std::string& event_name : event_names_) {
    std::string full_event_name =
        base::StringPrintf("%s.%s", api_name_.c_str(), event_name.c_str());
    v8::Local<v8::Object> event =
        event_handler->CreateEventInstance(full_event_name, context);
    DCHECK(!event.IsEmpty());
    v8::Maybe<bool> success = object->CreateDataProperty(
        context, gin::StringToSymbol(isolate, event_name), event);
    DCHECK(success.IsJust());
    DCHECK(success.FromJust());
  }

  return object;
}

void APIBinding::HandleCall(const std::string& name,
                            const APISignature* signature,
                            gin::Arguments* arguments) {
  std::string error;
  v8::Isolate* isolate = arguments->isolate();
  v8::HandleScope handle_scope(isolate);
  std::unique_ptr<base::ListValue> parsed_arguments;
  v8::Local<v8::Function> callback;
  {
    v8::TryCatch try_catch(isolate);
    parsed_arguments =
        ParseArguments(signature, arguments, *type_refs_, &callback, &error);
    if (try_catch.HasCaught()) {
      DCHECK(!parsed_arguments);
      try_catch.ReThrow();
      return;
    }
  }
  if (!parsed_arguments) {
    arguments->ThrowTypeError("Invalid invocation");
    return;
  }

  // Since this is called synchronously from the JS entry point,
  // GetCurrentContext() should always be correct.
  method_callback_.Run(name, std::move(parsed_arguments), isolate,
                       isolate->GetCurrentContext(), callback);
}

}  // namespace extensions


text:
Croydon Town Centre
Police want to trace these two men after mass brawl outside nightclub in Surrey Street in Croydon
Police have today (Wednesday, January 24) released CCTV images of two men they are looking to identify following the fight on December 16
Olivia Tobin
Detectives in Croydon are trying to trace these two men after a mass brawl in Surrey Street last December.
On December 16, at about 3.45am, a large fight took place where suspects, believed to be in their late teens and early 20s, were armed with an "array of weapons", according to the police.
Three men received stab wounds in the brawl, while two other men received knife wounds.
Following the fight, police cordoned off a large section of Surrey Street, between Iceland and the steps leading to The Arcade, for most of the Saturday (December 16), which disrupted trade at the historic market street, while they completed investigations at the scene.
Latest Croydon news
Caterham crash
Bashment ban
Selhurst Park redevelopment
Teen found guilty
All five injured men received treatment following the brawl and have since been discharged from hospital.
A 23-year-old was taken to hospital with a stab wound to the leg and a 24-year-old was found in Wandle Road with stab wounds to the chest, though his condition was not life-threatening.
The men police want to identify (Image: The Metropolitan Police)
Another 23-year-old man and a 19-year-old man also took themselves to a hospital where they were treated for a cut on the left hand and a stab wound to the right arm respectively.
The fifth victim of the fight was found in Denmark Road with a "slash wound to the face".
The ugly scenes took places outside nightclub 34 which was threatened with closure after the incident.
It has, though, been allowed to remain open, but will have its opening hours cut.
When officers reached the club glass bottles were thrown at them by clubbers leaving the venue, according to a spokesman for the Met Police.
The spokesman said that the disturbance then escalated and that several people drew weapons, including large knives and a machete.
Croydon property developer found not guilty of attacking woman when he went to change home's locks
No arrests have so far been made in relation to the brawl and police are appealing for witnesses or anyone with any information to come forward.
Police have today (Wednesday, January 24) released CCTV images of two suspects they are looking to identify.
Anyone who recognises the men in the images is asked to call Croydon CID on 101 or get in touch via Twitter @MetCC.
To give information anonymously call Crimestoppers on 0800 555 111 or go online to crimestoppers-uk.org

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Solving particular integral $R(x)$ has $a^x$ term. I am supposed to solve the below differential equation question.
$$
\left(D^2+4\right) y = \cosh(2x - 1) + 3^x
$$
I am not aware of a case where $R(x)$ has a term of form $a^x$. How should I proceed from here?

A: As usual, solve the homogeneous one first and then try a linear combination of $\sinh, \cosh$ and $3^x$ for a particular solution.
Note that
$$
\frac{d}{dx} \left[a^x\right]
 = \frac{d}{dx} \left[e^{x \ln a}\right]
 = e^{x \ln a} \ln a
 = a^x \ln a.
$$

A: $$(D^2+4)y_p=3^x$$
$$(D^2+4)y_p=e^{\ln 3x}$$
Note that $(u=\ln 3 x)$:
$$D=\dfrac {d}{dx}=\dfrac {d}{d \ln 3 x}\dfrac {d \ln 3 x
}{dx}=\ln 3\dfrac {d}{du} $$
So that the DE becomes:
$$(\ln^2 3D^2+4)y_p=e^{u}$$
$$y_p=\dfrac {1}{\ln^23 D^2+4}e^u=\dfrac {e^u}{\ln^23+4}$$
$$ \implies y_p=\dfrac {3^x}{\ln^23+4}$$

redpajama_set_name: RedPajamaStackExchange


text:
TITAN COMICS REVEALS CREATIVE TEAM FOR NEW TEKKEN SERIES!
by First Comics News on February 17, 2017 TITAN COMICS REVEALS CREATIVE TEAM FOR NEW TEKKEN SERIES!2017-02-17T23:39:43-07:00 - News
New original comic adventures based in the world of the best-selling fighting video game!
Titan Comics are thrilled to announce the phenomenal creative team of writer Cavan Scott (Doctor Who, Vikings: Uprising) and artist Andie Tong (Teenage Mutant Ninja Turtles) who kick-off the exciting new TEKKEN comic series in May, 2017!
Based on the smash hit video game series, Titan is working closely with BANDAI NAMCO Entertainment to deliver exciting in-universe tales. Hitting stores in May 2017, sensational new comic series will tap into TEKKEN's rich history, and will feature memorable characters from the celebrated video game canon including Jin Kazama, Heihachi Mishima, Yoshimitsu, Nina Williams and Paul Phoenix.
"I'm so excited to be bringing the explosive world of TEKKEN back to comics." said writer Cavan Scott, "I've been a fan of the series ever since I slipped the first game into my PlayStation back in 1995. We're having great fun delving into the series' history to bring back some familiar faces, along with all the fan-favorite fighters you'd expect. There's going to be intrigue, danger and, of course, knock-out battles galore!"
This hotly anticipated debut issue comes with five art covers to collect by artists Alex Ronald (Mycroft Holmes), Andie Tong, Anton Kokarev (Evil Dead 2: Beyond Dead By Dawn), Jimbo Salgado (Assassin's Creed® Uprising), and Nick Percival (Dark Souls).
Titan Comics' TEKKEN #1 hits stores and digital platforms in May, 2017 and is available for order from the March edition of PREVIEWS.
https://www.firstcomicsnews.com/titan-comics-reveals-creative-team-for-new-tekken-series/https://www.firstcomicsnews.com/wp-content/uploads/2017/02/Tekken-logo-600x257.pnghttps://www.firstcomicsnews.com/wp-content/uploads/2017/02/Tekken-logo-150x64.png 2017-02-17T23:39:43-07:00 First Comics NewsNews
New original comic adventures based in the world of the best-selling fighting video game! Titan Comics are thrilled to announce the phenomenal creative team of writer Cavan Scott (Doctor Who, Vikings: Uprising) and artist Andie Tong (Teenage Mutant Ninja Turtles) who kick-off the exciting new TEKKEN comic series in May,...


