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quantum states encoded in optical fields are an obvious way to implement quantum communication protocols @xcite .
the optical approach offers also a route towards scalable quantum computing , with the particularly promising linear - optics scheme of knill , laflamme , and milburn @xcite .
optical fields are distinguished from standard qubit models of quantum information processing in two ways .
first , photons are bosons that can occupy field modes in arbitrary numbers .
although a qubit can be implemented as a superposition of a single photon in two orthogonal modes , the entire hilbert space describing optical fields has plenty of room to go beyond this standard , dual - rail representation .
secondly , most important error mechanisms that affect optical fields have a specific form .
this enables one to optimize strategies for shielding quantum information from their deleterious effects .
the above features can be illustrated with the example of photon loss , also referred to in literature as amplitude damping .
such a mechanism can be modeled as a transmission of optical fields through a partly reflecting beam splitter .
this attenuates the transmitted field , leading to a random removal of photons from the initial state .
the effects of photon loss can be dealt with by adopting the strategy of quantum error correction @xcite , which consists in designing so - called code subspaces in which qubits are protected from dominant errors .
such codes can be constructed from states that contain more than one photon per mode @xcite , which makes them more efficient in terms of required numbers of photons and modes .    a natural method to manipulate optical codes
is to use linear optics networks @xcite . in the simplest scenario ,
such networks are passive , i.e.  they do not involve auxiliary photons , conditional detection , or feed - forward operations . in this contribution
we review the recent proof @xcite that even the simplest codes , which correct for just a single photon loss , can not be universally processed using passive linear optics only .
as we will see below , the same properties that make the codes correctable for photon loss , also prohibit a range of linear - optics manipulations .
logical qubits can be protected from the effects of errors by preparing more complex states of physical systems @xcite .
such states span the so - called code subspace , within which an arbitrary quantum superposition remains preserved despite the occurrence of an error from a certain class . in the most elementary case ,
the subspace is spanned by two orthogonal states which we shall denote by @xmath0 and @xmath1 .
the specific scenario we shall consider here is shielding logical qubits carried by optical fields from amplitude damping by encoding them in suitable multiphoton states .
let us restrict our attention to errors induced by the loss at most one photon from the field .
if we make two assumptions :    * the code is constructed in a subspace with a fixed total photon number , * the damping parameter is identical for all the modes involved ,    then the necessary and sufficient conditions for a code constructed in a system of bosonic modes to be robust against one - photon loss are given by @xcite : @xmath2 where @xmath3 and @xmath4 denote respectively the annihilation and creation operators of the field modes , and the indices @xmath5 run over all of @xmath6 bosonic modes .
the action of an operator @xmath7 on the code states can be interpreted as an event when a third party has observed in the leaked portion of the field one photon in the @xmath8th mode .
the third party belongs to the external environment , and her actions , as well as the outcome of her measurement are of course unknown to the owner of the qubit .
the equation @xmath9 implies a simple constraint : it must not be possible to pass between two orthogonal code states by moving just one photon between the modes , i.e.  annihilating it from the mode @xmath7 and creating one in the mode @xmath3 . starting from this observation , it is easy to construct two elementary examples of photon - loss codes using four photons distributed between two modes @xcite : @xmath10 are mutually orthogonal and have identical norms .
a code subspace can be conveniently characterized with the help of the corresponding projection operator @xmath11 , which in our two - dimensional case takes the form @xmath12 where @xmath13 is a certain @xmath14 matrix .
this matrix is hermitian , which is easily proven by considering the hermitian conjugation of eq .
( [ eq : correctpc ] ) .
a general passive linear optics transformation of a system of @xmath6 modes can be written as : @xmath15 where the operators @xmath3 describe the input representation , the operators @xmath16 refer to the output representation , and @xmath17 is a unitary @xmath14 matrix .
the unitarity of the matrix @xmath18 guarantees the preservation of commutation relations for the field operators .
any linear - optics transformation can be reversed , therefore the labels of input and output representations are purely conventional .
let us now write the correctability condition given in eq .
( [ eq : correctpc ] ) in the representation of the output modes @xmath19 .
a straightforward calculation shows that @xmath20 where @xmath21 and @xmath22 denote respectively the transposition and the complex conjugation of the matrix @xmath18 .
this means that after the application of the network @xmath18 the subspace @xmath11 remains a photon - loss code , and that the only change is the transformation of the matrix @xmath23 on the right hand side of the correctability condition .
this property reflects the fact that in our error model the correctability condition is independent of the specific modal decomposition .
indeed , the third party monitoring the leaked field can decompose it in an arbitrary basis of modes and measure them individually for the presence of a photon . because the damping coefficients are assumed to be identical for all the modes , such a procedure performed on the leaked field
does not alter the error model .
the fact that the photon - loss code is preserved by passive linear - optics transformations has important implications for encoding and decoding .
suppose that we start from a qubit in the standard , dual - rail representation , with the aim of mapping it onto the encoded subspace . a simple way to accomplish this would be to combine it with auxiliary modes prepared in a certain state and apply a passive linear optics transformation .
however , the reversed version of the argument presented above implies that if the output is a photon - loss code , then the input needs to be such a code as well .
this means that the input qubit itself is protected against photon loss , which obviously is not the case for the dual - rail representation .
the same applies to decoding : if we know a priori that no photon loss occurred , we can not convert the encoded qubit back into the dual - rail representation using a passive network .
therefore , there can not exist a passive network that would work more universally for the input affected by errors and provide a decoded qubit with the error syndrome contained in the state of auxiliary modes .
we will now consider single - qubit gates operating on the encoded qubit that can be implemented with passive networks .
thus we are looking for networks that do not mix the code subspace @xmath11 with the remaining complement of the hilbert space .
let us consider a linear - optics transformation of the annihilation operators given by eq .
