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quantum entanglement ( qe ) is the central feature of quantum mechanics that distinguishes a quantum system from its classical counterpart .
it is also the corner stone on which many of the novel applications of quantum mechanics -to quantum computation , quantum information theory , quantum cryptography and quantum teleportation - are based .
indeed , it is this promise that has led to a renewal of interest in qe in recent years .
the simplest example of qe is afforded by a bipartite system of two spin half ( or pseudo spin half ) particles , where the hilbert space has the minimal dimension four .
this system is also called a two qubit system ( 2qs ) in the context of quantum computations . for a 2qs ,
there is essentially only one measure of entanglement ; it may be given _
e.g. _ , by the von - neumann entropy @xmath0 , or expressed in terms of the sum of bilinears in the eigen values as given by the quantity @xmath1 , or in terms of det@xmath2 .
@xmath3 is the reduced density matrix obtained by taking a marginal trace over one of the spin degrees of freedom .
states with a vanishing entanglement are separable ( in the strong sense ) : they admit a factorisation @xmath4 . on the other hand , fully entangled states have the maximum correlation , and are collectively designated as bell states .
consider now a n - qubit system ( nqs ) , realised as a multipartite system of @xmath5 spin halfs .
nqs upto @xmath6 have been prepared experimentally with photons , for example @xcite .
it is easy to check that for an nqs , @xmath0 , @xmath7 and @xmath8 are not equivalent .
further , none of them is exhaustive .
a natural question that arises is how one may identify a complete set of entanglement measures for any nqs .
there have been several proposals @xcite that attempt at answering this question .
one note worthy proposal is from sixia yu _
et al _ @xcite who posit a hierachy of @xmath9 classes of entanglement .
the total number grows exponentially with increasing @xmath5 .
an alternative approach is to look for physically interesting measures , apart from @xmath0 and @xmath7 , with ( potential ) applicability to quantum computation .
this simpler approach would also give direction to the experimentalists to prepare nqs in specified states .
in this context , a useful question to ask is what the entanglement property of a 2qs is , when looked upon as a subsystem of an nqs , when @xmath10 ( there would be no new information when @xmath11 ) .
there would be @xmath12 such subentanglements , as analogues of two particle correlations . in defining the new measure
, we seek to determine the proximity of a given 2qs to the classic bell states mentioned above .
wootters @xcite has developed a closely related concept called concurrence ( hereafter denoted @xmath13 ) for 2qs , which is required to characterise the so called entanglement of formation ( eof ) @xcite .
indeed , eof is defined to be the minimum entropy carried by a mixed state , under all possible resolutions @xmath14 of a reduced density matrix of a 2qs ; the set \{@xmath15 } does not _ necessarily _ form a basis - much less an orthonormal set .
@xmath13 , which measures this eof is then defined as follows :    the concurrence for a mixed 2qs is given by @xmath16 where the @xmath17 are the eigenvalues , arranged in decreasing order of magnitude , of the operator + @xmath18    the so called eof is itself given by    the entanglement of formation is given by + @xmath19    the above definitions are attractive , and have been used to estimate bipartite entanglement in spin systems @xcite .
there are also reports that states with nonvanishing @xmath13 have been prepared experimentally @xcite .
attempts have been made to generalise the definition to higher spin sectors @xcite .
there is , thus , quite an interest in @xmath13 .
the definition of @xmath13 , rather involved in its form , crucially hinges on the concept of eof .
however , it is well known that there is no way of inferring how a system is prepared @xcite , no matter which property of the density matrix we look at , unless it corresponds to a pure state .
thus the physical interpretation of eof is rather dubious , and does not seem to have any operational significance . indeed
, suppose the expansion of @xmath20 in eq.1 is replaced by a more general decomposition @xmath21 p_{\{\alpha_i\ } } \vert \psi(\{\alpha_i \})><\psi(\{\alpha_i\ } ) \vert$ ] , where the states are now labelled by a set of continuous parameters @xmath22 with @xmath23 $ ] being an appropriate integration measure .
the parameters could be taken to be the coordinates of the associated compact phase space , for instance .
the probabilities @xmath24 have as much physical significance as the discrete variables @xmath25 in eq.1 .
but the new expansion would change the criterion for eof , leading to a different expression for concurrence .    with this critique of @xmath13 in mind
, we propose an alternative definition of concurrence ( hereafter called @xmath26 ) , with a two fold aim : to attain conceptual and computational simplicity .
@xmath26 is required merely to capture the entanglement of a 2qs which is a subsystem of a larger nqs , in a manner that suggests a generalisation to higher spin sectors .
we take this up in the next section .
as a warm up , and for motivation , consider a 2qs first . let a 2qs be in the state @xmath27 the key step in defining concurrence is in recognising that @xmath28 is a measure of entanglement , equivalent to the standard measures listed above .
@xmath29 is much simpler to evaluate , though .
our notion of concurrence is inspired by , and is very close in its definition to , @xmath29 .
although @xmath29 does _ not _ carry any new information , it does exhibit a symplectic structure which is not apparent in the other two measures .
further , as observed by hill and wootters @xcite , it can also be written in the form @xmath30 , the inner product of @xmath31 with its conjugate ( equivalently , time reversed ) state which we define thus : @xmath32 the main thrust of the paper is that this form of entanglement for a 2qs needs _ little _ modification in defining concurrence in nqs , @xmath10 .
the definition will be given in steps , so as not to have a cluttered notation .        which is obtained by a partial conjugation  on the sector ( 12 ) .
the definition of concurrences @xmath36 and @xmath37 follow similarly . by definition @xmath38 take values in the range
@xmath39 $ ] .
