Quantum error correction of observables

Quantum error correction of observables

Cédric Bény,¹Achim Kempf,¹and David W. Kribs^2,3

1Department of Applied Mathematics, University of Waterloo, Ontario, Canada, N2L 3G1

2Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada, N1G 2W1

3Institute for Quantum Computing, University of Waterloo, Ontario Canada, N2L 3G1 共Received 10 May 2007; published 2 October 2007兲

A formalism for quantum error correction based on operator algebras was introduced by us earlier关Phys.

Rev. Lett. 98, 10052共2007兲兴via consideration of the Heisenberg picture for quantum dynamics. The resulting theory allows for the correction of hybrid quantum-classical information and does not require an encoded state to be entirely in one of the corresponding subspaces or subsystems. Here, we provide detailed proofs for our earlier results, derive more results, and elucidate key points with expanded discussions. We also present several examples and indicate how the theory can be extended to operator spaces and general positive operator-valued measures.

DOI:10.1103/PhysRevA.76.042303 PACS number共s兲: 03.67.Pp, 03.67.Lx, 03.67.Hk

I. INTRODUCTION

A new framework for quantum error correction was de- rived in 关1兴 through a Heisenberg picture reformulation of the Schrödinger approach to error correction, and an expan- sion of the notion of a quantum code to allow for codes determined by algebras generated by observables. As the ap- proach generalizes standard quantum error correction共QEC兲 关2–6兴and operator quantum error correction共OQEC兲 关7,8兴, we called the resulting theory “operator algebra quantum er- ror correction”共OAQEC兲. An important feature of OAQEC is that it provides a formalism for the correction of hybrid quantum-classical information关9,10兴.

In this paper we provide proofs for the results stated in 关1兴, and we establish a number of new results. In addition, we expose some of the finer points of the theory with dis- cussions and several examples. We also outline how the theory can be extended to the case of operator spaces gener- ated by observables and general positive operator-valued measures共POVMs兲.

We continue below by establishing notation and describ- ing requisite preliminary notions. In the next section we present a detailed analysis of passive quantum error correc- tion within the OAQEC framework. The subsequent section does the same for active quantum error correction. This is followed by an expanded discussion of the application to information flow from关1兴, and we conclude with a section on the operator space and POVM extension.

Preliminaries

Given a 共finite-dimensional兲 Hilbert space H, we let L共H兲 be the set of operators onHand letL1共H兲be the set of density operators onH. We shall write␳^,␴^,␶for density operators andX, Y, etc., for general operators. The identity operator will be written as1.

Noise models in quantum computing are described共in the Schrödinger picture兲by completely positive trace-preserving 共CPTP兲 mapsE: L共H兲→L共H兲 关11兴. We shall use the term quantum channelto describe such maps. Every mapEhas an operator-sum representation E共␳兲=兺aE_a␳^Ea

†, where the op- eratorsE_aare called the operation elementsornoise opera-

tors for E. The Hilbert-Schmidt dual map E^† describes the corresponding evolution of observables in the Heisenberg picture. A set of operation elements forE^†is given by 兵E_a^†其. Trace preservation ofEis equivalent to the requirement that E^†isunital; that is, E^†共1兲=1.

A quantum system A 共or B兲 is a subsystem of H if H decomposes asH=共A丢B兲丣K. Subspaces of Hcan clearly be identified as subsystems A with one-dimensional ancilla 共dimB= 1兲. An algebraA of operators on H that is closed under Hermitian conjugation is called a共finite-dimensional兲 C^* algebra, what we will simply refer to as an “algebra.”

Algebras of observables play a key role in quantum mechan- ics关12兴and recently it has been shown that they can be used to encode hybrid quantum-classical information 关9,10兴. Be- low we shall discuss further the physical motivation for con- sidering algebras in the present setting. Mathematically, finite-dimensionalC^* algebras have a tight structure theory that derives from their associated representation theory关13兴.

In particular, there is a decomposition ofHinto subsystems H=丣k共Ak丢B_k兲丣K such that with respect to this decompo- sition the algebra is given by

A=兵丣

k关L共Ak兲丢1B_k兴其丣0_K. 共1兲 The algebrasL共Ak兲丢1B_kare referred to as the “simple” sec- tors ofA. We shall write Mn for the set of n⫻n complex matrices, and identify Mn with the matrix representations for elements of L共A兲 when dimA=n and an orthonormal basis forAis fixed.

II. PASSIVE ERROR CORRECTION OF ALGEBRAS As the terminology suggests, the existence of a passive code for a given noise model implies that no active operation is required共beyond decoding兲to recover quantum informa- tion encoded therein. Mathematically, it is quite rare for a generic channel to have passive codes. However, many of the naturally arising physical noise models include symmetries that do allow for such codes关14–25兴.