Sunrise and sunset times, civil twilight start and end times as well as solar noon, and day length for every day of February 2019 in Island, KY.
In Island, KY, the first day of February is 10 hours, 22 minutes long. The last day of the month is 11 hours, 21 minutes, so the length of the days gets 59 minutes longer in February 2019.


text:
genealogically speaking
Research and Records
Family Documents
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Charlotte A. Davis[1, 2]
Name Charlotte A. Davis
Born 4 Aug 1829 Doddridge, West Virginia, United States [1, 3, 4]
Reference Number 6930
_AMTID 190111188409:1030:113173531
Died 26 Apr 1891 Salem, Harrison, West Virginia, United States [4, 5]
Buried Doddridge, West Virginia, United States [5]
Greenbrier Seventh Day Baptist Cemetery
Person ID I6899 Megan's Big Tree
Last Modified 17 Oct 2016
Father Absalom Davis, b. 9 Dec 1809, Harrison, West Virginia, United States
, d. 29 Apr 1888, Doddridge, West Virginia, United States
(Age 78 years)
Mother Polina S. W. Davis, b. 8 Feb 1811, d. 3 Feb 1901, Doddridge, West Virginia, United States
Married 9 Oct 1828
Relationship First Cousins
Family Henry J. Meek, b. Between 1828 and 1829, Virginia, United States
, d. Y
1. Lewis H. Meek, b. Jan 1852, Doddridge, West Virginia, United States , d. Y
2. Affa Jane Meek, b. 21 Jun 1853, Doddridge, West Virginia, United States , d. 8 Jun 1941, Petroleum, Ritchie, West Virginia, United States (Age 87 years)
3. Alfred N. Meek, b. 9 May 1856, Doddridge, West Virginia, United States , d. 19 Mar 1940, Parkersburg, Wood, West Virginia, United States (Age 83 years)
4. Almeda L. Meek, b. 22 Jul 1857, Doddridge, West Virginia, United States , d. Between 1920 and 1930 (Age 62 years)
5. B. Meek, b. Between 1858 and 1859, Virginia, United States , d. Y
6. Julia Meek, b. 22 Apr 1860, Doddridge, West Virginia, United States , d. 7 Jul 1942, Clarksburg, Harrison, West Virginia, United States (Age 82 years)
7. Polina S. Meek, b. Nov 1861, West Virginia, United States , d. Y
8. Charles J. Meek, b. Nov 1864, Doddridge, West Virginia, United States , d. Between 1930 and 1940 (Age ~ 65 years)
9. Mary Helen Meek, b. 1 Jul 1867, Doddridge, West Virginia, United States , d. 28 Apr 1888, Doddridge, West Virginia, United States (Age 20 years)
10. Anderson G. Meek, b. 2 Aug 1869, Doddridge, West Virginia, United States , d. Y
11. Missouri Meek, b. 8 Aug 1871, Doddridge, West Virginia, United States , d. 27 Sep 1910, Doddridge, West Virginia, United States (Age 39 years)
Born - 4 Aug 1829 - Doddridge, West Virginia, United States
Died - 26 Apr 1891 - Salem, Harrison, West Virginia, United States
Buried - Address:
Greenbrier Seventh Day Baptist Cemetery - - Doddridge, West Virginia, United States
[S8] 1850 Census .
[S12] 1870 Census .
[S1634] West Virginia Vital Research Records .
[S797] Find-a-Grave Website .


With numerous preferences, strategies, and practices of post workout recovery, any athlete can relate to the intricacies of post workout nutrition. And the fitness industry is full of options. When it comes to nutrition All Pro Science seeks to capture recovery better than the rest of the field.
All Pro Science is a supplement company with roots in athletics and supporting athletic performance. The APS product I recently tested myself was APS Recovery. The product I tried was a great tasting pineapple-orange flavor powder that mixes well with water. This product is built for those aforementioned precious post workout moments.
APS prides themselves on quality. APS Recovery is compromised of all-natural ingredients and includes no artificial flavors. APS Recovery is made from one hundred percent grass-fed whey and dons a 3:1 carbohydrate to protein ratio, which provides adequate carbohydrate to replenish depleted glycogen stores.
With the aim to extend endurance, APS Recovery provides dietary energy to last through longer training sessions.
With regards to recovery, the product is designed to reduce recovery time, soreness, and fatigue, as well as aid in the growth of muscle.
Lastly, any athlete can benefit from the rehydration that APS offers during and after training sessions.
As someone who doesn't utilize recovery supplements regularly, I experienced APS Recovery as a bit of a novice. I found the pineapple-orange flavor to actually be quite good. In some ways it even felt like a post workout treat. It's not as filling as a big meal after training, but I understand there are a good number of athletes who find it difficult to stomach real food shortly after training. The product seemed to do the job.
I know if I miss my post workout nutrition window of 30-45 minutes, I start to get tunnel vision and my only purpose on Earth becomes finding food. The ease of consuming APS Recovery offers a simple solution to others who have this same experience as me. The carbohydrate load of APS Recovery offers the glycogen to maintain energy levels and top off glycogen stores depleted from exercise. The protein component, then, is ideal for synthesizing this very recovery.
In a perfect world, one would always benefit most from real food. In the real world, however, we often need to rely on brands like APS with the knowledge and power to create products that support high performance training. Having APS Recovery means we can maximize those valuable post workout minutes to recover.
A common jab at similar products is that they can be largely dominated by simple sugars and sold as just what the body needs to recover. With 8 grams of sugar per serving, APS Recovery has surprising moderate sugar levels, when compared to similar products by Endurox and CytoSport, for example.
One could argue that athletes engaged in training and performances outside of the norm have needs that lie outside of the norm, as well. I think that's where a credible company like All Pro Science steps in. For athletes looking for a quality protein source after training, APS has the quality products and the experience in the industry to fulfill those needs. Plus, it tastes great.
APS Recovery is available online at www.allproscience.com for $39.99.


text:
Juventus to rival Barcelona for Lyon's Depay
Dec, 22nd 2020 – Barcelona now face competition from Juventus to sign Lyon forward Memphis Depay. Ronald Koeman, who worked with Depay during his time in charge of the Holland national team, identified him as a key target in the summer but a deal never materialized.
Depay's contract expires at the end of this season and could be available for as little as £4.5million when the transfer window opens next month.
According to latest information, Lyons are willing to sell Depay for just €5 million. Any potential deal would also need approval from the interim board as the Barcelona presidential elections are not set to be held for another month.
On Barcelona's side, things haven't changed much. If they are to get approvals to sign Depay, they would then need to sell a couple of players for both sporting reasons and financial reasons.
Depay wants to come back to Barcelona but other clubs could be interested in him too since Lyon is eager to sell. He and Lyon are on rush to facilitate a move and if he is at the top of their wish list, it's time for Barca to make the next move.


text:
\section{Introduction}\label{section1}

The discrimination of quantum states is an area that has received
considerable attention. For a recent review see \cite{springer}, for example.
Pure state discrimination has proven to be useful for both quantum
cryptography and for quantum algorithms \cite{bennett,bergou1}.  The basic
protocol is the following.  One has a list of possible quantum
states, and one is presented with a system that is guaranteed to be in
one of them.  Our task is
to determine which.  If the states are not orthogonal, they cannot be
discriminated perfectly,
and it is necessary to specify what kind of strategy we wish to use.  One
possibility is
minimum error discrimination \cite{helstrom}.  In this procedure, we always
specify a state, but, because the
states are not orthogonal, there is some chance that we will make a mistake.
Minimum-error procedures minimize the chance of making a mistake.  Another
alternative is
unambiguous discrimination.  This procedure never makes a mistake, but it can
sometimes fail.
If the list of states contains $N$ possibilities, then this procedure has
$N+1$ possible outputs, one for each
of the states on list plus a failure output.  If we receive an output that
specifies one of the states, then we
know what the state of our system was, and if we receive the failure output,
then we have no
idea what it was. An optimum unambiguous procedure is one that minimizes the
probability of
receiving the failure output.  We shall be considering unambiguous
discrimination here.

