Maps for general open quantum systems and a theory of linear quantum error correction
aa r X i v : . [ qu a n t - ph ] F e b Maps for general open quantum systems and a theory of linear quantum error correction
Alireza Shabani and Daniel A. Lidar
1, 2 Department of Electrical Engineering-Systems, University of Southern California, Los Angeles, CA 90089, USA Departments of Chemistry and Physics, University of Southern California, Los Angeles, CA 90089, USA
We show that quantum subdynamics of an open quantum system can always be described by a Hermitianmap, irrespective of the form of the initial total system state. Since the theory of quantum error correctionwas developed based on the assumption of completely positive (CP) maps, we present a generalized theoryof linear quantum error correction, which applies to any linear map describing the open system evolution. Inthe physically relevant setting of Hermitian maps, we show that the CP-map based version of quantum errorcorrection theory applies without modifications. However, we show that a more general scenario is also possible,where the recovery map is Hermitian but not CP. Since non-CP maps have non-positive matrices in their range,we provide a geometric characterization of the positivity domain of general linear maps. In particular, we showthat this domain is convex, and that this implies a simple algorithm for finding its boundary.
PACS numbers: 03.67.Pp, 03.67.Hk, 03.67.Lx
I. INTRODUCTION
The problem of the formulation and characterization of thedynamics of quantum open systems has a long and extensivehistory [1, 2, 3]. This problem has become particularly rel-evant in the context of quantum information processing [4],where a remarkable theory of quantum error correction (QEC)was developed in recent years to address the problem of howto process quantum information in the presence of decoher-ence and imperfect control [5]. A key assumption common tomany previous QEC studies is that the evolution of the quan-tum information processor can be described by a successionof completely positive (CP) maps [6], interrupted by unitarygates or measurements [7]. However, it is well known thatif the initial total system state is entangled, quantum dynam-ics is not described by a CP map [8, 9, 10, 11, 12]. In fact,we showed very recently in Ref. [13] that a CP map arises ifand only if the initial total system state has vanishing quantumdiscord [14], i.e., is purely classically correlated. One is thusnaturally led to ask whether this impacts the applicability ofQEC theory under circumstances where non-classical initialstate correlations play a role. Here “initial state” does not re-fer exclusively to the “ t = 0 ” point, but also to intermediatetimes where the recovery map is applied, since this map wasalso assumed to be CP in standard quantum error correctiontheory [7]. Motivated by this fact we here critically revisit theCP maps assumption in QEC, and show that it can be relaxed . To do so, we first consider the problem of characterizingthe type of map that describes open system evolution givenan arbitrary initial total system state (Section II). We showthat this map is always a linear, Hermitian map (of whichCP maps are a special case). We then argue that the genericnoise map describing the evolution of a quantum computeras it undergoes fault tolerant quantum error correction (FT- Note that this is issue is entirely distinct from the critique of Marko-vian fault tolerant QEC expressed in [15], which was concerned with thecompatibility of other assumptions of fault-tolerant QEC (specifically, fastgates and pure ancillas) with rigorous derivations of the Markovian limit.
QEC) is indeed not a CP map, but rather such a Hermitian,linear map (Section III). The reason is, essentially, that imper-fect error correction results in residual non-classical correla-tions between the system and the bath, as the next QEC cycleis applied. To deal with this, we develop a generalized the-ory of QEC which we call “linear quantum error correction”(LQEC), which applies to arbitrary linear maps on the system(Section IV). Then we show that, fortunately, the CP-mapbased version of QEC theory applies without modificationsin the physically relevant setting of Hermitian maps. How-ever, we show that a more general scenario is also possible,where the recovery map is Hermitian but not CP. This is usefulsince it obviates the unrealistic assumption that the recoveryancillas enter the QEC cycle as classically correlated with theother system qubits. Our results significantly extend the realmof applicability of QEC, in particular to arbitrarily correlatedsystem-environment states. We conclude in Section V.
II. QUANTUM DYNAMICAL PROCESSES AND MAPS
In this section we prove a basic new result, that a quantumdynamical process can always be represented as a linear, Her-mitian map from the initial to the final system-only state. Indoing so we rely heavily on our previous work [13].The dynamics of open quantum systems can be describedas follows. Consider a quantum system S coupled to anothersystem B , with respective Hilbert spaces H S and H B , suchthat together they form one isolated system, described by thejoint initial state (density matrix) ρ SB (0) . Their joint time-evolved state is then ρ SB ( t ) = U ( t ) ρ SB (0) U † ( t ) , (1)where U ( t ) is the unitary propagator of the joint system-bathdynamics from the initial time t = 0 to the final time t , i.e.,the solution to the Schrodinger equation ˙ U = − ( i/ ~ )[ H, U ] ,where H is the joint system-bath Hamiltonian. The object ofinterest is the system S , whose state at all times t is governedaccording to the standard quantum-mechanical prescriptionby the following quantum dynamical process (QDP): ρ S ( t ) = Tr B [ ρ SB ( t )] = Tr B [ U SB ( t ) ρ SB (0) U SB ( t ) † ] . (2) Tr B represents the partial trace operation, corresponding toan averaging over the bath degrees of freedom [3].The QDP (2) is a transformation from ρ SB (0) to ρ S ( t ) .However, since we are not interested in the state of the bath, itis natural to ask:Under which conditions on ρ SB (0) is theQDP a map Φ Q ( t ) , ρ S ( t ) = Φ Q ( t )[ ρ S (0)] , (3)and what are the properties of this map?In general, a map is an association of elements in the rangewith elements in the domain. Here we use the term “map”solely to indicate a state-independent transformation betweentwo copies of the same Hilbert space, in particular H S S . Then, a well-known partial answer is that if ρ SB (0) is atensor product state, i.e., ρ SB (0) = ρ S (0) ⊗ ρ B (0) , then theQDP (2) is a CP map. A more general answer was providedin [13]. To explain this answer we must first introduce someterminology. A. Various linear maps
A map
Φ : B ( H )