The following is the standard definition of a noiseless subsystem 共and decoherence-free subspace when dimB= 1兲 1050-2947/2007/76共4兲/042303共9兲 042303-1 ©2007 The American Physical Society

in the Schrödinger picture. Suppose we have a decomposi- tion of the Hilbert space H as H=共A丢B兲丣K. As a nota- tional convenience, we shall write␳丢␶for the operator on Hdefined by共␳丢␶兲丣0_K.

Definition 1. We say thatAis a noiseless共or decoherence- free兲subsystem forEif for all␳苸L1共A兲and␴苸L1共B兲there exists␶苸L1共B兲such that

E共␳丢␴兲=␳丢␶^. 共2兲 The “decoherence-free” terminology is usually reserved for subspaces共when dimB= 1兲.

A. Decoherence-free and noiseless subspaces and subsystems in the Heisenberg picture

The following theorem gives an equivalent formulation of this definition in the Heisenberg picture, that is in terms of the evolution of observables, given by the dual channelE^†. We introduce the projectorPofHonto the subspaceA_丢B.

Theorem 2.Ais a noiseless subsystem forEif and only if PE^†共X丢1兲P=X丢1 共3兲 for all operatorsX苸L共A兲.

Proof. IfA is a noiseless subsystem forE, then for allX 苸L共A兲we have

Tr关PE^†共X丢1兲P共␳丢␴兲兴

= Tr关E^†共X丢1兲共␳丢␴兲兴

= Tr关共X丢1兲E共␳丢␴兲兴

= Tr共X␳丢␶兲

= Tr共X␳兲Tr共␶兲

= Tr共X␳兲

= Tr关共X丢1兲共␳丢␴兲兴.

This is true for all␳苸L1共A兲and all␴苸L1共B兲. By linearity it follows thatPE^†共X丢1兲P=X丢1 for allX苸L共A兲.

Reciprocally, if we assume Eq. 共3兲 to be true for all X 苸L共A兲, then for all␥苸L1共A丢B兲 we have

Tr关XTrB„PE共␥兲P…兴= Tr关共X丢1兲PE共␥兲P兴

= Tr关PE^†共X丢1兲P␥兴

= Tr关共X丢1兲␥兴

= Tr„XTrB共␥兲…,

where we have freely used the facts P共X丢1兲P=X丢1 and P␥^P=␥. Since the above equation is true for allX, we have TrB(PE共␥兲P)= TrB共␥兲 for all ␥苸L1共A丢B兲, which was shown in关8兴to be equivalent to the definition of A being a

noiseless subsystem forE. 䊏

Note that Eq.共3兲 can be satisfied even if part of an ob- servableX丢1 spills outside of the subspace PH under the action ofE^†. The projectorsPin Eq.共3兲show that the noise- less subsystem condition in the Heisenberg picture is only concerned with the “matrix corner” ofE^†共X_丢1兲 partitioned byP.

In some cases the equivalence of Eqs.共2兲and共3兲can be seen from a different perspective. For a bistochastic or unital channel关those for whichE共1兲=1兴, the dualE^†is also a chan- nel. Then structural results for unital channels from关27兴can be used to give an alternate realization of this equivalence, and passive codes may be computed directly from the com- mutant of the operation elements for E. The simplest case would be for “self-dual” channels, those for which E^†=E. Clearly, any E with Hermitian operation elements is self- dual. In particular, self-dual channels include all Pauli noise models, which are channels with operation elements belong- ing to the Pauli group, the group generated by tensor prod- ucts of unitary Pauli operatorsX,Y,Z.

As a further illustrative共nonunital兲example, consider the single-qubit spontaneous emission channel 关11兴 given by E共␴兲= Tr共␴兲兩0典具0兩. Here P=兩0典具0兩. This channel is imple- mented by operation elementsE₀=P and E₁=兩0典具1兩. Hence the dual channel is given byE^†共␴兲=P␴^P+E1

†␴^E1. The sub- spaceAspanned by the ground state兩0典is a decoherence-free subspace forE共though it cannot be used to encode quantum information since dimA= 1兲. In this case, Eq.共3兲is equiva- lent to the statement PE^†共P兲P=P, which may be readily verified.