( [ eq : linopttransf ] ) .
the transformation of the modes induces a certain unitary operator @xmath24 in the hilbert space of the multimode bosonic system .
the condition that the operator @xmath24 does not take us beyond the code subspace can be written as : @xmath25 the set of all networks that preserve the code subspace forms a group .
let us now suppose that this group is continuous .
this means that we can find a one - parameter subgroup composed of elements @xmath26 parameterized with a real parameter @xmath27 according to : @xmath28 where @xmath29 is an @xmath14 hermitian matrix .
mathematically , @xmath30 is an element of the lie algebra associated with the lie group of unitary @xmath14 matrices .
the unitary operator @xmath31 can then be written as : @xmath32 where @xmath33 is the representation of the matrix @xmath30 for the multimode bosonic system .
it is given by a bilinear combination of the creation and annihilation operators @xcite : @xmath34 the correctness of this expression can be verified by considering operators @xmath35 and writing differential equations for @xmath36 , whose solution recovers eq .
( [ eq : linopttransf ] ) with the transformation matrix given by eq .
( [ eq : gammas = exp - slambda ] ) .
let us now consider an infinitesimal transformation of the form @xmath37 .
the second term , given explicitly in eq .
( [ eq : rlambda ] ) , comprises a sum of expressions of the form @xmath38 that appear also in the correctability condition given in eq .
( [ eq : correctpc ] ) .
it is easy to see that : @xmath39 where we introduced a real coefficient @xmath40 . inserting this result to eq .
( [ eq : prgammap = rgammap ] ) yields : @xmath41 this means that for general , not necessarily infinitesimal , operators @xmath31 we have @xmath42 .
therefore the operator @xmath31 restricted to the code subspace generates only an irrelevant , uniform phase factor .
consequently , there does not exist a continuous group of linear transformations that would produce non - trivial gates on the encoded qubit .
we demonstrated in the preceding section that groups of transformations which can be implemented on photon - loss codes using passive networks must be discrete . for two examples of codes presented in section  [ sec : photonlosscodes ]
these groups can be found analytically . in the case of the four - photon code defined in eq .
( [ eq : fourphoton ] ) , one can use the parameterization of special unitary @xmath43 matrices in terms of euler angles to show easily that all gates preserving the code subspace are generated by two transformations given , up to overall phase factors , by : @xmath44 the above transformations are realized respectively by a balanced beam splitter and a @xmath45 phase shift . in order to gain an insight into the structure of the set of gates , it is helpful to consider transformations of the state @xmath46 .
the application of @xmath47 and @xmath48 generates two other states given , up to phase factors , by @xmath49 and @xmath50 , which together form an equilateral triangle in the bloch sphere of the encoded qubit , shown in figure  1(a ) .
gates that can be implemented with passive linear optics form the rotation group of this triangle .
a more lengthy , but still elementary reasoning @xcite shows that for the three - photon code defined in eq .
( [ eq : threephoton ] ) the transformations preserving the code subspace are obtained from two generators : @xmath51 which correspond respectively to a tritter @xcite and a @xmath52 phase shift on one of the modes .
starting from the initial state @xmath53 , these transformations generate a regular tetrahedron in the bloch sphere of the encoded qubit , with vertices corresponding to the states @xmath54 , and @xmath55 , shown in figure  1(b ) . as before ,
passive networks realize the rotation group of this solid .    ) and ( b ) the three photon code defined in eq .
( [ eq : threephoton ] ) .
the points represent states that can be produced from the logic state @xmath1 using passive networks , and dashed lines depict rotations that are generated by unitary transformations defined respectively in eqs .
( [ eq:4phtr ] ) and ( [ eq:3phtr ] ) . ]
we have shown that restrictions on manipulating photon - loss codes with linear optics are intimately linked to the correctability condition itself .
the invariance of the correctability condition with respect to unitary transformations realized by passive networks prohibits their use for encoding and decoding .
furthermore , passive linear - optics networks are obtained from infinitesimal generators of the form @xmath56 that either move one photon between modes , or introduce linear phase shifts . however , in the code subspace these expressions need to reduce to @xmath57-numbers to ensure correctability , which severely limits available manipulations on encoded qubits .
this work has been supported by the european commission under the integrated project qubit applications ( qap ) funded by the ist directorate as contract number 015848 , polish mnisw grant 1  p03b  011  29 and afosr under grant number fa8655 - 06 - 1 - 3062 .
we discuss codes for protecting logical qubits carried by optical fields from the effects of amplitude damping , i.e.  linear photon loss . 
 we demonstrate that the correctability condition for one - photon loss imposes limitations on the range of manipulations than can be implemented with passive linear - optics networks .    ,    quantum error correction , photon loss , amplitude damping