7→ B ( H ) [space of bounded operatorson H ] is linear if Φ[ aρ + bρ ] = a Φ[ ρ ] + b Φ[ ρ ] for anypair of states ρ , ρ : H 7→ H , and constants a, b ∈ C . Alinear map is called Hermitian if it maps all Hermitian opera-tors in its domain to Hermitian operators. We first present anoperator sum representation for arbitrary and Hermitian linearmaps, that generalizes the standard Kraus representation forCP maps [6]. The proof is presented in Appendix A. Theorem 1
A map Φ L : M n M m (where M n is the spaceof n × n matrices) is linear iff it can be represented as Φ L ( ρ ) = X α E α ρE ′† α (4) where the “left and right operation elements” { E α } and { E ′ α } are, respectively, m × n and n × m matrices. Φ H is a Hermitian map iff Φ H ( ρ ) = X α c α E α ρE † α , c α ∈ R . (5)We will sometimes denote a linear map by listing its ele-ments, as in Φ L = { E α , E ′ α } rα =1 . Note that a linear map Φ L = { E α , E ′ α } rα =1 is trace preserving if P rα =1 E ′† α E α = This is meant to exclude claims that system state-dependent transforma-tions qualify as CP maps, as in Ref. [16]. In such cases the elements of thetransformation (the “Kraus operators”) depend on the system input state,which contradicts our notion of a map. I . Also note that the two sets of operation elements { E α , E ′ α } ri =1 and { F β , F ′ β } rβ =1 , where F β = P rα =1 u αβ E α and F ′ β = P rα =1 v αβ E ′ α , represent the same linear map Φ L ifthe matrices u and v satisfy uv † = I .As a simple example of a non-CP, Hermitian map, con-sider the inverse-phase-flip map . The well-known CP phase-flip map is [4]: Φ PF ( ρ ) = (1 − p ) ρ + pσ z ρσ z , where ≤ p ≤ and σ z is a Pauli matrix. Solving for Φ − from Φ − [Φ PF ( ρ )] = ρ , we find that Φ − ( ρ ) = c ρ + c σ z ρσ z ,where c = p/ (2 p − and c = 1 − c , and c , c have op-posite sign for < p < . Moreover, Tr[Φ − ( ρ )] = Tr( ρ ) .Therefore Φ − is a trace-preserving, Hermitian, non-CP map.A linear map is called “completely positive” (CP) if it isa Hermitian map with c α ≥ ∀ α . CP maps play a keyrole in quantum information and quantum error correction [4],though they have a much earlier origin [6, 17]. There are otheruseful characterizations of CP maps – see, e.g., Refs. [3, 4]. Itturns out that there is a tight connection between CP and Her-mitian maps [10, 11]: a map is Hermitian iff it can be writtenas the difference of two CP maps.The definition of a CP map Φ CP implies that it can be ex-pressed in the Kraus operator sum representation [6]: ρ S ( t ) = X α E α ( t ) ρ S (0) E † α ( t ) = Φ CP ( t )[ ρ S (0)] . (6)If the operation elements E α satisfy P α E † α E α = I then Tr[ ρ S ( t )] = 1 . B. Special linear states
Following Ref. [13], we define the class of “special-linear”(SL) states for which the QDP (2) always results in a linear,Hermitian map. An arbitrary bipartite state on H S ⊗ H B canbe written as ρ SB = X ij ̺ ij | i ih j | ⊗ φ ij , (7)where {| i i} dim H S i =1 is an orthonormal basis for H S , and { φ ij } dim H S i,j =1 : H B
7→ H B are normalized such that if Tr[ φ ij ] = 0 then Tr[ φ ij ] = 1 . The corresponding reducedsystem and bath states are then ρ S = P ( i,j ) ∈C ̺ ij | i ih j | ,where C ≡ { ( i, j ) | Tr[ φ ij ] = 1 } , and ρ B (0) = P i ̺ ii φ ii .Hermiticity and normalization of ρ SB , ρ S , and ρ B imply ̺ ij = ̺ ∗ ji , φ ij = φ † ji , and P i ̺ ii = 1 . Definition 1
A bipartite state ρ SB , parametrized as in Eq.(7), is in the SL-class if either Tr[ φ ij ] = 1 or φ ij = 0 , ∀ i, j . Thus a non-SL state is a state for which there exist indexes i and j such that Tr[ φ ij ] = 0 but φ ij = 0 . The followingresult proven in Ref. [13] (generalizing an earlier result inRef. [12]) provides an almost complete answer to the questionposed above: Theorem 2 (Theorem 2 of [13]) If ρ SB (0) is an SL-classstate then the QDP (2) is a linear, Hermitian map Φ H : ρ S (0) ρ S ( t ) . A further result proven in Ref. [13] (Theorem 3 there) pro-vides necessary and sufficient conditions on ρ SB (0) for theQDP (2) to be a CP map, namely, ρ SB (0) should be a statewith vanishing quantum discord [14]. Such a state cannotcontain any quantum correlations. This clearly illustrates thelimitations of CP maps in describing quantum dynamics. Atthe same time one may wonder as to the generality of theSL-class employed in Theorem 2. Non-SL states are sparse[13], so it is in this regard that we stated that Theorem 2 pro-vides an almost complete answer to the question posed above.However, we can go further. As mentioned without proof inRef. [13], in fact the QDP (2) is a linear, Hermitian map from ρ S (0) ρ S ( t ) for any initial state ρ SB (0) . We next provethis key fact. C. Hermitian maps for arbitrary initial states
We split the general initial state representation (7) into asum over SL and non-SL terms (thus splitting { ̺ ij } and { φ ij } into two sets): ρ SB (0) = X ij ∈ (SL) α ij | i ih j | ⊗ ϕ ij + X ij ∈ (nSL) β ij | i ih j | ⊗ ψ ij . (8)In accordance with the definition of SL states, in the first sumwe include only terms α ij | i ih j | ⊗ ϕ ij for which Tr[ ϕ ij ] = 0 or ϕ ij = 0 , in the second only terms β ij | i ih j | ⊗ ψ ij with bathoperators { ψ ij } satisfying ψ ij = 0 and Tr[ ψ ij ] = 0 . By virtueof this decomposition only the first term contributes to the ini-tial system state: ρ S (0) = Tr B [ ρ SB (0)] = P ij (SL) α ij | i ih j | .This is because the condition Tr[ ψ ij ] = 0 eliminates any con-tribution from the second term in the decomposition (8) to theinitial system state. Consequently Eq. (3) assumes an affineform: Φ Q ( t )[ ρ S (0)] = Φ SL ( t )[ ρ S (0)] + K nSL ( t ) , (9)with the term K nSL ( t ) being a shift that is independent of ρ S (0) .As shown in Ref. [13], the linear map Φ SL is constructedas a function of the bath operators { ϕ ij } : Φ SL ( t )[ ρ S (0)] ≡ X ( i,j ) ∈ (SL); k,α λ ijα V αkij P i ρ S (0) P j ( W αkij ) † , (10)where P i ≡ | i ih i | are projectors, λ ijα are the singular valuesin the singular value decomposition φ ij = P α λ ijα | x αij ih y αij | ,and the operators V αkij ≡ h ψ k | U | x αij i and W αkij ≡ h ψ k | U | y αij i act on the system only, with {| ψ k i} being an orthonormal ba-sis for the bath Hilbert space H B .In addition, the non-SL terms in Eq.(8) generate the shiftterm K nSL ( t ) = X ij ∈ (nSL) β ij Tr B [ U SB ( t ) | i ih j | ⊗ ψ ij U † SB ( t )] . (11)This shows explicitly that K nSL ( t ) does not depend on theinitial system state, since the latter is fully parametrized by the coefficients { α ij } ij ∈ (SL) , while K nSL ( t ) depends only uponthe coefficients { β ij } ij ∈ (nSL) .Now we take a further step to argue that the affine map (9)is actually a linear, Hermitian map if the map acts only on thespace of density matrices. This is a direct application of theresult in Ref. [10]. Theorem 3
The QDP (2) is representable as a linear, Hermi-tian map Φ H ( t ) : ρ S (0) ρ S ( t ) for any initial system-bathstate. Proof.