One can consider more general spontaneous emission channels with nontrivial decoherence-free subspaces. For in- stance, consider a qutrit noise model that describes sponta- neous emission from the second excited state to the ground state. The corresponding channel is defined by E共␴兲=P␴^P +E␴^{E†, whereP=}兩0典具0兩+兩1典具1兩andE=兩0典具2兩. The subspace PC³= span兵兩0典,兩1典其 is a single-qubit decoherence-free sub- space forEsinceE共␴兲=␴^{for all}␴^=P␴P. It is also easy to see in this case that Eq.共3兲is satisfied sincePE^†共X兲P=Xfor allX苸L共PC³兲. Notice also in this example thatE^† can in- duce “spillage” fromP toP^⬜. Indeed, this can be seen im- mediately from the operator-sum representation forE^†; spe- cifically, for allX苸L共PC³兲we have

E^†共X兲=PXP+E^†XE=X+␣兩2典具2兩, 共4兲 where␣⁼具0兩X兩0典.

B. Conserved algebras of observables

A POVM determined by a set of operators X=兵X_a其 evolves via the unital CP mapE^†in the Heisenberg picture. If for allawe haveXa=E^†共Xa兲, then all the statistical informa- tion aboutX=兵Xa其has been conserved. Indeed, for any initial state ␳, we have Tr共␳^Xa兲= Tr(␳E^†共Xa兲)= Tr(E共␳兲Xa). More- over, if we have control on the initial states, an expected feature in quantum computing, we can ask which elements are conserved if the state starts in a certain subspace PH;

that is, which elements satisfy PE^†共Xa兲P=PXaP, or equiva- lently, Tr(X_aE共P␳^P兲)= Tr共X_aP␳^P兲 for all ␳苸L1共H兲. This, together with the Heisenberg characterization of Eq.共3兲, mo- tivates the following definition.

Definition 3. We shall say that a setSof operators onHis conserved by E for states in PH if every element of S is conserved; that is, if

PE^†共X兲P=PXP ∀X苸S. 共5兲

The focus of the present work is error correction for al- gebras generated by observables. Let us consider in more detail the physical motivation for considering algebras. In the Heisenberg picture a set of operators兵Xa其evolves according to the unital CP mapE^†with elementsE_a^†instead ofEa. If for all values of the label a we have X_a=E^†共X_a兲 then all the statistical information about X has been conserved by E as noted above. In such a scenario we say thatXais conserved by E. In particular, if X defines a standard projective mea- surement,X=兺ap_aX_a withX_a²=X_a for alla, then the projec- torsXalinearly span the algebra they generate. Hence, in this caseE conserves an entire commutative algebra. Therefore, focusing on the conservation共and more generally, correction defined later兲of sets of operators that have the structure of an algebra, apart from allowing a complete characterization, is also sufficient for the study of all the correctableprojective observables.

Further observe that Eq. 共5兲 applied to a set of observ- ables that generate an共arbitrary兲algebra gives a generaliza- tion of noiseless subsystems. Indeed, by Theorem 2 any sub- algebra A of L共PH兲 for which all elements X苸A satisfy Eq. 共5兲 is a direct sum of simple algebras, each of which encodes a noiseless subsystem 关when dimAk⬎1 as in Eq.

共1兲兴. It is also important to note that given an algebra A conserved for states inPH共or more generally, correctable as we shall see兲 quantum information cannot, in general, be encoded into the entire subspacePHfor safe recovery, but rather into subsystems of PH determined by the splitting induced from the algebra structure ofA. These points will be further expanded upon in the discussion of Sec. III B.

We now establish concrete testable conditions for passive error correction. Namely, these conditions are stated strictly in terms of the operation elements for a channel. This result was stated without proof in关1兴. The special case of simple algebras in the Schrödinger picture was obtained in 关7,8兴. Techniques of关24兴are used in the analysis. We first present a simple lemma that will be used below.

Lemma 4. Let F be a CP map with elements Fa. If A 艌0 is such that F共A兲= 0 then it follows that AF_a= 0 for every elementF_a.

Proof. If 兺aF_a^†AFa= 0 then for any state 兩␺典, 兺a具␺兩F_a^†AF_a兩␺典= 0. Since each operator F_a^†AF_a is positive, this is a sum of non-negative terms. Therefore each indi- vidual term must equal zero;具␺兩F_a^†AFa兩␺典= 0 for alla. This means that the vector

冑

AFa兩␺典is of norm zero, and therefore is the zero vector. This being true for all states兩␺典, we must have that

冑

AFa= 0, from which it follows thatAFa= 0. 䊏 Theorem 5. LetAbe a subalgebra ofL共PH兲. The follow- ing statements are equivalent:

共1兲Ais conserved byEfor states in PH.

共2兲 关E_aP,X兴= 0 for allX苸Aand alla.