The problem of optimal unambiguous discrimination between two pure quantum states was solved by Ivanovic, Dieks and Peres \cite{ivanovic,dieks,peres}.  They assumed that each of the two states were equally probable.  The case in which they are not was treated by Jaeger and Shimony \cite{jaeger} (for a somewhat simpler derivation of their result see \cite{bergou2}).  The case of more than two pure states is not simple and has been considered by a number of authors.  There are a few general results, but explicit procedures are available only for special cases \cite{terno,sun1}.
One important result is that for unambiguous discrimination to be possible, the states on the list must be linearly independent \cite{chefles}.  There are also lower bounds on the failure probability \cite{zhang}.

The discrimination of two mixed states has only been considered more recently.
Before we cite the previous results it will be useful to introduce some
terminology at this point. The support $S_{j}$ of the density operator
$\rho_{j}$, describing a quantum state, is the subspace of the entire
Hilbert space $\mathcal{H}$ spanned by the eigenvectors of $\rho_{j}$
belonging to nonzero eigenvalues (for  $j=1,2$). The rank of the density
operator is equal to the dimension of its support. The subspace orthogonal
to its support, ${\bar S}_{j}$, is the kernel of the density operator
$\rho_{j}$ such that $\mathcal{H}=S_{j} \oplus {\bar S}_{j}$. We shall
denote the projector onto $S_{j}$ by $P_{j}$ and the projector onto
${\bar S}_{j}$ by ${\bar P}_{j}$. Equipped with these definitions we can
now make the following general statement. For two mixed quantum states
unambiguous discrimination is possible with a finite probability of
success if and only if the supports of their density operators are not
identical. Indeed, in such a case the kernel of at least one of them is
not empty and a projective measurement along this kernel unambiguously
identifies the other state. On the other hand, if the supports are
identical, then so are their kernels,
and there is no direction in ${\mathcal H}$ that
could unambiguously identify at least one of the density operators.

We can now return to a listing of earlier results. The POVM for unambiguously
discriminating between a pure state and a rank two mixed state was derived in
\cite{sun2} and subsequently generalized to the case of unambiguously
discriminating a pure
state from any mixed state \cite{bergou1}.  Lower bounds on the failure
probability for the unambiguous discrimination of two mixed states were
derived by Rudolph \emph{et al}.
\cite{rudolph} and for an arbitrary number of mixed states by
Feng \emph{et al}. \cite{feng}.
Raynal \emph{et al}. \cite{raynal} proved two theorems that make it
possible to reduce the problem of
unambiguously discriminating between two arbitrary mixed states of rank
$k_{1}$ and $k_{2}$ to the discrimination of
two states of the same rank, $k \leq min(k_{1},k_{2})$, in a $2k$-dimensional
space.  Building on the results of \cite{rudolph} and \cite{feng}, Herzog and Bergou \cite{HB} found  explicit solutions for some special cases along with necessary conditions for the
saturation of the lower bound. In particular, these results showed that unlike
in the case of two pure states whether or not the lower bound can be attained
depends not only on the value of the prior probabilty of the states but also
on their structure. There are mixed states for which the lower bound can not
be reached for any value of the prior probability. In \cite{raynal2} the
optimal measurement operators were constructed explicitly for some special
cases.

In this paper we shall consider the unambiguous discrimination between two
subspaces.  What this means is the following.  A state is chosen from one
of two subspaces, and we wish to determine to which of the subspaces the
state belongs.  Within each subspace each state is equally likely, though
one subspace may be more likely than the other.  One place in which this
type of problem has arisen is in the consideration of programmable
discriminators \cite{bergou3}.  In this case, one is given three qubits,
the first two are
arbitrary but the third is guaranteed to be identical to either the first
or the second qubit, and the problem is to determine which two qubits
are identical.  The problem
can be solved by realizing that one is, in fact, discriminating between
two subspaces, the first being the subspace of three-qubit states that is
symmetric in the first and third qubits, and the second being the subspace
that is symmetric in the second and third qubits.

Subspace discrimination is a special case of the discrimination of two
mixed states; in this case the density matrices
are just proportional to the projection operators onto the subspaces.  Making use of the results of Raynal \emph{et al}. \cite{raynal}, we can restrict our attention to the case of two subspaces of dimension $k$
in a $2k$ dimensional space.  In particular, let $S_{1}$, and $S_{2}$ be $k$-dimensional subspaces of the entire Hilbert space, $\mathcal{H}$, which has dimension $2k$.
We can assume that the intersection of $S_{1}$ and $S_{2}$ is just the zero vector, a situation  which we henceforth refer to as general position.
We assume that $\rho_{1}= (1/k)P_{1}$ occurs with probability $\eta$, and
$\rho_{2}= (1/k)P_{2}$ occurs with probability $1-\eta$, where $P_{j}$ is the projection
onto $S_{j}$, for $j=1,2$.  The POVM that distinguishes them has three
elements, $\Pi_{1}$, $\Pi_{2}$, and $\Pi_{0}=I-\Pi_{1}-\Pi_{2}$, all of which are positive operators.  The probability of identifying $\rho_{j}$ if we are given $\rho_{j}$ is $p_{j}={\rm Tr}(\rho_{j}\Pi_{j})$,
and the probability of failing to identify it is $q_{j}={\rm Tr}(\rho_{j}\Pi_{0})$, for $j=1,2$.  The
condition that a state never be misidentified implies that $\Pi_{1}\rho_{2}=\Pi_{2}\rho_{1}=0$.
The average failure probability is
\begin{equation}
\label{Qave}
Q=\eta q_{1}+(1-\eta )q_{2} \ ,
\end{equation}
and our object is to find, for a given $\eta$,  a POVM that minimizes $Q$.  From the results in
\cite{rudolph} and \cite{feng}, we have that
\begin{equation}
\label{fidbound}
Q\geq 2\sqrt{\eta (1-\eta )} F(\rho_{1},\rho_{2}) \ ,
\end{equation}
where the fidelity between the two density matrices is given by
\begin{equation}
F(\rho_{1},\rho_{2})=\rm{Tr}((\rho_{1}^{1/2}\rho_{2}\rho_{1}^{1/2})^{1/2}) \ .
\label{fidelity}
\end{equation}
The conditions, under which this bound can be saturated, have been investigated in \cite{HB}. In \cite{raynal2} the optimum POVM has been given explicitly for certain special cases.
As we shall see, unlike for the case of two pure states, this bound cannot always be reached for two mixed states, in agreement with \cite{HB}.  The optimal measurement procedure
depends on the value of $\eta$,  a feature it has in common with the procedure for discriminating two pure states. For two pure states there is always a range of $\eta$ where equality in (\ref{fidbound}) can be reached. In the case of two mixed states, however, the optimal measurement procedure also depends on the structure of the two density operators, a distinctive  feature that has no equivalent in the case of two pure states. For $\eta$ near $0$ or $1$ the optimal measurements are projective ones.
In the intermediate regime, the optimal measurements are intermittently POVM's or projective measurements and, in general, their failure probability is higher than the fidelity bound (\ref{fidbound}). Only under very special conditions shall we find that the fidelity bound can be saturated.