Let N ≡ dim H S . Let F ≡ I and let { F µ : Tr( F µ ) =0 } N − µ =1 be a basis for the set of traceless Hermitian matriceswhich are mutually orthogonal with respect to the Hilbert-Schmidt inner product, i.e., Tr( F µ F ν ) = N δ µυ . Hence theinitial system state ρ S (0) can be expanded as ρ S (0) = 1 N ( I + N − X µ =1 b µ F µ ); b µ = Tr[ ρ S (0) F µ ] ≡ h F µ i ρ S (0) , (12)and the final system state is found to be ρ S ( t ) = 1 N [Φ SL ( I ) + N − X µ =1 b µ Φ SL ( F µ )] + K nSL = Φ H ( t )[ ρ S (0)] , (13)where the equivalent Hermitian map Φ H is constructed by set-ting Φ H ( I ) = Φ SL ( I ) + N K nSL and Φ H ( F µ ) = Φ SL ( F µ )1 ≤ ∀ µ ≤ N − . That this map is Hermitian is simple toverify, for all the components are Hermitian.Theorem 3 provides a complete, and perhaps surprising an-swer to the question posed at the beginning of this section.Namely, the most general form of a quantum dynamical pro-cess, irrespective of the initial system-bath state (in particulararbitrarily entangled initial states are possible) is always re-ducible to a Hermitian map from the initial system to the finalsystem state. The surprising aspect of this result is that it wasnot known previously whether QDP could always even be re-duced to a map between system states.Of course, this result does not resolve the more difficultquestion of ensuring the positivity of the final system state.That is, a Hermitian map may transform an initially positivesystem state to a non-positive one, violating the postulate ofpositivity of quantum states. To resolve this one must identifythe “positivity domain” of Φ H , i.e., the set of initial systemstates (positive by definition) which are mapped to positivestates by Φ H [10]. We address this in the next subsection. D. Geometric characterization of the Positivity Domain
In this subsection we prove the convexity of the positivitydomain and propose a geometric method for characterizing it.Let S ( H ) ≡ { ρ ∈ L ( H ) : ρ > , Tr ρ = 1 } , where L ( H ) isthe set of all linear operators on H . The positivity domain ofa linear map Φ L : S ( H )
7→ B ( H ) is: P Φ ≡ { ρ ∈ S ( H S ) :Φ L ( ρ ) > } .Following earlier work [18, 19, 20], in Ref. [21], a com-plete geometric characterization of density matrices was givenby using the Bloch vector representation for an arbitrary N -dimensional Hilbert space H . This works as follows: let { F µ } N − µ =1 be a basis set as in the proof of Theorem 3,whence the expansion (12) applies again. The vector b =( b , ..., b N − ) ∈ R N − of expectation values is known asthe Bloch vector, and knowing its components is equivalentto complete knowledge of the corresponding density matrix,via the map b ρ = N ( I + P N − µ =1 b µ F µ ) . Let n denote aunit vector, i.e., n ∈ R N − and P N − i =1 n i = 1 , and de-fine F n ≡ P N − µ =1 n µ F µ . Let the minimum eigenvalue ofeach F n be denoted m ( F n ) . The “Bloch space” B ( R N − ) is the set of all Bloch vectors and is a closed convex set, sincethe set S ( H ) is closed and convex, and the map b ρ islinear homeomorphic. As shown in Theorem 1 of Ref. [21],the Bloch space is characterized in the “spherical coordinates”determined by { F n } as: B ( R N − ) = (cid:26) b = r n ∈ R N − : r ≤ | m ( F n ) | (cid:27) . (14)It is hard to imagine a more intuitive or simpler geometricpicture.Next we show that the positivity domain is a convex set aswell. Proposition 1
The positivity domain P Φ of a linear map Φ L is a convex set. Proof.
Consider two density matrices ρ and ρ ′ as inte-rior points of P Φ with corresponding Bloch vectors b =( b , ..., b N − ) and b ′ = ( b ′ , ..., b ′ N − ) . The claim is thata third density matrix ρ ′′ with corresponding Bloch vector b ′′ ( α ) = α b +(1 − α ) b ′ , with ≤ α ≤ , is then also interiorto P Φ . This follows directly by linearity of the map Φ L . First,by assumption Φ L [ ρ ] = Φ L [ N ( I + P N − µ =1 b µ F µ )] > and Φ L [ ρ ′ ] = Φ L [ N ( I + P N − µ =1 b ′ µ F µ )] > , so that α Φ L [ ρ ] +(1 − α )Φ L [ ρ ′ ] > . Second, α Φ L [ ρ ] + (1 − α )Φ L [ ρ ′ ] =Φ L [ N I ]+ α P N − µ =1 b µ Φ L [ F µ ]+(1 − α ) P N − µ =1 b ′ µ Φ L [ F µ ] =Φ L [ N ( I + P N − µ =1 b ′′ µ F µ )] = Φ L [ ρ ′′ ] . Therefore indeed Φ L [ ρ ′′ ] > .We are now ready to describe an algorithm for finding theboundary of the positivity domain P Φ . We know at this pointthat P Φ is convex and that P Φ is a subset of the Bloch space,itself a closed convex set. Pick a unit vector n and draw a linethrough the origin of the Bloch space along n . If P Φ includesthe origin, i.e., the maximally mixed state, then convexity im-plies that this line intersects the boundary of P Φ once. If P Φ does not include the origin then convexity implies that thisline either intersects the boundary of P Φ twice or not at all.I.e., it follows from convexity that the line may not re-enterthe positivity domain once it exited. In order to determine thisboundary we may thus compute the eigenvalues of Φ L [ ρ n ( r )] as a function of r , where r is the parameter in Eq. (14), andwhere ρ n ( r ) is the density matrix determined via the mapping b = r n ρ . The computation should start from r = 0 andgo up to at most r = 1 / | m ( F n ) | . The boundary is identi-fied as soon as the eigenvalues of Φ L [ ρ n ( r )] go from all pos-itive semi-definite to at least one negative, or vice versa. Foreach unit vector n , the corresponding point on the border ofthe positivity domain can be found in this way. Then the al-gorithm constructs the boundary of the positivity domain byfinding the boundary points in all directions n . Of course, inpractice one can only sample the space of unit vectors n andfactors r . In principle this yields a complete geometrical de-scription of the positivity domain of a given linear map. III. CP MAPS AND FAULT TOLERANT QUANTUMERROR CORRECTIONA. CP maps: pro and con
We have already mentioned that a QDP (2) becomes a CPmap iff the initial system-bath state has vanishing quantumdiscord, i.e., is purely classically correlated [13]. The stan-dard argument in favor of CP maps is that since the system S may be coupled with the bath B , the maps describing phys-ical processes on S should be such that all their extensionsinto higher dimensional spaces should remain positive, i.e., Φ CP ⊗ I n ≥ ∀ n ∈ Z + , where I n is the n -dimensionalidentity operator. However, one may question whether this isthe right criterion for describing quantum dynamics [8]. Analternative viewpoint is to seek a description that applies to arbitrary ρ SB (0) , as we have done above. We now argue thatthis viewpoint is the correct one for fault-tolerant quantum er-ror correction (FT-QEC). B. (In)validity of the CP map model in FT-QEC