Proof. If关E_aP,X兴=关E_aP,PXP兴= 0 for alla, then PE^†共X兲P=

兺

_a ^PEa†XEaP

=

兺

_a ^PEa†PXPEaP

=

兺

_a ^{PEa†EaPXP=PE†共1兲PXP=PXP=X.}

Reciprocally, we assume that each X苸A satisfies PE^†共X兲P=PXP=X. Consider a projector Q苸A. We have PE^†共Q兲P=Qand hence PE^†共Q^⬜兲P=PQ^⬜. Therefore

Q^⬜PE^†共Q兲PQ^⬜=Q^⬜QQ^⬜= 0,

and similarlyQPE^†共Q^⬜兲PQ= 0. By Lemma 4, this implies, respectively, QEaPQ^⬜= 0 and Q^⬜EaPQ= 0 for all a. To- gether these conditions imply

QEaP=QEaPQ=EaPQ,

and thus关E_aP,Q兴= 0 for alla. Finally, note that sinceAis an algebra, then a generic element Y苸A can be written as a linear combination of projectors in A. Therefore we also have 关EaP,Y兴= 0 for all Y苸A, and this completes the

proof. 䊏

This theorem allows us to identify the largest conserved subalgebra of L共PH兲 conserved on the subspace PH.

Namely, a direct consequence of Theorem 5 is that the共nec- essarily †-closed兲commutant insideL共PH兲of the operators 兵EaP,PE_a^†其is the largest such algebra.

Corollary 6. The algebra A=兵X苸L共PH兲:∀ a关X,EaP兴

=关X^†,EaP兴= 0其is conserved on states inPHand contains all subalgebras ofL共PH兲 conserved on statesPH.

Note thatPitself may not belong to this algebra, unless it satisfiesE_aP=PE_aP. This special case was the case consid- ered in关24兴. With other motivations in mind, the special case P=1 was also derived in 关26兴 where it was shown that the full commutant of 兵E_a,E_a^†其 is the largest algebra inside the fixed point set of a unital CP map. 共This in turn may be regarded as a weaker form of the fixed point theorem for unital channels关27兴.兲

Likely the most prevalent class of decoherence-free sub- spaces are the stabilizer subspaces for abelian Pauli groups, which give the starting point for the stabilizer formalism关6兴. On then-qubit Hilbert space, lets⬍nand consider the group S generated by the Pauli phase flip operators Z₁, . . . ,Zs, where we have written Z₁=Z丢1丢¯丢1, etc. The joint eigenvalue-1 space forS is a 2^n−s-dimensional decoherence- free subspace for 兵Z_j: 1艋j艋s其, called the “stabilizer sub- space” forS. In the stabilizer formalism, any of the 2^s共nec- essarily 2^n−s-dimensional兲 mutually orthogonal joint eigenspaces for elements ofS could be used individually to build codes, but the eigenvalue-1 space is used as a conve- nience. Each of these eigenspaces supports a full matrix al- gebra Mk, where k= 2^n−s. In the present setting these decoherence-free subspaces may be considered together, as they are defined by the structure of the commutant S

⬘

⬵M_k^共2s兲. In particular, this entire algebra is conserved by any channel defined with operation elements given by linear combinations of elements taken fromS.

Allowing other Pauli operators into the error group can result in nonconserved scenarios. For instance, consider the casen= 3 and s= 2. The algebra S

⬘

^⬵M2

共4兲 is conserved by channels determined by elements ofS as above. However, it isnot conserved by channels determined by elements of G

=兵Z1,Z₂,X₁X₂其, even though the individual eigenvalue-1 sta- bilizer space is still a correctable code for G. This follows from Theorem 5 sinceGproperly containsS, and henceG

⬘

is properly contained inS

⬘

^.共In this caseP=1, and the largest conserved algebra is the full commutant G

⬘

of the error group.兲 Interestingly, there are still algebras conserved by channels determined by elements ofG, since its commutant has the structureG

⬘

^{= Alg兵G其}

⬘

^⬵共I2丢M2兲丣共I2丢M2兲.

Let us reconsider the special case in which the projectorP itself is one of the correctable observables. Most importantly, this guarantees that after evolution all states are back into the codePH. Indeed, the probability that a state␳initially in the code is still in the code after evolution is then given by

p= Tr„E共␳兲P…= Tr„␳PE^†共P兲P…= Tr共␳P兲= 1. 共6兲 We note that this also guarantees a repetition of the noise map will be conserved.

A refinement of the proof of Theorem 5 yields the follow- ing result when the projectorP is conserved and belongs to the algebra of observables in question.

Theorem 7. LetAbe an algebra containing the projector P. The following statements are equivalent:

共1兲Ais conserved byEfor states in PH.

共2兲 关EaP,X兴= 0 for allX苸PAP and alla.