@xmath40 also has an in built symplectic structure ; to see this , consider the quadruplet of vectors @xmath41 the expansion is in the ordered basis @xmath42 .
the concurrence is now simply written as @xmath43 which naturally generalises the definition of entanglement in a 2qs in a manner appropriate to our purpose @xcite .
@xmath44 is the transpose of @xmath45 .
one can straight away construct quadruplets for @xmath46 by inspection .
it is also straight forward to generalise the definition when the parent state is an nqs in a pure state .
we write the expression for the @xmath47 sector :      the computational simplicity of @xmath26 is evident , from its very form , if only we demonstrate its viability .
a convenient method is to use @xmath13 as a benchmark to test the new definition against .
for , the definition of @xmath13 is also motivated by the entanglement measure @xmath29 .
figure 1 . gives a comparison of the two definitions , where the states are generated randomly .
one sees an overall agreement so long as @xmath13 remains positve , without having to impose the minimality condition by hand . in those regions where @xmath13 has to be declared as vanishing ( when @xmath51 ) in _ definition 1 _ , @xmath26 differs from @xmath13 : indeed
, these states are distinguishable , in the sense that each of them has a preferential proximity to one of the bell states which is more clear in fig.2 .
the computational advantage is of course evident from the form of @xmath26 . for ,
if we start with a nqs , there are @xmath12 such concurrences . even in regions where @xmath13 has no interpretational problem , _ definition 1 _ requires an evaluation of three reduced density matrices , a determination of their partner states , and an evaluation of the eigenvalues of @xmath52 .
the necessity of these computations gets obviated in the determination of @xmath26 .
we proceed to enlarge the definition of concurrence to higher spin sectors .
note that there is no analogue of @xmath13 here , since the criterion for eof which was evolved in @xcite is specific to a 2qs .
the generalisation in our case is naturally suggested by the form for @xmath26 written above .
indeed , let @xmath53 be the state in which an nqs is .
we define the conjugate state @xmath54      the above definition is unfortunately restrictive ; it works only when the subsystem of the nqs is bosonic ; the fermionic states ( @xmath55 odd ) are always orthogonal to their conjugates .
apart from this , there is generality with respect to the choice of the @xmath55 qubits since any @xmath55 qubits may be chosen to be in the order given , by a permutation .
the number of independent concurrence measures is given by @xmath57 .
figure 3 displays the 4q concurrences for states generated from a 6qs . as an illustration , and as an example of the discerning capability of the higher order concurrences which we have defined here ,
the shor code will be compared with the steane code , both of which are used in quantum error correction @xcite .
the shor code @xmath58 is a nine qubit state , written as a direct product of three 3q states each of which have the form @xmath59 , in writing which we have employed the qubit notation : @xmath60 . this entangled state has vanishing concurrences in _ all _ orders .
in contrast , the steane code , constructed with seven qubits has a more involved structure ; it is not seperable the way the shor state is . , and is given by @xmath61 . as a manifestation , although all the 2q concurrences vanish , the 4q concurrences survive in the sectors @xmath62 , and all of them attain the maximum allowed value 1 .
the 6q concurrences vanish .
these features highlight quantitatively the manner in which the two codes differ from each other . at this point
, it is good to remember that a complete separability in the strong sense , where the multiqubit state is a direct product of single qubit states , necessarily implies a vanishing concurrence of all orders .
the converse is not necessarily true , as exemplified by the state generating the shor code .
finally , the question of handling mixed states still remain .
not getting into the general case , only the 2q concurrence will be discussed .
the general case only involves rewriting the argument with more indices .
indeed , given a 2qs in a mixed state @xmath20 , the method is to look upon @xmath20 as having descended from a pure parent state of a higher dimension @xmath5 .
significantly , @xmath63 .
the determination of parent state is done by simple inspection .
it also follows that if @xmath20 has only real components , then @xmath64 it is not difficult to check that although the parent state is not unique , the concurrence determined will be so - reflecting the fact that it is a property of the system , and not of the parent - introduced purely as an intermediate step .
this final remark completes the programme undertaken .    in conclusion ,
we have redefined in this note the concept of concurrence which has a much simpler form , and is computationally trivial - compared to @xmath13 and its generalisations thereof .
it is not based on the notion of eof .
while concurrences do not completely characterise a system - which was not our aim - it appears that they do constitute an important subset of entangled states , especially in the two qubit sector .
the higher order concurrences also have a significance , as illustrated by the concurrence properties of shor and the steane codes .
the true import of entanglements of this kind would manifest with only a more geometric approach , which will be taken up in a subsequent publication .
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, elements of @xmath3 in the ( 12 ) sector are given by @xmath65 which , incidentally gives a three parameter family of three qubit states all of which reduce to the same density matrix in the ( 12 ) sector .
the @xmath66 structure is obvious .
alexander wong _ et al _ phys
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michael a. nielsen and isaac i. chuang , quantum computation and quantum information ( cambridge university press , 2000 ) chapter 10 .
we refine the notion of concurrence in this paper by a redefinition of the concept . 
 the new definition is simpler , computationally straight forrward , and allows the concurrence to be directly read off from the state . 
 it has all the positive features of the definition given by wootters over and above which it can discriminate between different systems to which the wootters prescription would assign the same value . 
 finally , the definition leads to a natural extension of the notion to multiqubit concurrence , which we illustrate with examples from quantum error correction codes .    0.5 cm    0.5 cm    pacs:03.65.ud , 03.67.-a , 03.67.lx