The main technical device that we shall use to find the optimal measurements is that of Jordan
bases.  These bases take the following form.  The states $|\psi_{1}\rangle , \ldots |\psi_{k}\rangle$
form an orthonormal basis for $S_{1}$, $|\psi_{k+1}\rangle ,\ldots |\psi_{2k}\rangle$ form an orthonormal basis for $S_{2}$, and, in addition these states have the property that
\begin{equation}
\langle\psi_{i}|\psi_{k+j}\rangle = \delta_{ij}\cos\theta_{i} ,
\end{equation}
where $\cos\theta_{1}\geq\cos\theta_{2}\geq\ldots\cos\theta_{k}$, and $1\leq i,j\leq k$.  The
states $|\psi_{1}\rangle , \ldots, |\psi_{k}\rangle$ and $|\psi_{k+1}\rangle ,\ldots, |\psi_{2k}\rangle$
are called Jordan bases and the angles $\theta_{i}$ are called the Jordan angles.  Bases
satisfying these conditions can be constructed for any two subspaces \cite{jordan}.   Note that
the basis vectors $|\psi_{i}\rangle$ and $|\psi_{k+i}\rangle$ are eigenvectors of the operators
$P_{1}P_{2}P_{1}$ and $P_{2}P_{1}P_{2}$, respectively, where
\begin{eqnarray}
P_{1}P_{2}P_{1} |\psi_{i}\rangle & = & \cos^{2}\theta_{i}|\psi_{i}\rangle \ ,  \nonumber \\
P_{2}P_{1}P_{2}|\psi_{k+i}\rangle & = & \cos^{2}\theta_{i}|\psi_{k+i}\rangle \ ,
\end{eqnarray}
where $1\leq i \leq k$.

The paper is organized as follows. Sec. \ref{section2} is devoted to the
derivation of the general results. Besides giving the general theory for
distinguishing two $k$ dimensional subspaces in a $2k$ dimensional Hilbert
space, these results also hold for a rather general class of density
operators, so they are directly relevant to the problem of optimal
unambiguous discrimination between two mixed states. In Sec. \ref{section3}
we present some possible applications of the results.
Finally, in Sec. \ref{section4} we give a brief summary and
outlook for future research.

\section{Distinguishing subspaces and its relation to the discrimination of mixed states}\label{section2}

We shall actually solve a somewhat more general problem than the
discrimination of two subspaces.  Let $S_{1}$ and $S_{2}$ be $k$ dimensional
subspaces of a $2k$
dimensional complex Hilbert space $\mathcal{H}$ which are in general
position, as discussed in the introduction, and
let $\{|\psi_{1}\rangle,\ldots,|\psi_{k}\rangle \}$ and
$\{|\psi_{k+1}\rangle,\dots,|\psi_{2k}\rangle\}$ be Jordan bases associated
to these subspaces.  Consider the two density matrices
\begin{eqnarray}
\rho_{1}&=&\sum_{i=1}^{k}\alpha_{i}|\psi_{i}\rangle\langle\psi_{i}| \ ,
\nonumber \\
\rho_{2}&=&\sum_{i=1}^{k}\beta_{i}|\psi_{i+k}\rangle\langle\psi_{i+k}| \ ,
\end{eqnarray}
where $\alpha_{i}>0$, $\sum_{i}\alpha_{i}=1$, $\beta_{i}>0 $, and
$\sum_{i}\beta_{i}=1$.
Clearly, $\rho_{1}$ has support in $S_{1}$ and $\rho_{2}$ has support
in $S_{2}$.  In this
case the orthonormal frames for the density matrices given by the spectral
theorem coincide with the Jordan frames of the supports. If
\begin{eqnarray}
\rho_{1} &  = & (1/k)\sum_{i=1}^{k}|\psi_{i}\rangle\langle\psi_{i}|
\nonumber \\
\rho_{2} &  = & (1/k)\sum_{i=1}^{k}|\psi_{i+k}\rangle\langle\psi_{i+k}| ,
\end{eqnarray}
we say that  $\rho_{1}$ and $\rho_{2}$ are uniformly mixed states, and
discriminating between uniformly mixed states corresponds to the case of
discriminating
between the subspaces $S_{1}$ and $S_{2}$.  Inserting the weights
$\alpha_{i}$ and
$\beta_{i}$ will allow us to address several issues in the general
theory of mixed state discrimination.

We will now construct an optimal POVM to unambiguously discriminate between $\rho_{1}$ and $\rho_{2}$.  The POVM elements are $\Pi_{1}$, $ \Pi_{2}$ and $\Pi_{0}=I-\Pi_{1}-\Pi_{2}$ with the properties as discussed in the introduction.  $\Pi_{1}$ and $\Pi_{2}$ are self adjoint operators supported on ${\bar S}_{2}$ and
${\bar S}_{1}$, respectively.  In order for them to form an optimal POVM they must be positive and, crucially, the rank of $\Pi_{0}$ must not exceed $k$ \cite{raynal}.  We wish to choose $\Pi_{1}$ and $\Pi_{2}$ so that the expression
$P(\eta )=\eta {\rm Tr}(\Pi_{1} \rho_{1})+(1 - \eta){\rm Tr}(\Pi_{2}\rho_{2})$ is maximized. This is the same as minimizing the average failure probability $Q(\eta)=\eta {\rm Tr}(\Pi_{0}\rho_{1})+(1-\eta ){\rm Tr}(\Pi_{0}\rho_{2})$ that was introduced in (\ref{Qave}).

Let $T_{i}$ be the linear subspace
spanned by $|\psi_{i}\rangle$ and $|\psi_{i+k}\rangle$. The
$T_{i}$'s, with $1\leq i\leq k $, are mutually orthogonal, two dimensional
subspaces, which are invariant under both $\rho_{1}$ and $\rho_{2}$.
Let us define states $|z_{i}\rangle$ and $|y_{i}\rangle$ as
\begin{eqnarray}
|\psi_{i}\rangle & = & \sin\theta_{i}|z_{i}\rangle +
\cos\theta_{i}|\psi_{i+k}\rangle \nonumber \\
|\psi_{i+k}\rangle & = & \sin\theta_{i}|y_{i}\rangle +
\cos\theta_{i}|\psi_{i}\rangle  ,
\end{eqnarray}
where $|z_{i}\rangle$ $(|y_{i}\rangle)$
is the orthogonal complement of $|\psi_{i+k}\rangle$
$(|\psi_{i}\rangle)$ in $T_{i}$. Furthermore
$ \{ |z_{1}\rangle,\ldots ,|z_{k}\rangle \}$ forms an orthonormal basis for
${\bar S}_{2}$, and
$\{ |y_{i}\rangle,\ldots ,|y_{k}\rangle \}$ forms an orthonormal basis for
${\bar S}_{1}$.

If we write $\Pi_{1}=\sum_{i,j=1}^{k} a_{ij}|z_{i}\rangle\langle z_{j}|$ and
$\Pi_{2}=\sum_{i.j=1}^{k} b_{ij}|y_{i}\rangle\langle y_{j}|$,  then
\begin{equation}
p_{1}={\rm Tr}(\Pi_{1}\rho_{1})=\sum_{i=1}^{k} a_{ii} \alpha_{i}
\sin^{2} \theta_{i} \ ,
\end{equation}
and
\begin{equation}
p_{2}={\rm Tr}(\Pi_{2}\rho_{2})=\sum_{i=1}^{k} b_{ii} \beta_{i}
\sin^{2} \theta_{i} \ .
\end{equation}
These equations do not depend upon the off-diagonal terms of $\Pi_{1}$ and $\Pi_{2}$.
If $\Pi_{1}$ and $\Pi_{2}$ are to be elements of a POVM they must be positive
so $a_{ii}, b_{ii} \geq 0$ is a minimum requirement. The presence of off-diagonal elements imposes additional restrictions on the diagonal elements if we wish to ensure positivity. Since the off-diagonal
 elements do not play a role in $p_{1}$ and $p_{2}$, it suffices to search for our optimal POVM among the
  diagonal operators.