Proof. If 关EaP,X兴=关EaP,PXP兴= 0 for all a, then PE^†共X兲P=PXPfollows as above. On the other hand, if each X苸A satisfies PE^†共X兲P=PXP, then note that forX=P this yieldsPE^†共P兲P=P, which implies PE^†共P^⬜兲P= 0. Therefore, by Lemma 4,P^⬜E_aP= 0; that is,E_aP=PE_aPfor alla. Hence for allX苸A we also have PE^†共PXP兲P=PXP. But the set PAPis itself an algebra, sincePbelongs toA, and hence is spanned by its projectors. The rest of the proof proceeds as

above. 䊏

III. ACTIVE ERROR CORRECTION OF ALGEBRAS More generally, active intervention into a quantum system may be required for error correction. In particular, we should be able to protect a set of operators兵Xa其 共which could define a POVM for instance兲by acting with a channelRsuch that each Xa is mapped by R^† to one of the operators Ya

=E^†共Xa兲. That is, R^†共Xa兲=Ya, and thus 共RⴰE兲^†共Xa兲=Xa. This, together with the previous discussions, motivates the following definition.

Definition 8. We say that a set S of operators on H is correctableforEon states in the subspacePHif there exists a channelR such thatS is conserved by RⴰE on states in PH; in other words,

P共RⴰE兲^†共X兲P=PXP ∀ X苸S. 共7兲 This equation is the same as the one we used to define passive error correction, except that we now allow for an arbitrary “correction operation” R after the channel has acted. In particular, as in other settings, the passive case may be regarded as the special case of active error correction for which the correction operation is trivial,R=id.

This notion of correctability is more general than the one addressed by the framework of OQEC, and extends the one introduced in关1兴from algebras to arbitrary sets of observ- ables. Here we shall continue to focus on algebras, and in Sec. V we consider an extension of the notion to general

operator spaces. Whereas OQEC focuses on simple algebras L共A兲丢1^B, here correctability is defined for any set, and in particular, for any finite-dimensional algebra. See Sec. III B for an expanded discussion on the form of OAQEC codes.

A. Testable conditions for OAQEC codes

A set of operation elements for a given channel are the fundamental building blocks for the associated physical noise model. Thus, a characterization of a correctable OAQEC code strictly in terms of the operation elements for a given channel is of immediate interest. The following result was stated without proof in 关1兴. It generalizes the central result for both QEC关3兴and OQEC 关7,8,29兴.

Theorem 9. LetAbe a subalgebra ofL共PH兲. The follow- ing statements are equivalent:

共1兲Ais correctable forEon states in PH.

共2兲 关PE_a^†E_bP,X兴= 0 for allX苸Aand alla,b.

Proof. We write Ra for the elements of R. According to Theorem 5, the conservation of A by RⴰE implies RaEbX

=XRaEbP for all X苸A and all a,b. But we also have R_aE_bX^†=X^†R_aE_bP, so that XE_b^†R_a^†=PE_b^†R_a^†X. Therefore PE_c^†EbX=兺aPE_c^†R_a^†RaEbX=兺aPE_c^†R_a^†XRaEbP=XE_c^†EbP.

We will prove the sufficiency of this condition by explic- itly constructing a correction channel. Fork艌1, letP_kbe the projector onto thekth simple sector of the algebraA. Also let P₀=P−1_A, where 1_A is the unit element of the algebra A.

We have PkE_b^†EcPk=1丢Abc for some operators Abc and all k艌0. Hence the theory of operator error correction guaran- tees that each subspace can be individually corrected. Here, however, we have the additional property PkE_b^†EcPk⬘= 0 whenever k⫽k

⬘

, which allows the correction of the state even if it is in a superposition between several of the sub- spaces Pk. Explicitly, we have共1丢具l兩兲Pk共Eb†Ec兲Pk⬘共1丢兩l

⬘

^典兲

=␦kk⬘␭bc

kll⬘1 for some ␭bc

kll⬘苸C, where we denote 1=1m_k. According to the standard theory of error correction this condition guarantees the existence of channelsRkcorrecting the error operators Fckl=EcPk共1丢兩l典兲 for all l and all c, or any linear combination of them. In particular, we will con- sider linear combinations of the form F˜

ck=兺n具n兩␺典Fckn

=EcPk共1丢兩␺典兲 for any normalized vector 兩␺典. Furthermore, from the standard theory we know the elements of the cor- rection channels Rk can be assumed to have the form R_cl^共k兲

=兺bj␣clbj

共k兲 共1丢具j兩兲PkE_b^†for some complex numbers␣clbj 共k兲 . We now show that the trace-decreasing channelR with elementsRkcl=Pk共1丢兩l典兲R_cl^共k兲corrects the algebraAon states PH for the channel E. First note that R_cl^共k兲EaP=兺bj␣clbj