Let us show that system-environment correlations impose asevere restriction on the applicability of CP maps in FT-QEC.The CP map model used in FT-QEC [22, 23, 24, 25, 26, 27,28, 29] can be described as follows (see, e.g., Eq. (8.1) in[28]): ρ S ( T ) = Φ totCP ( T, t )[ ρ S ( t )] where Φ totCP ( T, t ) = O Ni =1 Φ U ( t i )Φ CP ( t i , t i − ) , (15)where T ≡ t N is the total circuit time, and where Φ U [ ρ S ] = U S ρ S U † S is a unitary map (automatically CP) that describesan ideal quantum logic gate. This represents the idea usedrepeatedly in FT-QEC, that the noisy evolution at every timestep can be decomposed into “pure noise” Φ CP ( t i , t i − ) fol-lowed by an instantaneous and perfect unitary gate Φ U ( t i ) .More precisely, in FT-QEC one assumes that the evolutionstarts ( t = t = 0 ) from a product state, then undergoes a CP In this subsection we denote noise maps by their initial and final times, todistinguish them from the instantaneous unitary maps. map Φ CP ( t , t ) due to coupling to the environment, followedby an instantaneous error correction step Φ U ( t ) . If the latterwere perfect then the post-error-correction state would againbe a product state ρ S ( t ) ⊗ ρ B ( t ) . However, FT-QEC allowsfor the fact that the error correction step is almost never per-fect, which means that there is a residual correlation betweensystem and bath at t . Hence, according to Ref. [13], the mapthat describes the evolution of the system is a CP map if andonly if the residual correlation is purely classical. Otherwise itis a Hermitian map. To make this point more explicit, considera sequence of two noise time-steps, interrupted by one errorcorrection step. In the ideal scenario, where the error correc-tion step Φ U ( t ) works perfectly (i.e., reduces the system-bathcorrelations to purely classical), we would have Φ (2)CP ( t , t ) = Φ CP ( t , t )Φ U ( t )Φ CP ( t , t ) , (16)where Φ CP ( t , t ) is again a CP noise map. However, in re-ality Φ U ( t ) works imperfectly [system-bath correlations arenot purely classical after the action of Φ U ( t ) ], and the actualmap obtained is Φ (2)H ( t , t ) = Φ H ( t , t )Φ U ( t )Φ CP ( t , t ) , (17)where Φ H ( t , t ) is now a Hermitian map. Note that, in fact,even the assumption that the first noise map is CP will notbe true in general, due to errors in the preparation of the ini-tial state, leading to non-classical correlations between systemand bath. We conclude that in general the CP map model (15)should be replaced by Φ totH ( T, t ) = O Ni =1 Φ U ( t i )Φ H ( t i , t i − ) , (18)where Φ H ( t i , t i − ) are Hermitian maps , not necessarily CP. It is worth emphasizing that this distinction between purelyclassical and other correlations, and the resulting differencebetween CP and Hermitian evolution, is not a distinction thathas thus far been made in FT-QEC theory. Rather, in FT-QEC one distinguishes between “good” and “bad” fault paths,where the former (latter) contain only a few (too many) errors.Quoting from [30]: “There are good fault paths with so-calledsparse numbers of faults which keep being corrected duringthe computation and which lead to (approximately) correct an-swers of the computation; and there are bad fault-paths whichcontain too many faults to be corrected and imply a crash ofthe quantum computer.” This leads to a splitting of the totalmap (15) into a sum over good and bad paths. One then showsthat the computation can proceed robustly via the use of con-catenated codes, provided the “bad” paths are appropriatelybounded. In [28](p.1272) it was pointed out that the sum over“good” paths need not be a CP map, but can be decomposedinto a new sum over CP maps [Eq. (8.13) there]. This new de-composition can then be treated using standard FT-QEC tech-niques. However, this assumes again that the total evolution isa CP map, which in fact it is not [Eq. (18)]. Note that Eq. (18) applies also to non-Markovian noise, and is hence com-plementary to Hamiltonian FT-QEC [30, 31, 32].
These observations motivate a generalized theory of QEC,which can handle non-CP noise maps. This is the subject ofthe next section.
The main result of this theory is reassuring:in spite of the invalidity of the CP map model in FT-QEC, theCP-map based results apply because the same encoding andrecovery that corrects a Hermitian map can be used to correcta closely related CP map, whose coefficients are the absolutevalues of the Hermitian map . This is formalized in Corollary1.
IV. LINEAR QUANTUM ERROR CORRECTION
Having argued that non-CP Hermitian maps arise naturallyin the study of open systems, and in particular FT-QEC, wenow proceed to develop the theory of Linear QEC. For gen-erality we do this for arbitrary linear maps, i.e., maps of theform (4). We then specialize to the physically relevant case ofHermitian maps.Let us first recall the fundamental theorem of “standard”QEC (for CP noise and CP recovery maps) [7]: Let P be aprojection operator onto the code space. Necessary and suf-ficient conditions for quantum error correction of a CP map, Φ CP ( ρ ) = P i F i ρF † i are P F † i F j P = λ ij P ∀ i, j. (19)An elegant proof of this theorem and a construction of thecorresponding CP recovery map was given in Refs. [4, 33];we use some of their methods in the proofs of Theorems 4,5. A. CP-recoverable linear noise maps
While general (non-Hermitian) linear maps of the form (4)do not arise from quantum dynamical processes [Eq. (2)], itis still interesting from a purely mathematical standpoint toconsider QEC for such maps. Moreover, we easily recoverthe physical setting from these general considerations.Theorem 4 shows that there is a class of linear noise mapswhich are equivalent to certain non-trace-preserving CP noisemaps when it comes to error correction using CP recoverymaps.
Theorem 4
Consider a general linear noise map Φ L ( ρ ) = P Ni =1 E i ρE ′† i and associate to it an “expanded” CP map ˜Φ CP ( ρ ) = P Ni =1 E i ρE † i + P Ni =1 E ′ i ρE ′† i . Then any QECcode C and corresponding CP recovery map R for ˜Φ CP arealso a QEC code and CP recovery map for Φ L . Proof.