Let ${\bar p}_{i}=\langle\psi_{i}|\Pi_{1}|\psi_{i}\rangle$ and
 ${\bar p}_{i+k}=\langle \psi_{i+k}|\Pi_{2}|\psi_{i+k}\rangle$ be the individual success probabilities of
 the Jordan basis states for $1\leq i \leq k$. Let
${\bar q}_{i}=\langle\psi_{i}|\Pi_{0}|\psi_{i}\rangle$ and
 ${\bar q}_{i+k}=\langle\psi_{i+k}|\Pi_{0}|\psi_{i+k}\rangle$ be the individual failure probabilities of
 the Jordan basis states for $1 \leq i \leq k$. Here we introduced the overbar notation in order to
 distinguish the partial success and failure probabilities, ${\bar p}_{i}$ and ${\bar q}_{i}$, of
  $|\psi_{i}\rangle$ from the total success and failure probabilities, $p_{j}$ and $q_{j}$, of $\rho_{j}$
  for $i \leq k$ and $j=1,2$. Obviously, ${\bar p}_{i} + {\bar q}_{i} = 1$
holds.  We can then express the POVM operators as $\Pi_{1}=\sum_{i=1}^{k}
\Pi_{1,i}$ and $\Pi_{2}=\sum_{i=1}^{k}\Pi_{2,i}$
where
\begin{equation}
\Pi_{1,i}=\frac{1-\bar
q_{i}}{\sin^{2}\theta_{i}}|z_{i}\rangle\langle z_{i}| ,
\end{equation}
and
\begin{equation}
\Pi_{2,i}=\frac{1-\bar
q_{i+k}}{\sin^{2}\theta_{i}}|y_{i}\rangle\langle y_{i}| .
\end{equation}
We can also set $\Pi_{0}=\sum_{i=1}^{k}\Pi_{0,i}$, where
$\Pi_{0,i}=I_{T_{i}}- \Pi_{1,i}- \Pi_{2,i}$, and $I_{T_{i}}$ is the
identity in $T_{i}$.

We now need to determine the values of ${\bar q}_{i}$ and ${\bar q}_{i+k}$.
This can be done by noticing that what we have done is to reduce our problem
to $k$ problems of optimally discriminating two vectors,
in particular the vector $|\psi_{i}\rangle$ from the vector
$|\psi_{i+k}\rangle$ in $T_{i}$.  In more detail the situation is the
following.  In our overall ensemble, with $\rho_{1}$ occurring with
probability $\eta$ and $\rho_{2}$ with probability $1-\eta$,
the probability of occurrence for $|\psi_{i}\rangle$ is
$\eta\alpha_{i}$ and the probability for the occurrence of $|\psi_{i+k}\rangle$
is $(1-\eta )\beta_{i}$.  Therefore, the probability for the occurrence
of a vector in $T_{i}$ is just the sum of these probabilities,
\begin{equation}
p(T_{i})=\eta\alpha_{i}+(1-\eta )\beta_{i}
\end{equation}
Now, the probability that $|\psi_{i}\rangle$ occurs given that $T_{i}$ has
ocurred is $p(i|T_{i})=\eta\alpha_{i}/p(T_{i})$ and the probability
that $|\psi_{i+k}\rangle$ occurs given that $T_{i}$ has occurred is
$p(i+k|T_{i})=(1-\eta )\beta_{i}/p(T_{i})$. Consequently, in $T_{i}$
we want to unambiguously discriminate
$|\psi_{i}\rangle$ occurring with probability $p(i|T_{i})$ and
$|\psi_{i+k}\rangle$ occuring with probability $p(i+k|T_{i})$ so as to
minimize the failure probability
\begin{equation}
\label{Qi}
Q_{i}(\eta )=p(i|T_{i}){\bar q}_{i}+p(i+k|T_{i}){\bar q}_{i+k} .
\end{equation}
This problem was first solved by Jaeger and Shimony \cite{jaeger}, (a
somewhat simpler solution is given in \cite{bergou2}), and we can now make use
of that solution.  Let
\begin{equation}
I_{i}= \left[ \frac{\beta_{i} \cos ^{2} \theta_{i}}{\alpha_{i}+
\beta_{i} \cos^{2}\theta_{i}}, \frac{\beta_{i}}{\beta_{i} +
\alpha_{i} \cos^{2}\theta_{i}} \right]  = [c_{i},d_{i}]
\end{equation}
and, in addition, let
\begin{equation}
 {\bar q}_{i}^{opt} (\eta )=\sqrt{\frac{(1-\eta )\beta_{i}}{\eta \alpha_{i}}}
\cos \theta_{i} \ .
\end{equation}
For a given $\eta$, the value of $\bar q_{i}$ which
minimizes $Q_{i}(\eta)$ is
\begin{equation}
\label{qiopt}
q_{i}( \eta )=\left\{\begin{array}{ll} 1 & {\eta \leq c_{i}}\\
 {\bar q}_{i}^{opt} (\eta) & \mbox{ if $ \eta $ is in $I_{i}$}\\
 \cos ^{2}\theta_{i}& {\eta\geq d_{i}}\end{array} \right.
\end{equation}
and $q_{k+i}(\eta )=\cos^{2}\theta_{i}/q_{i}(\eta )$. Furthermore the rank of
$\Pi_{0,i}$ is one, with the nonzero eigenvalue given by
\begin{equation} \lambda_{i}= \left(
\frac{\cos^{2}\theta_{i}}{{\bar q}_{i}} -2 \cos^{2} \theta_{i} +
{\bar q}_{i} \right) \frac{1}{\sin^{2}\theta_{i}} \ ,
\end{equation}
with the corresponding eigenstate
\begin{equation}
\label{zeta1} |\zeta_{i}\rangle=\frac{\cos\theta_{i}(1-{\bar
q}_{i})}{\sin^2\theta_{i}} |\psi_{i+k}\rangle +
  \frac{{\bar q}_{i}-\cos^{2}\theta_{i}}{\sin^{2}\theta_{i}}
|\psi_{i}\rangle \ .
\end{equation}
This specifies the POVM within $T_{i}$.

The optimal overall failure probability can now be expressed as
\begin{eqnarray}
Q^{opt} & = & \sum_{i=1}^{k} Q_{i}^{opt}p(T_{i})  \nonumber \\
 & = & \sum_{i=1}^{k}[\eta \alpha_{i}{\bar q}_{i}(\eta)
+(1-\eta) \beta_{i}{\bar q}_{k+i}(\eta)] \ ,
\end{eqnarray}
where $Q_{i}^{opt}$ is the failure probability that results when
Eq.\ (\ref{qiopt}) is substituted into Eq.\ (\ref{Qi}).
Its explicit expression is given by
\begin{equation}
Q_{i}^{opt}p(T_{i}) = \left\{\begin{array}{ll} \eta \alpha_{i}+(1-\eta)
\beta_{i}\cos^{2}\theta_{i} & \mbox{if $\eta \leq c_{i}$}\\
 2\sqrt{\eta(1-\eta)\alpha_{i}\beta_{i}}|\cos\theta_{i}| & \mbox{if $c_{i} \leq \eta \leq d_{i}$}\\
 \eta\alpha_{i}\cos ^{2}\theta_{i}+(1-\eta)\beta_{i} & \mbox{if $\eta \geq d_{i}$}\end{array} \right. \ .
\label{Qiopt}
\end{equation}
The center line is the geometric mean of the two terms in either the first or the last line and, therefore, represents an absolute minimum for $Q_{i}$.  We obtain the absolute possible minimum of the total failure probability if we sum the center lines for all $1 \leq i \leq k$. The summation yields
\begin{equation}
\label{Qopt}
Q^{opt} = 2\sqrt{\eta(1-\eta)} \sum_{i=1}^{k}
\sqrt{\alpha_{i}\beta_{i}}|\cos\theta_{i}| \ .
\end{equation}
Clearly, this absolute minimum can only be realized if and only if the intersection of all of the intervals $I_{i}$ is not empty and the operating value of $\eta$ is in this intersecton.