共k兲 共1

丢具j兩兲P_kE_b^†E_aP_k=R_cl^共k兲E_aP_k. Hence, for a general operator X

=兺kA_k丢1 in the algebra we have P共E^†ⴰR^†兲共X兲P

=兺akclPkE_c^†R_al^共k兲†AkR_al^共k兲EcPk. Considering each term k sepa- rately, for any state兩␺典we have

兺

acl

共1丢具␺兩兲PkE_c^†R_al^共k兲†AkR_al^共k兲EcPk共1丢兩␺典兲

=Ak=共1丢具␺兩兲共Ak丢1兲共1丢兩␺典兲,

where we have used the dual of the fact thatRkcorrects the error operators F˜

ck. Therefore 兺aclP_kE_c^†R_al^共k兲†A_kR_al^共k兲E_cP_k

=共Ak丢1兲 and summing those terms over k yields P共E^†

ⴰR^†兲共X兲P=兺k共Ak丢1兲=X. 䊏

As an immediate consequence of Theorem 9 we have the following.

Corollary 10. The algebra A=兵X 苸L共PH兲:∀a,b关X,PE_a^†EbP兴= 0其 is correctable on states in PH and contains all subalgebras of L共PH兲 correctable on states inPH.

To further explain the structure of the correction channel in Theorem 9 let us show how it is constructed from OQEC correction channels. This will also give an alternative proof of the sufficiency of the correctability condition.

For simplicity, we will in fact build a channel which cor- rects the larger algebra BªA丣C共1_A−P兲. Remember that P_kis the projector onto the kth simple sector of the algebra A, assuming a decomposition as in Eq.共1兲. Also we include P₀=1_A−P which projects onto the additional sector in B.

Our correctability condition guarantees that each of those sectors is an OQEC code. Let Rk be a OQEC correction channel for thekth simple sector. We use the “raw” subunital version of the subsystem correction channels whose elements are all linear combinations of the operatorsPkE_a^†. They have the property that Q_kªRk

†共1兲=Rk

†共P_k兲 is a projector. Since the elements of the channelRkare linear combinations of the operatorsPkE_a^†, we haveQkQl=Rk†共Pk兲Rl†共Pl兲= 0 ifk⫽l, be- cause all the terms contain a factor of the form P_kE_a^†E_bP_l

= 0. This means the the projectorsQkare mutually orthogo- nal and sum to a projectorQª兺kQk. The channels Rkalso have the property that Rk共␳兲=Rk共Qk␳^Qk兲 for any state ␳^. From these “local” channels we can construct a trace- preserving correction channel for the full algebra as follows:

R^†共X兲ª

兺

_k ^{Rk†共X兲+Tr共PX兲}_Tr共P兲 ^{Q⬜. 共8兲}

This CP map is a channel becauseR^†共1兲=兺kQk+Q^⬜=1. We have to check that it corrects the algebraB. First, concerning the effect ofE^† on the last term of the correction channel, note that

PE^†共Q兲P=

兺

_k ^{PE†„Rk†共Pk兲…P=}

兺

_k ^{PkE†„Rk†共Pk兲…Pk}

=

兺

_k ^Pk=P.

HencePE^†共Q^⬜兲P= 0. It follows that for anyX苸Bwe have, keeping in mind that the channel elements of Rkare linear combinations of the operatorsPkE_a^†,

PE^†„R^†共X兲…P=

兺

_kl ^{PlE†„Rk†共X兲…Pl}

=

兺

k

PkE^†„Rk

†共X兲…Pk=

兺

k

PkXPk=X, which is the desired property for the correction channelR.

As in the passive case, we can consider the situation in which the projector is correctable and belongs to the algebra.

In this case, the observables and states do not spill out from

PHunder the action ofRⴰE, and so the channel followed by the correction operation is repeatable. The previous proof can be readily refined for this purpose.

Theorem 11. LetAbe an algebra containing the projector P. The following statements are equivalent:

共1兲Ais correctable forEon states in PH.

共2兲 关PEa

†EbP,X兴= 0 for allX苸PAP and alla,b.

Proof. Since P苸A then PAP is a subalgebra of A.