The operation elements of ˜Φ CP are { F i } Ni =1 = { √ E i } Ni =1 and { F N + i } Ni =1 = { √ E ′ i } Ni =1 , whence ˜Φ CP ( ρ ) = P Ni =1 F i ρF † i . The standard quantum error cor-rection conditions (19) for ˜Φ CP , where λ ≡ (cid:18) α γγ † α ′ (cid:19) = λ † , (20)become three sets of conditions in terms of the E i and E ′ i :(i) P E † i E j P = 2 α ij P, (ii) P E ′† i E ′ j P = 2 α ′ ij P, (iii) P E † i E ′ j P = 2 γ ij P, (21)where i, j ∈ { , ..., N } and α ij = λ ij , γ ij = λ i,N + j , α ′ ij = λ N + i,N + j . The existence of a projector P which sat-isfies Eqs. (21)(i)-(iii) is equivalent to the existence of a QECcode for ˜Φ CP . Assuming that a code C has been found (i.e., P C = C ) for ˜Φ CP , we use this as a code for Φ L and show thatthe corresponding CP recovery map R CP is also a recoverymap for Φ L . Indeed, let G j ≡ P Ni =1 u ij F i be new operationelements for ˜Φ CP , where u is the unitary matrix that diago-nalizes λ , i.e., u † λu = d . Then ˜Φ CP = P Nj =1 G j ρG † j . Let R CP = { R k } be the CP recovery map for ˜Φ CP . Assume that ρ is in the code space, i.e., P ρP = ρ . We now show that R CP [Φ L ( ρ )] = ρ , i.e., we have CP recovery. First, R CP [Φ L ( ρ )] = X k R k N X i =1 F i ρF † N + i ! R † k = N X i =1 2 N X j,j ′ =1 u ∗ ij u N + i,j ′ × X k ( R k G j P ) ρ (cid:16) P G † j ′ R † k (cid:17) . (22)Now, note that P G † k G l P = X ij u ∗ ik u jl P F † i F j P = X ij u ∗ ik λ ij u jl P = d k δ kl P. (23)Then the polar decomposition yields G k P = U k ( P G † k G k P ) / = p d k U k P. (24)The recovery operation elements are given by R k = U † k P k , (25)where P k = U k P U † k . Therefore P k = G k P U † k / √ d k . Thisallows us to calculate the action of the k th recovery operatoron the l th error [4, 33]: R k G l P = U † k P † k G l P = U † k ( U k P G † k / p d k ) G l P = δ kl p d k P. (26)Therefore, R CP [Φ L ( ρ )] = N X i =1 2 N X j,j ′ =1 u ∗ ij u N + i,j ′ × X k (cid:16) δ kj p d k P (cid:17) ρ (cid:16) P p d k δ kj ′ (cid:17) = ρ N X i =1 (cid:0) udu † (cid:1) N + i,i = ρ N X i =1 λ N + i,i = 2 ρ Tr γ † . (27) Next note that, using condition (21)(iii) and trace preservationby Φ L : P E ′† i E i P = 2 γ † ii P = ⇒ γ † P = P X i E ′† i E i P = P = ⇒ Tr γ † = 12 . (28)Hence, finally: R CP [Φ L ( ρ )] = ρ (29)for any ρ in the codespace.Note that ˜Φ CP ( ρ ) need not be tracepreserving: Tr[ ˜Φ CP ( ρ )] = Tr[( P Ni =1 E † i E i + P Ni =1 E ′† i E ′ i ) ρ ] , and while P Ni =1 E ′† i E i = I if Φ L istrace preserving, we do not have conditions on P Ni =1 E † i E i and P Ni =1 E ′† i E ′ i .We define the class of “CP-recoverable linear noise maps” { Φ CPR } as those Φ L for which CP recovery is always possi-ble. By Theorem 4 this includes all Φ L for which P can befound satisfying conditions (21)(i)-(iii). However, these con-ditions are not necessary. B. Non-CP-recoverable linear noise maps
We now define “non-CP-recoverable linear noise maps” { Φ nCPR } as those Φ L for which non-CP-recovery is alwayspossible. Theorem 5 shows constructively that { Φ nCPR } in-cludes all linear noise maps Φ L for which P can be foundsatisfying only conditions (21)(i) and (ii). Clearly, { Φ CP } ⊂{ Φ CPR } ⊂ { Φ nCPR } ⊂ { Φ L } . Theorem 5
Let Φ L = { E i , E ′ i } i be a linear noise map. Thenevery state ρ = P ρP encoded using a QEC code defined by aprojector P satisfying only Eqs. (21)(i) and (ii) can be recov-ered using a non-CP recovery map. Proof.
Let G k = P i u ik E i and G ′ k = P i u ′ ik E ′ i , where theunitaries u and u ′ respectively diagonalize the Hermitian ma-trices α and α ′ : d = u † αu and d ′ = u ′† α ′ u ′ . Define a recov-ery map R = { R k , R ′ k } (not necessarily CP) with operationelements R k = U † k P k , R ′ k = U ′† k P ′ k . (30)Here P k = U k P U † k , P ′ k = U ′ k P U ′† k are projection opera-tors, and U k and U ′ k arise from the polar decomposition of G k P and G ′ k P , i.e., G k P = U k ( P G † k G k P ) / and G ′ k P = U k ( P G ′† k G ′ k P ) / . The proof is entirely analogous to theproof of Theorem 4, except that we must keep track of boththe primed and unprimed operators. Following through thesame calculations we thus obtain R k G l √ ρ = √ d k δ kl √ ρ and R ′ k G ′ l √ ρ = p d ′ k δ kl √ ρ . Using this in the recovery map ap-plied to the linear noise map, we find: R [Φ( P ρP )] = X kl R k E l P ρP E ′† l R ′† k = X kl R k ( X j u ∗ lj G j ) P ρP ( X i u ′ li G ′† i ) R ′† k = F L P ρP ∝ ρ, (31)where F L ≡ X ijkl u ∗ lj u ′ li q d k d ′∗ k δ kj δ ki = X kl u ∗ lk u ′ lk q d k d ′∗ k = Tr[ u ′ d ′† du † ] = Tr[ u ′ u † αα ′† ] (32)is a “correction factor” for non-CP recovery of linear noisemaps, which was in the case of CP recovery, above.Gathering the expressions derived in the last proof, we havethe following explicit expressions for the left and right recov-ery operations: R k = U † k P † k = 1 √ d k P X i u ∗ ik E † i , R ′ k = 1 p d ′ k P X i u ′∗ ik E ′† i . (33)This also shows that, in general, R k need not equal R ′ k , i.e.,the recovery map is linear but not necessarily CP.Note that standard QEC can also be interpreted as “errorcorrection by inversion”, in the following sense: when thenoise map is CP and recovery is also CP, recovery is the in-verse of the noise map restricted to the code space (TheoremIII.3 in Ref. [7]). The same is true for our LQEC results above,which relax the restriction to CP noise maps. C. The physical case: Hermitian maps
The general physical case is the case of Hermitian noisemaps, to which any quantum dynamical process can be re-duced, as follows from Theorem 3. We can specialize Theo-rems 4 and 5 to this case.