The interpretation of this expression is straightforward. Making use of
the structure of the Jordan bases, we obtain $\rho_{1}^{\frac{1}{2}}
\rho_{2}\rho_{1}^{\frac{1}{2}} = \sum_{i}\alpha_{i}\beta_{i}
\cos^{2}\theta_{i}|\psi_{i}\rangle\langle \psi_{i}|$, after a simple
calculation. Comparing this expression with Eq.\ (\ref{fidelity}) tells us
immediately that Eq.\ (\ref{Qopt}) can be cast to the form
\begin{equation}
 Q^{opt}= 2\sqrt{\eta (1-\eta)}F(\rho_{1},\rho_{2}) \ ,
\end{equation}
where $F(\rho_{1},\rho_{2}) = \sum_{i}\sqrt{\alpha_{i}\beta_{i}}
|\cos\theta_{i}|$. Here $F(\rho_{1},\rho_{2})$ is the fidelity between
$\rho_{1}$ and $\rho_{2}$, constructively proving that the fidelity bound,
Eq.\ (\ref{fidbound}), can be saturated. To obtain an explicit expression
for the fidelity would be a hopeless task, in general. What made it possible
here is the fact that the density operators are diagonal in the Jordan bases
of their support and we could take full advantage of the ensuing Jordan
structure. Furthermore, the above expression for the optimum failure
probability holds only if $\eta$ is an element of the intersection of all
of the $I_{i}$ intervals, $I_{0}=\bigcap_{i=1}^{k}I_{i}$. $I_{0}$ may be
empty, it often is. Note, however, that in the case where $\rho_{1}$ and
$\rho_{2}$ are uniformly mixed states, which is the case of subspace
discrimination,
\begin{equation}
I_{i}=\left[ \frac{\cos^{2}\theta_{i}}{1+\cos^{2}\theta_{i}},
\frac{1}{1+\cos^{2}\theta_{i}} \right]
\subseteq I_{i+1}
\end{equation}
so $I_{0}=I_{1} \neq \emptyset$ and the fidelity result holds in the
entire $I_{1}$ interval.

For the case when $k = 2$ there are only two such intervals, $I_{1}$ and $I_{2}$, and  we will now give a complete classification of their intersection pattern. To this end, we first introduce a one-parameter characterization of $\rho_{1}$ and $\rho_{2}$. Let us set $\alpha_{1}=\alpha$ and, consequently, $\alpha_{2}=1-\alpha$.  Similarly, we set $\beta_{1}=\beta$ and, consequently, $\beta_{2}=1-\beta$. We can now introduce a two-dimensional parameter plane, $\alpha \beta$, where the square in the first quadrant, bounded by $0 \leq \alpha \leq 1$ and $0 \leq \beta \leq 1$,  corresponds to physically acceptable choices for mixed states. So our task is reduced to finding the regions within this square with qualitatively different overlap patterns. When $\cos^{2}\theta_{1}>\cos^{2}\theta_{2}$, which is the convention that we adopted at the beginning, the patterns can be sorted into five categories: \newline
i) $d_{1}<c_{2}$, i.\ e. $I_{1}$ is to the left of $I_{2}$ and their intersection is empty. This happens when
\begin{equation}
\beta \leq {\bar \beta}_{1}(\alpha)=\frac{\alpha \cos^{2}\theta_{1}
\cos^{2}\theta_{2}}{1-\alpha(1-\cos^{2}\theta_{1}\cos^{2}\theta_{2})} \ .
\end{equation}
${\bar \beta}_{1}(\alpha)$ is a hyperbola in the $\alpha \beta$ plane
and it is the divider between this region and the next, when \newline
ii) $c_{1}<c_{2}<d_{1}<d_{2}$, i.\ e. the right end of $I_{1}$ partially
overlaps with the  left end of $I_{2}$ and their intersection is the overlap.
This happens when
\begin{equation}
\beta \leq {\bar \beta}_{2}(\alpha)=\frac{\alpha
\cos^{2}\theta_{2}}{\cos^{2}\theta_{1}-\alpha(\cos^{2}\theta_{1}-\cos^{2}\theta_{2})} \ .
\end{equation}
${\bar \beta}_{2}(\alpha)$ is a hyberbola in the $\alpha \beta$ plane
and it is the divider between this region and the next, when \newline
iii) $c_{2}<c_{1}$ and $d_{1}<d_{2}$, i.\ e. $I_{1}$ is inside $I_{2}$ and the intersection coincides with $I_{1}$. This happens when
\begin{equation}
\beta \leq {\bar \beta}_{3}(\alpha)=\frac{\alpha
\cos^{2}\theta_{1}}{\cos^{2}\theta_{2}+ \alpha(\cos^{2}\theta_{1}-
\cos^{2}\theta_{2})} \ .
\end{equation}
${\bar \beta}_{3}(\alpha)$ is a hyberbola in the $\alpha \beta$ plane
and it is the divider between this region and the next, when \newline
iv) $c_{1}<c_{2}<d_{2}<d_{1}$, i.\ e. the left end of $I_{1}$ partially overlaps with the right end of $I_{2}$ and the intersection is the overlap. This happens when
\begin{equation}
\beta \leq {\bar \beta}_{4}(\alpha)=\frac{\alpha}{\cos^{2}\theta_{1}\cos^{2}
\theta_{2} + \alpha(1-\cos^{2}\theta_{1}\cos^{2}\theta_{2})} \ .
\end{equation}
${\bar \beta}_{4}(\alpha)$ is a hyberbola in the $\alpha \beta$ plane and it
is the divider between this region and the next, when, finally \newline
v) $d_{2}<c_{1}$, i.\ e. $I_{1}$ is to the right of $I_{2}$ and the intersection is empty. This happens when
\begin{equation}
\beta \geq {\bar \beta}_{4} \ .
\end{equation}

We note that when $\cos^{2}\theta_{1} = \cos^{2}\theta_{2}$ the two
inner dividers, ${\bar \beta}_{2}$ and ${\bar \beta}_{3}$, both
degenerate into the diagonal of the square, $\beta = \alpha$. Our
findings are summarized in Fig. \ref{Fig1} where the five regions of
the parameter space, resulting from the four dividers ${\bar \beta}_{1},
\ldots,{\bar \beta}_{4}$, are displayed for the representative
values $\cos^{2}\theta_{1} = \frac{3}{4}$ and $\cos^{2}\theta_{2}
= \frac{1}{4}$.

\begin{figure}[!ht]
\epsfig{file=Fig1.eps, height=7cm}
\caption{Regions of the parameter space. Dotted line:
${\bar \beta}_{1}(\alpha)$, short dashed line: ${\bar \beta}_{2}(\alpha)$,
medium dashed line: ${\bar \beta}_{3}(\alpha)$, long dashed line:
${\bar \beta}_{4}(\alpha)$. In the region below ${\bar \beta}_{1}(\alpha)$
and above ${\bar \beta}_{4}(\alpha)$ the intersection of $I_{1}$ and $I_{2}$
is empty, the fidelity bound can not be reached; in the regions between
${\bar \beta}_{1}(\alpha)$ and ${\bar \beta}_{2}(\alpha)$, and between
${\bar \beta}_{3}(\alpha)$ and ${\bar \beta}_{4}(\alpha)$ the intervals
$I_{1}$ and $I_{2}$ partially overlap, the fidelity bound can be reached
in these overlaps; and in the region between ${\bar \beta}_{2}(\alpha)$
and ${\bar \beta}_{3}(\alpha)$ the interval $I_{1}$ is inside $I_{2}$,
the fidelity bound can be reached in the entire $I_{1}$. For the figure we
used the values $\cos \theta_{1} = \frac{\sqrt{3}}{2}$ and
$\cos \theta_{2} = \frac{1}{2}$.}
\label{Fig1}
\end{figure}

Next we give two illustrative examples for $k=2$. In these examples we prescribe $\Pi_{0}$ by giving its eigenvectors $|\zeta_{i}\rangle$ and eigenvalues $\lambda_{i}$ for the possible $\eta$ interval configurations. We adopt the notation $|\zeta_{i}(\eta)\rangle$ ($\lambda_{i}(\eta)$) for the eigenvector (eigenvalue) corresponding to  ${\bar q}_{i}(\eta) = {\bar q}^{opt}_{i}$.