Therefore correctability ofAimplies correctability ofPAP which from Theorem 9 implies that关PE_a^†E_bP,X兴= 0 for all X苸PAP. Reciprocally, if this condition is satisfied then by Theorem 9 there exists a channel R correcting the algebra PAP. In fact this channel corrects all ofA. Indeed, remem- ber that the channel R built in the proof is such that R^†共X兲=R^†共PXP兲 for all X. Therefore for all X苸A, P共R ⴰE兲^†共X兲P=P共RⴰE兲^†共PXP兲P=PXP 䊏. In practice, the operation elements for a channel are usu- ally not known precisely; often it is just the linear space they span that is known关28兴. Thus, for the explicit construction of correction operations in Theorems 9 and 11 to be of practical value, one has to show that the correction channel R also corrects any channel whose elements are linear combinations of the elementsEa. This is indeed the case. A simple way to see this is to note that if the testable conditions for conserved algebras of Theorems 5 and 7 are satisfied for RⴰE, then they are also satisfied forRⴰE

⬘

where the operation elements ofE

⬘

are linear combinations of those forE.

B. Schrödinger picture

In order to illustrate how OAQEC generalizes OQEC we restate a special case of the above results in the Schrödinger picture: Suppose we have a decomposition

H=

关

^{丣k共Ak丢Bk兲}

兴

^{丣K, 共9兲}

withPthe projector ofHontoK^⬜=丣kAk丢Bk. The algebra in question includes P as its unit and is given by A

=兵丣k关L共Ak兲丢1B_k兴其丣0_K. Observe that the hypotheses of both results Theorems 9 and 11 are satisfied. It follows that A is correctable forE for states in PHif and only if there exists a channel R such that for any density operator ␳

=兺k␣k共␳k丢␶k兲 with ␳k苸L1共A_k兲, ␶k苸L1共B_k兲, and non- negative scalars兺k␣k= 1, there are operators␶k

⬘

苸L1共Bk兲for which

共RⴰE兲共␳兲=

兺

_k ^{␣kR关E共␳k丢␶k兲兴=}

兺

_k ^{␣k共␳k丢␶k}

⬘

^兲.

共10兲 Experimentally, each of the subsystemsA_kcan be used indi- vidually to encode quantum information. An extra feature of this OAQEC code is the fact that an arbitrary mixture of encoded states, one for each subsystem, can be simulta- neously corrected by the same correction operation.

By Theorem 9共or Theorem 11兲, there is a correction op- erationRfor which Eq.共10兲is satisfied if and only if for all a,b there are operatorsXabk苸L共Bk兲such that

PE_a^†E_bP=

兺

_k ^{1Ak丢Xabk. 共11兲}

Note that contrary to the Heisenberg formulation of Eq.

共7兲, the formulation of Eq. 共10兲implicitly relies on the rep- resentation theory for finite-dimensionalC^*algebras. As the representation theory for arbitraryC^*algebras is intractable, this suggests the Heisenberg picture may be more appropri- ate for an infinite-dimensional generalization of this frame- work.

Let us consider a qubit-based class of examples to illus- trate the equivalence established in Theorem 9. A specific case was discussed in关1兴. Suppose we have a hybrid quan- tum code whereind qubit codes 兩␺j典, 1艋j艋d, are each la- beled by a classical “address”兩j典, 1艋j艋d. In this case P

=兺^d_j=11₂丢兩j典具j兩=1₂丢1d and the algebra is A=丣_j=1^d L共C²兲

丢兩j典具j兩. A generic density operator for this code is of the form ␳⁼兺_j=1^d ␣j␳j丢兩j典具j兩, where ␳j=兩␺j典具␺j兩, ␣j艌0, and 兺_j=1^d ␣j= 1. This hybrid code determined byA andP is cor- rectable forEif and only if for alla,b,jthere are scalars␭abj

such that

PE_a^†E_bP=

兺

j=1 d

␭abj共1₂丢兩j典具j兩兲.

As the ancilla for each individual qubit兩␺j典 is one dimen- sional, in this case the correction operation will correct the code precisely,共RⴰE兲共␳兲=␳^.

In the Schrödinger picture, the correction channel built in the proofs of Theorems 9 and 11 is equal to

R共␳兲=

兺

_k ^{Rk共Qk␳Qk兲+Tr共Q}_Tr^⬜_P^␳兲P.

In words, one first measures the observable defined by the complete set of orthogonal projectorsQ_kandQ^⬜. If the state is found to be in one of the subspaces Q_k then the OQEC correction channelRkis applied to correct the corresponding subsystem of the algebra. Otherwise, if the state happens to be in the subspaceQ^⬜, this means that the initial state of the system was not in the code. Therefore what we do in this case does not matter. In the channel R defined above, we chose for simplicity to set the state toP/ TrP.

IV. APPLICATION TO INFORMATION FLOW Consider the interaction between a “system” S and an

“apparatus” A where the initial state of the apparatus is known. For any state 兩␺S典苸HS, we define V兩␺S典⬟U共兩␺S典

丢兩␺A典兲for a unitaryUacting onHS丢HAand a fixed initial state 兩␺A典苸HA. The operatorV is an isometry between the spaceHSand the spaceHS丢HA. Tracing over the final state of the apparatus gives us a channel fromB共HS兲 toB共HS兲: ESS共␳兲= TrA共V␳^V†兲whose dual is

ESS† 共X兲=V^†共X丢1兲V.