Corollary 1
Consider a Hermitian noise map Φ H ( ρ ) = P Ni =1 c i K i ρK † i and associate to it a CP map ˜Φ CP ( ρ ) = P Ni =1 | c i | K i ρK † i . Then any QEC code C and correspondingCP recovery map R CP for ˜Φ CP are also a QEC code and CPrecovery map for Φ H . The important conclusion we can draw from Corollary 1 isthat standard QEC techniques apply whether the noise map isCP or, as it will almost always be due to non-classical corre-lations, Hermitian. This is because Corollary 1 tells us thatit is safe to replace all negative c i coefficients by their abso-lute values, and thus replace the actual noise map by its CPcounterpart. Proof.
We have Φ H ( ρ ) = P Ni =1 E i ρE ′† i with { E i = √ c i K i } Ni =1 and { E ′ i = ( √ c i ) ∗ K i } Ni =1 , whence we can applythe construction of Theorem 4. Indeed, the “expanded” CPmap becomes ˜Φ CP ( ρ ) = P Ni =1 E i ρE † i + P Ni =1 E ′ i ρE ′† i = P Ni =1 | c i | K i ρK † i , as claimed, and hence a QEC code and CPrecovery for ˜Φ CP is also a QEC code and CP recovery for Φ H .In particular, R CP [Φ H ( ρ )] = ρ .Note that ˜Φ CP need not be trace preserving even in the Her-mitian map case: Tr[ ˜Φ CP ( ρ )] = Tr[ P Ni =1 | c i | K † i K i ρ ] , but if Φ H is trace preserving then we only have P Ni =1 c i K † i K i = I ,hence cannot conclude more about Tr[ ˜Φ CP ( ρ )] . Also notethat substitution of E i = √ c i K i and E ′ i = ( √ c i ) ∗ K i into theQEC conditions (21)(i)-(iii) yields α ′ ij = q c i c j (cid:16)q c j c i (cid:17) ∗ α ij and γ ij = ( √ c j ) ∗ √ c j α ij , i.e., unlike in the general linear mapscase, the matrices α ′ and γ in Eq. (20) are not independentfrom α . In fact, as shown in Appendix B we can give a directproof of Corollary 1 which only invokes a single block of the λ matrix.
1. Example of CP recovery: Inverse bit-flip map
Consider “diagonalizable maps”, i.e., Φ D ( ρ ) ≡ P i c i K i ρK † i , where c i ∈ C . The expanded CP map is ˜Φ CP = P i | c i | K i ρK † i . Now consider as a specific in-stance an independent-errors inverse bit-flip map on threequbits: Φ IPF ( ρ ) = c ρ + c P n =1 X n ρX n , where X n isthe Pauli σ x matrix applied to qubit n , where c and c are real, have opposite sign, and c + 3 c = 1 (a Her-mitian map). Then ˜Φ CP = | c | ρ + | c | P n =1 X n ρX n ,which is a non-trace preserving version of the wellknown independent-errors CP bit-flip map. The codeis C = span {| L i ≡ | i , | L i ≡ | i} , and P = | L ih L | + | L ih L | , which satisfies Eq. (B1)with F = p | c | I and F , , = p | c | X , , . Then byCorollary 1 the same code (and corresponding CP recoverymap) also corrects Φ IPF . The CP recovery map R CP hasoperation elements R = P and { R n = √ P X n } n =1 ;indeed, it is easily checked that R CP [Φ IPF ( P ρP )] =
P ρP for any state ρ ∈ C .
2. Hermitian recovery maps
Since Hermitian maps are the most general physical maps,it is natural to consider Hermitian recovery of Hermitian noisemaps. We thus define “Hermitian recovery maps” {R H } asthose Hermitian maps that correct a Hermitian noise map Φ H ,i.e., R H ◦ Φ H ( ρ ) ∝ ρ . The following result presents a possibleset of Hermitian recovery maps. Corollary 2
Consider a Hermitian noise map Φ H ( ρ ) = P Ni =1 c i K i ρK † i with error operators { K i } satisfying the re-lations P K † i K j P = α ij P . Any Hermitian map R H ( ρ ) = P k h k R k ρR † k with recovery operators { R k } as in Eq. (25)and { h k } ∈ R corrects the noise map Φ H . The proof is given in Appendix C, and employs a methodsimilar to that of the proof of Theorem 5. BS t t T ( ) S T data qubitencodingancillas recoveryancillas ( ) SB t R A encoding U AB bath … ! B U S U S UU … R S R S ( ) SRB t t recovery SR U H H DE transmission orcomputation U SRB U S UU RB …… FIG. 1: The initial system-bath state is the generically non-VQDstate ρ SB ( t ) . The encoded system S = D + E consists of dataqubits D and encoding ancillas E . We also include the recoveryancillas R , which are assumed to be completely isolated until theyare brought into contact with S and B at a later time. Thus thefull initial state is ρ SB ( t ) ⊗ ρ R ( t ) . The overall evolution is gov-erned by the unitary U SRB which acts on the system S , the bath B ,and eventually the recovery ancillas R , and is denoted by the largegrey box. The state of the data qubits is ρ | ψ i = Tr E,B [ ρ SB ( t )] ,a state which is as close as possible (by isolating the system) to thedesired pure data state | ψ i . The state of each of the encoding ancil-las is ρ | i = Tr E ′ ,B [ ρ SB ( t )] , a state which is as close as possible(again, by isolating the system) to the desired pure encoding ancillastate | i . Here Tr E,B denotes a partial trace over all encoding an-cillas and the bath, Tr E ′ ,B denotes a partial trace over all but oneof the encoding ancillas, and the bath. Ideally, the encoding uni-tary U S is then applied to the encoded system. This is of course anidealization since in reality the encoding operation will not be a per-fect unitary; instead what is really applied is U SRB ( t , t ) , whichis supposedly close to the ideal U S ⊗ I R ⊗ I B . Thus, after theencoding the total state is ρ SRB ( t ) = U SRB ( t , t )[ ρ SB ( t ) ⊗ ρ R ( t )] U † SRB ( t , t ) and the encoded system state is ρ S ( t ) =Tr R,B [ ρ SRB ( t )] . The system is then passed through the noisechannel for the purpose of either computation or communication,i.e., ρ SRB ( t ) = U SRB ( t , t ) ρ SRB ( t ) U † SRB ( t , t ) , whence ρ S ( t ) = Tr R,B [ ρ SRB ( t )] = Φ H [ ρ S ( t )] , where Φ H is a Her-mitian noise map since ρ SRB ( t ) is generically a non-VQD statedue to the initial non-classical correlations between S and B . Thegoal of the error correction procedure is to recover the original en-coded system state from ρ S ( t ) , and to this end we introduce re-covery ancillas R at t . Similarly to the encoding ancillas, theserecovery ancillas are each in the state ρ | i = Tr S,R ′ ,B [ ρ SRB ( t )] , astate which is as close as possible to the desired pure recovery ancillastate | i . Next, ideally the recovery unitary U SR ⊗ I B is applied. Inreality what is applied is U SRB ( T, t ) , which is supposedly closeto the ideal U SR ⊗ I B . Then the recovery ancillas are discardedand possibly recycled, leaving the encoded system in the final state ρ S ( T ) = Tr R,B [ ρ SRB ( T )] = R S [ ρ S ( t )] , which can be measured.Since ρ SRB ( t ) is generically not a VQD state (due to non-classicalcorrelations between S and R , mediated by their mutual interactionwith B ), it is clear that the recovery map R S is generically a non-CP Hermitian map. We recover the CP recovery map scenario if,for example, ρ SRB ( t ) = ρ SB ( t ) ⊗ ρ R ( t ) . The assumption thatthis is not the case is consistent with the working premise of this pa-per and is equivalent in that regard to the assumption that the initialsystem-bath state is not of the form ρ SB ( t ) = ρ S ( t ) ⊗ ρ B ( t ) .