In our first example we consider the case where $\rho_{1}$ and $\rho_{2}$ are uniformly mixed, corresponding to subspace discrimination. Then $I_{1}=[c_{1},d_{1}] \subseteq I_{2}=[c_{2},d_{2}]$, where
\begin{equation}
c_{i}=\frac{\cos^{2}\theta_{i}}{1+\cos^{2}\theta_{i}} \ , \hspace{0.5cm} d_{i}=\frac{1}{1+\cos^{2}\theta_{i}} \ .
\end{equation}
The interval $0 \leq \eta \leq 1$ is divided into five subintervals, $[0,c_{2}]$, $[c_{2},c_{1}]$, $[c_{1},d_{1}]$, $[d_{1},d_{2}]$, and $[d_{2},1]$. Table \ref{table1} summarizes the behavior of $\Pi_{0}$ in each of the subintervals.
\begin{table}[h!]
\caption{Eigenvalues and eigenvectors of $\Pi_{0}$ in the various intervals of $\eta$ for Example 1. Both states are uniformly mixed and the intersection of $I_{1}$ and $I_{2}$ is the entire $I_{1}$. The fidelity bound can be reached in the intersection.}
\begin{tabular}{||c||c|c|c|c||}
\hline
$\eta$ & $\lambda_{1}$ & $\lambda_{2}$ &  $|\zeta_{1}\rangle$ & $|\zeta_{2}\rangle$ \\ \hline \hline
$[0,c_{2}]$ & $1$  & $1$  & $|\psi_{1}\rangle$ & $|\psi_{2}\rangle$  \\ \hline
$[c_{2},c_{1}]$& $1$ & $\lambda_{2}(\eta)$  & $|\psi_{1}\rangle$ & $|\zeta_{2}(\eta)\rangle$ \\ \hline
$[c_{1},d_{1}]$ & $\lambda_{1}(\eta)$ & $\lambda_{2}(\eta)$  & $|\zeta_{1}(\eta)\rangle$ & $|\zeta_{2}(\eta)\rangle$ \\ \hline
$[d_{1},d_{2}]$ & $1$ & $\lambda_{2}(\eta)$ & $|\psi_{3}\rangle$ & $|\zeta_{2}(\eta)\rangle$ \\ \hline
$[d_{2},1]$ & $1$ & $1$ & $|\psi_{3}\rangle$ & $|\psi_{4}\rangle$ \\ \hline
\end{tabular}
\\
\label{table1}
\end{table}

For our second example we choose $ \rho_{1}$ and $\rho_{2}$ so that
$d_{1}\leq c_{2}$, which can be easily arranged by picking $\beta <
{\bar \beta}_{1}(\alpha)$. We then have  intervals $[0,c_{1}]$,
$[c_{1},d_{1}]$, $[d_{1},c_{2}]$, $[c_{2},d_{2}]$, and $[d_{2},1]$.  This
is a situation in which the fidelity bound for the failure probability,
Eq.\ (\ref{fidbound}), can never be achieved. The behavior of $\Pi_{0}$
is summarized in Table \ref{table2}.
\begin{table}[h!]
\caption{Eigenvalues and eigenvectors of $\Pi_{0}$ in the various intervals of $\eta$ for the second example in the text. The intersection of $I_{1}$ and $I_{2}$ is empty and the fidelity bound cannot be reached in this case.}
\begin{tabular}{||c||c|c|c|c||}
\hline
$\eta$ & $\lambda_{1}$ & $\lambda_{2}$ &  $|\zeta_{1}\rangle$ & $|\zeta_{2}\rangle$ \\ \hline \hline
$[0,c_{1}]$ & $1$  & $1$  & $|\psi_{1}\rangle$ & $|\psi_{2}\rangle$  \\ \hline
$[c_{1},d_{1}]$& $\lambda_{1}(\eta)$ & $1$  & $|\zeta_{1}(\eta)\rangle$ & $|\psi_{2}\rangle$ \\ \hline
$[d_{1},c_{2}]$ & $1$ & $1$ & $|\psi_{3}\rangle$ & $|\psi_{2}\rangle$ \\ \hline
$[c_{2},d_{2}]$ & $1$ & $\lambda_{2}(\eta)$ & $|\psi_{3}\rangle$ & $|\zeta_{2}(\eta)\rangle$ \\ \hline
$[d_{2},1]$ & $1$ & $1$ & $|\psi_{3}\rangle$ & $|\psi_{4}\rangle$ \\ \hline
\end{tabular}
\\
\label{table2}
\end{table}

The trend is clear from these two examples. For each of the five regions of the $\alpha\beta$ parameter plane the operating value of $\eta$ can have five possibilities. It can be outside of the intervals $I_{1}$ and $I_{2}$ (three such intervals if $I_{1}$ and $I_{2}$ do not intersect and two if they do), it can be in the nonoverlapping regions of $I_{1}$ and $I_{2}$ (two such intervals) and, finally, it can be in the intersection of $I_{1}$ and $I_{2}$ (zero or one such interval). The five parameter regions and the five possibilities for $\eta$ in each of these regions give us altogether twenty five characteristically different cases. In only three of them can the fidelity bound be reached. In twelve cases the optimum measurement is a standard von Neumann projection and in the remaining ten cases it is a combination of projections in some dimensions and POVMs in the others.

We believe that these trends are general and they hold for the
discrimination of any Rank $2$ mixed states not just the ones where
the Jordan basis coincides with the spectral representation but it will
be much harder to find explicitly the different regions in the parameter space and
the different intervals of the prior probability $\eta$. We also conjecture
that for the discrimination of two Rank $N$ mixed states there are
$(2N+1)^{N}$ cases altogether and the fidelity bound can be reached in
only $2N-1$ of them. Since the growth in the number of possibilities
as a function of $N$ is faster than exponential it seems as though it
would be extremely difficult to give a complete classification of
the cases for $N > 2$. Furthermore, since the number of cases when the
fidelity bound can be reached grows only linearly with $N$, the weight of
the density operators for which the fidelity bound can be attained quickly
becomes negligible with increasing $N$.