3. How does non-CP, Hermitian recovery arise?
In standard QEC theory the recovery map is considered CP.The reason for this is that the recovery ancillas are introducedafter the action of the noise channel so that they enter in a ten-sor product state with the encoded qubits that underwent thenoise channel. The recovery map is obtained in the standardsetting by first applying a unitary over the encoded qubits plusrecovery ancillas, then tracing out the recovery ancillas. Thisis manifestly a CP map over the encoded qubits.Since we know that the recovery map experienced by theencoded qubits is CP if and only if the initial state of the en-coded and recovery ancilla qubits has vanishing quantum dis-cord [13], it is clear how a non-CP recovery map can be im-plemented: the recovery ancillas should have non-vanishingquantum discord with the encoded qubits. Since this will stillbe a QDP, the resulting recovery map will be Hermitian ac-cording to Theorem 3.Such a situation can come about in various ways. For ex-ample, a scenario which is particularly relevant for quantumcomputation and communication, is one where the environ-ment causes the recovery ancillas to become non-classicallycorrelated with the encoded qubits before the recovery opera-tion can be applied. This is a reasonable scenario since, whilethe recovery ancillas are presumably kept pure and isolatedfrom the environment for as long as possible, at some pointthey must be brought into contact with the encoded qubits,and at this point all qubits (encoded and recovery ancillas) aresusceptible to correlations mediated by the environment. Thisis shown in Fig. 1.
V. CONCLUSIONS
This work aimed to fill two gaps: one in the theory of openquantum systems, and a resulting gap in the theory of quantumerror correction. The first gap had to do with the type of mapsthat describe open systems given arbitrary initial states of thetotal system. In fact, it was not a priori clear that there shouldeven be a linear map connecting the initial to the final opensystem state for arbitrary initial total system states. Build-ing upon the class of “special linear states” we introduced in[13] we showed here that in fact such a linear map descriptiondoes always exist, and moreover, for quantum dynamics themap is always Hermitian. The map reduces to the completelypositive type if and only if the initial total system state hasvanishing quantum discord [13]; in all other cases it is Her-mitian but not CP. This result, we argued, impacts the theoryof quantum error correction, where previously the assumptionof CP maps was taken for granted. In the second part of thiswork we filled this gap in QEC theory, by developing a theoryof Linear Quantum Error Correction (LQEC), which general-izes the CP-map-based standard theory of QEC. We showedthat to every linear map Φ L is associated a CP map which, ifcorrectable, also provides an encoding with corresponding CPrecovery map for Φ L (Theorem 4). Moreover, it is possible tofind a non-CP recovery for Φ L within a larger class of codes(Theorem 5). From a physical standpoint this result is actu-ally too general, since only Hermitian maps ever arise fromquantum dynamics [to the extent that the standard quantumdynamical process (2) is valid]. Hence we specialized LQECto the Hermitian maps case, and showed that in this case stan-dard QEC theory for CP maps already suffices, in the sensethat it is legitimate to replace a given Hermitian noise map bya corresponding CP map obtained simply by taking the abso-lute values of all the Hermitian map coefficients. Any QECcode which corrects this CP map will also correct the originalHermitian map (Corollary 1). Nevertheless, there is room fora genuine generalization when one considers Hermitian maps,since it is also possible to perform QEC using Hermitian re-covery maps (Corollary 2). We argued that, in fact, recov-ery maps will generically be non-CP Hermitian maps, sincerecovery ancillas that are introduced into a quantum circuitprior to the recovery step will become non-classically corre-lated with the environment and consequently with the rest ofthe system.An interesting open question for future studies is whetherthe results presented here have an impact on the threshold forfault tolerant quantum error correction. For example, note thatwhile CP recovery perfectly returns the encoded state [Eqs.(29) and (B8)], non-CP recovery only does so up to a propor-tionality factor which depends on the details of the noise andrecovery maps [ F L in Eq. (32) and F H in Eq. (C2)]. This pro-portionality factor – assuming non-CP recovery is applied –may differ for different terms in the fault path decomposition[28], an effect which may propagate into the value of the faulttolerance threshold. This requires careful analysis, which isbeyond the scope of this paper. Acknowledgments
Funded by the National Science Foundation under GrantsNo. CCF-0726439, PHY-0802678, and PHY-0803304, andby the United States Department of Defense (to D.A.L.). Partof this work was done while D.A.L. enjoyed the generous hos-pitality of the Institute for Quantum Information at the Cali-fornia Institute of Technology.
APPENDIX A: PROOF OF THEOREM 1
We use a method similar to Choi’s proof for a CP maprepresentation [34], recently clearly reviewed in Ref. [35].The main difference between the proofs in Refs. [34, 35] andour proof is that in the previous proofs positivity allowed forthe use of standard diagonalization, whereas in the absence ofpositivity we use the singular value decomposition [36].
Proof.
Eq. (4) immediately implies that Φ L is a linear map.For the other direction, let f M = P ni,j =1 | i ih j | ⊗ | i ih j | = n | φ ih φ | , where | i i is a column vector with at position i and ’s elsewhere, and | φ i = n − / P i | i i ⊗ | i i is a maximallyentangled state over H ⊗ H , where H is the Hilbert spacespanned by {| i i} ni =1 . f M is also an n × n array of n × n ma-trices, whose ( i, j ) th block is | i ih j | . Construct two equivalent expressions for ( I ⊗ Φ L )[ f M ] , where I is the ( n × n ) × ( n × n ) identity matrix. (i) ( I ⊗ Φ L )[ f M ] is an n × n array of m × m matrices, whose ( i, j ) th block is Φ L [ | i ih j | ] . (ii) Considera singular value decomposition: ( I ⊗
Φ)[ f M ] = U DV = P α λ α U | α ih α | V = P α λ α | u α ih v α | . Here U and V are uni-tary, D = diag( { λ α } ) is diagonal and λ α ≥ are the singularvalues of ( I ⊗ Φ L )[ f M ] . Divide the column (row) vector | u α i ( h v α | ) into n segments each of length m and define an m × n ( n × m ) matrix E α ( E ′ α ) whose i th column (row) is the i thsegment; then E α | i i ( h i | E ′† α ) is the i th segment of | u α i ( h v α | ).Therefore the ( i, j ) th block of | u α ih v α | becomes E α | i ih j | E ′† α .Equating the two expressions in (i) and (ii) for the ( i, j ) th block of ( I ⊗ Φ L )[ f M ] , we find Φ L [ | i ih j | ] = P α λ α E α | i ih j | E ′† α . Since λ α ≥ we can redefine E α as √ λ α E α and E ′ α as √ λ α E ′ α , which we do from now on. Fi-nally, the linearity assumption on Φ L , together with the factthat the set {| i ih j |} ni,j =1 spans M n , implies Eq. (4).Next let us prove Eq. (5) for Hermitian maps. For an oldproof that uses very different techniques see Ref. [37]. Eq. (5)immediately implies that Φ H is a Hermitian map. For the otherdirection, associate a matrix L Φ H with the Hermitian map Φ H : ρ ′ = Φ H ( ρ ) ⇐⇒ ρ ′ mµ = L mµnν ρ nν (summation over repeatedindices is implied). Hermiticity of ρ and its image ρ ′ implies ρ ′ µm = ρ ′∗ mµ = L mµ ∗ nν ρ ∗ nν = L mµ ∗ nν ρ νn , i.e., L mµ ∗ nν = L µmνn [38]. We can use this property of L Φ H to show that if Φ H is aHermitian map, then I ⊗ Φ H is Hermiticity preserving. Con-sider M = M nνkξ | k ih ξ |⊗| n ih ν | . Then M ′ = ( I ⊗ Φ H )[ M ] = M nνkξ | k ih ξ | ⊗ Φ H ( | n ih ν | ) = M mµkξ | k ih ξ | ⊗ L mµnν | n ih ν | . As-sume that M mµ ∗ kξ = M µmξk . This property holds for M = f M = | φ ih φ | where | φ i = dim( H ) − / P i | i i ⊗ | i i is a maxi-mally entangled state over H ⊗ H ( M µmξk ≡ ). Then M ′† = M mµ ∗ kξ | ξ ih k | ⊗ L mµ ∗ nν | ν ih n | = M µmξk | ξ ih k | ⊗ L µmνn | ν ih n | = M ′ . Therefore ( I ⊗ Φ H )[ | φ ih φ | ] is Hermitian, and in par-ticular invertible. It follows that the SVD used in the proofof Theorem 1 can be replaced by standard diagonalization( U = V † ). In this case the left and right singular vectors | u α i = h v α | † are the eigenvectors of ( I ⊗ Φ H )[ | φ ih φ | ] and c α = λ α are its eigenvalues. Then E α = E ′ α in Eq. (4) and c α ∈ R .We note that by splitting the spectrum of ( I ⊗ Φ H )[ | φ ih φ | ] into positive and negative eigenvalues, { c + α ≥ } and { c − α ≤ } , we have as an immediate corollary a fact that was alsonoted in [10]: Any Hermitian map can be represented asthe difference of two CP maps: Φ( ρ ) = P α c + α E + α ρE + † α − P α | c − α | E − α ρE −† α . APPENDIX B: DIRECT PROOF OF COROLLARY 1
Proof.
The operation elements of ˜Φ CP are { F i = p | c i | K i } Ni =1 , whence ˜Φ CP ( ρ ) = P Ni =1 F i ρF † i . The standardquantum error conditions (19) for ˜Φ CP is a set of conditionsin terms of the F i : P F † i F j P = β ij P, i, j ∈ { , . . . , N } . (B1)0The existence of a projector P which satisfies Eq. (B1) isequivalent to the existence of a QEC code for ˜Φ CP . Assum-ing that a code C has been found (i.e., P C = C ) for ˜Φ CP , weuse this as a code for Φ H and show that the corresponding CPrecovery map R CP is also a recovery map for Φ H . Indeed, let G j ≡ P Ni =1 u ij F i be new operation elements for ˜Φ CP , i.e., ˜Φ CP = P Nj =1 G j ρG † j , where u is the unitary matrix that di-agonalizes the Hermitian matrix β = [ β ij ] , i.e., u † βu = d .Let R CP = { R k } be the CP recovery map for ˜Φ CP . Assumethat ρ is in the code space, i.e., P ρP = ρ . We now show that R CP [Φ H ( ρ )] = ρ , i.e., we have CP recovery. First, R CP [Φ H ( ρ )] = X k R k N X i =1 c i | c i | F i ρF † i ! R † k = N X i =1 c i | c i | N X j,j ′ =1 u ∗ ij u ij ′ × X k ( R k G j P ) ρ (cid:16) P G † j ′ R † k (cid:17) . (B2)Now, note that, using Eq. (B1): P G † k G l P = X ij u ∗ ik u jl P F † i F j P = X ij u ∗ ik β ij u jl P = d k δ kl P. (B3)Then the polar decomposition yields G k P = U k ( P G † k G k P ) / = √ d k U k P . The recovery operationelements are given by R k = U † k P k ; P k = U k P U † k . (B4)Therefore P k = G k P U † k / √ d k . This allows us to calculatethe action of the k th recovery operator on the l th error: R k G l P = U † k P † k G l P = U † k ( U k P G † k / p d k ) G l P = δ kl p d k P. (B5)Therefore, R CP [Φ H ( ρ )] = N X i =1 c i | c i | N X j,j ′ =1 u ∗ ij u ij ′ × X k (cid:16) δ kj p d k P (cid:17) ρ (cid:16) P p d k δ kj ′ (cid:17) = ρ N X i =1 c i | c i | (cid:0) udu † (cid:1) ii = ( N X i =1 c i | c i | β ii ) ρ. (B6) Next note that, using condition (B1) and trace preservation by Φ H : P F † i F i P = β ii P = ⇒ N X i =1 c i | c i | β ii P = P N X i =1 c i | c i | F † i F i P = P N X i =1 c i K † i K i P = P = ⇒ N X i =1 c i | c i | β ii = 1 . (B7)Hence, finally: R CP [Φ L ( ρ )] = ρ (B8)for any ρ in the codespace. APPENDIX C: PROOF OF COROLLARY 2
Proof.
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