\section{Applications}\label{section3}
Let us now consider a simple example that we will be able to use as
the basis for applications
of subspace discrimination.  Let $\mathcal{H}$ be a four-dimensional
space with the orthonormal
basis $\{ |j\rangle\ |\ j=0,\ldots, 3\}$.  For the first subspace,
$S_{1}$, we choose the span of the
vectors $|0\rangle$ and $|1\rangle$, and for the second, $S_{2}$
we choose the span of the
vectors $|u_{0}\rangle = (|0\rangle +|2\rangle )/\sqrt{2}$ and
$|u_{1}\rangle = (|1\rangle +
|3\rangle )/\sqrt{2}$.  The states $\{ |0\rangle ,|1\rangle \}$
and $\{ |u_{0}\rangle , |u_{1}\rangle \}$
form Jordan bases for the subspaces $S_{1}$ and $S_{2}$.  Defining the vectors
\begin{eqnarray}
|y_{1}\rangle = |2\rangle &\hspace{3mm}& |y_{2}\rangle  = |3\rangle
\nonumber \\
|z_{1}\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |2\rangle )
&\hspace{3mm}& |z_{2}\rangle =
\frac{1}{\sqrt{2}}(|1\rangle - |3\rangle ) ,
\end{eqnarray}
we have that ${\bar S}_{1}$ is the span of $|y_{1}\rangle$ and $|y_{2}\rangle$, and
${\bar S}_{2}$ is the span of $|z_{1}\rangle$ and $|z_{2}\rangle$.  Application of the formulas
in the previous section gives us the POVM for discriminating between $S_{1}$ and $S_{2}$.
The POVM exists for $1/3\leq \eta \leq 2/3$, where the detection operators are given by
\begin{eqnarray}
\Pi_{1} & = & \sqrt{2}\left( \sqrt{2}-\sqrt{\frac{1-\eta}{\eta}}\right) {\bar P}_{2} \nonumber \\
\Pi_{2} & = & \sqrt{2}\left( \sqrt{2} - \sqrt{\frac{\eta}{1-\eta}}\right) {\bar P}_{1}  \ .
\end{eqnarray}
The corresponding failure probability is $Q=\sqrt{2\eta (1-\eta )}$. We also mention here that this solution was already derived in \cite{HB} using a slightly less general approach.

One possible application of this POVM is the following.  Suppose that Alice and Bob cannot
communicate with each other, but they can communicate with Charlie.  Charlie wants Alice
and Bob to share a secure bit string.  He sends to Alice and Bob one particle each from either of
the two-particle states
\begin{eqnarray}
|\Psi_{0}\rangle & = & \frac{1}{\sqrt{2}}(|0\rangle_{a}|1\rangle_{b} + |1\rangle_{a}|0\rangle_{b})
\nonumber \\
|\Psi_{1}\rangle & = & \frac{1}{\sqrt{2}}(|u_{0}\rangle_{a}|u_{1}\rangle_{b} + |u_{1}\rangle_{a}
|u_{0}\rangle_{b}) .
\end{eqnarray}
If Alice and Bob both succeed in identifying which state was sent, they share a bit,
$|\Psi_{0}\rangle$ corresponding to $0$ and $|\Psi_{1}\rangle$ corresponding to $1$.  The
reduced density matrices that Alice and Bob must distinguish are $\rho_{0}=(1/2)P_{1}$,
which results if $|\Psi_{0}\rangle$ is sent, and $\rho_{0}=(1/2)P_{2}$, which results if
$|\Psi_{1}\rangle$ is sent.  The above POVM does this optimally (we shall assume that
$\eta =1/2$).  The procedure would be the following.  Charlie sends one of the two states to
Alice and Bob (one particle to each).  They independently perform their measurements.  They
then tell Charlie whether they succeeded, and he tells each of them whether the bit is valid or
not.  The bit is valid when both Alice's and Bob's measurements succeeded, and invalid
otherwise.

The security comes from the fact that $|\Psi_{0}\rangle$ and $|\Psi_{1}\rangle$ are not orthogonal.
An eavesdropper, Eve, cannot perfectly distinguish these two states.  Her measurement procedure
will either sometimes produce errors or sometimes fail.  She must, however, send particles on to
Alice and Bob.  There is no state she can send them that will guarantee that one or both of their
measurements fail, so that sometimes Alice's and Bob's measurements will tell them that they
have received a state different from the one that Charlie sent.  By comparing some of their bits
with those of Charlie, they can tell whether this has occurred.  One possibility is that they can use
some of the invalid bits, in particular the ones for which one of the measurement succeeded and
the other did not.  For example, if Alice's measurement succeeded, then she can tell Charlie
the result of her measurement, and Charlie can see whether it corresponds to the state that he
sent.  If it does not, then they know that an eavesdropper was present.

A second example concerns operator discrimination \cite{acin}-\cite{chefles2}.
Alice starts with the two-qubit state, $|\Psi_{in}\rangle = |0\rangle
|0\rangle$.  She sends the state through one of two black boxes, each black
box performing an operation on the input state.  The first black box
performs an unknown, arbitrary single qubit rotation on the second qubit.
The second first performs an unknown, arbitrary single qubit rotation on the
second quibit and a Hadamard operation on the first, and this is followed by
sending both quibits through a C-NOT gate, with the first qubit as the
control and the second as the target.  Alice then sends the resulting
output state to Bob, who must decide which black box Alice used.  Note that
what is being done here is the discrimination between two sets of operators;
the first black box performs an abitrary operator from the first sent and
the second black box performs an arbitrary operator from the second set.

If the input state is sent through the first black box, the output state
that is sent to Bob is
\begin{equation}
|\Psi_{1out}\rangle = \alpha |0\rangle |0\rangle + \beta |0\rangle |1\rangle ,
\end{equation}
where $\alpha$ and $\beta$ are unknown.  If the input state was sent through
the second black box, Bob receives the state
\begin{eqnarray}
|\Psi_{2out}\rangle  & = & \frac{1}{\sqrt{2}}[\alpha (|0\rangle |0\rangle
+ |1\rangle |1\rangle ) \nonumber \\
 &{}& + \beta (|0\rangle |1\rangle + |1\rangle |0\rangle )  .
\end{eqnarray}
The state $|\Psi_{1out}\rangle$ lies in the subspace spanned by the vectors
$\{ |0\rangle |0\rangle ,
|0\rangle |1\rangle \}$ and the state $|\Psi_{2out}\rangle$ lies in the
space spanned by
$\{ (|0\rangle |0\rangle + |1\rangle |1\rangle ), (|0\rangle |1\rangle
+|1\rangle |0\rangle ) \}$.  That
means that distinguishing $|\Psi_{1out}\rangle$ and $|\Psi_{2out}\rangle$
reduces to the problem of
distinguishing these two supspaces.  Making the correspondence with our
previous example
\begin{eqnarray}
|0\rangle |0\rangle \rightarrow |0\rangle & \hspace{1cm} & |1\rangle
|1\rangle \rightarrow |2\rangle
\nonumber \\
|0\rangle |1\rangle \rightarrow |1\rangle & \hspace{1cm} & |1\rangle
|0\rangle \rightarrow |3\rangle ,
\end{eqnarray}
we see that the problem reduces to the one we have already solved.
The subspace in which
$|\Psi_{1out}\rangle$ lies corresponds to $S_{1}$ and the one in which
$|\Psi_{2out}\rangle$ lies
corresponds to $S_{2}$.  Therefore, the POVM we have already found will
optimally distinguish (assuming that the input state is $|0\rangle |0\rangle$)
 through which black box the input state was sent.

\section{Conclusion}\label{section4}
We have presented a POVM that optimally and unambiguously
discriminates between two
subspaces.  The construction of this POVM made use of the Jordan bases of
the two subspaces. The results are, in fact, more general than what is
stated in the title. They represent the complete solution to the problem
of optimal unambiguous discrimination between mixed states of a special
class, viz. between those states for which the spectral form coincides with
the Jordan representation.

We presented two applications of the measurement procedure,
discriminating between two-particle states if one has only one of the
particles and deciding to which of two sets an unknown quantum operation
belongs.

This procedure can be used to distinguish arbitrary mixed states by
discriminating between their
supports, but the results will not, in general, be optimal.  In order
to optimally discriminate
between mixed states the structure of the states within their supports
must be taken into account. Based, however, on the results of this paper
we believe that this is an extremely difficult task for density matrices of
Rank $>2$. The case of optimally discriminating between arbitrary Rank $2$
density matrices appears more tractable, however.  How it can be accomplished
is a problem that still remains open.

\begin{acknowledgments}
This research was partially supported by a grant from PSC-CUNY as well
as by a CUNY collaborative grant. JB gratefully acknowledges many helpful
discussions with Ulrike Herzog (Humboldt University, Berlin), on various
aspects of state discrimination.
\end{acknowledgments}

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