We can also trace out the final state of the system to get a channel from L共HS兲 toL共HA兲: ESA共␳兲= TrS共V␳^V†兲 where␳ 苸B共HS兲 共see Fig.1兲. The channelESAis uniquely defined by ESS, up to an arbitrary unitary operation on the apparatus, and is usually called thecomplementary channelofESS. Its dual is simply

ESA† 共Y兲=V^†共1丢Y兲V.

Using Theorem 9, we can determine which observables have been preserved by either ESS or ESA, irrespectively of the system’s initial state. The answers are given by two sub- algebras of L共HS兲: respectively, ASS andASA. The algebra ASS characterizes the information about the system’s initial state, which has been preserved by the system’s evolution, and ASA characterizes the information about the system’s initial state, which has been transferred to the environment.

Those algebras can be expressed in terms of the elements of one of the channels. For instance, if Ea are elements for ESS, then V can be expressed as V=兺bEb丢兩␾b

A典 for some orthonormal set of vectors兩␾b

A典ofHA. Hence for any choice of a basis兩a典 of HSwe obtain a family of elements for the channelESA, namely,

Fa=

兺

_b ^{兩␾bA典具a兩Eb.}

This means that the relevant operators entering Theorem 9 for the second channel are

F_a^†Fb=

兺

_c ^{Ec†兩a典具b兩Ec=ESS† 共兩a典具b兩兲.}

Note that the operators 兩a典具b兩 form a basis for the whole operator algebra L共HS兲. Hence the observables correctable for the apparatus form the algebra ASA= Alg共RanESS† 兲

⬘

^{: the} algebra of operators commuting with all operators in the range of ESS

† . Hence we see that a direct consequence of Theorem 9 is that in an open dynamics defined by a channel E, full information about a projective observable can escape the system if and only if it commutes with the range of the dual mapE^†, which is the set of observables with first mo- ment information conserved byE. This generalizes results in 关26兴.

We can characterize the observables representing infor- mation which has been “duplicated” between the system and the apparatus. They form the intersection

CªASS艚ASA.

From the correctability of ASS 关Eq. 共7兲兴 we have that ASS債RanESS

† , from which it follows that ASA

= Alg共RanESS† 兲

⬘

^債ASS

⬘

. Hence, the algebra of duplicated ob- FIG. 1. Interaction between a systemSand an apparatusA of known initial state. Tracing over one of the two final systems gives us one of two channelsESSorESA.

servables isC債ASS

⬘

艚ASS, whereASS

⬘

艚ASSis the center of ASS: those elements of the algebra which commute with all other elements. In particular, the duplicated algebraCiscom- mutative. Note that the contrary would have violated the no- cloning theorem after correction of both channels. Since the algebraCis commutative, it is generated by a single projec- tive observable which can be represented by a self-adjoint operatorC.

Given that, after the interaction, both the system and the apparatus contain information about the same observable C on the initial state of the system, we may expect that they are correlated. LetPi苸Cbe the projectors on the eigenspaces of C. There exists a POVM with elementsXion the system as well as a POVMYion the apparatus such that

ESS

†共Xi兲=ESA

† 共Yi兲=Pi.

Note that ifRSSandRSAare correction channels forESSand, respectively,ESA, thenXi=RSS

† 共Pi兲 andYi=RSA

† 共Pi兲.

We will show that the observables Xi and Yi are corre- lated. First note that Tr共PiPk兲=␦ki, which we can also write as

Tr共PiPk兲= Tr„PiESS

†共Xk兲…= Tr关PiV^†共Xk丢1兲V兴=␦ki.

This means that when k⫽i, 共Xk丢1兲VPi= 0, which can be seen by expending Pi in terms of eigenvectors. Also since 兺kX_k=1, then VP_i=共Xi丢1兲VPi. The same argument is true also for Y_k. Therefore 共X_k丢1兲VP_i=共1丢Y_k兲VP_i=␦ikVP_i. Hence

V^†共X_i丢Y_j兲V=

兺

_kl ^{PkV†共Xi丢Yj兲VPl=}

兺

_kl ^{␦ik␦jlPiPj=␦ijPk,}

which means that for any state␳ of the system Tr关V␳^V†共Xi丢Yj兲兴=␦ijTr共␳^Pi兲.

Hence the probability that the outcome of a measurement of X differs from that ofY is zero. This means that the infor- mation that the apparatus “learns” about the system and which is characterized by the observableCis correlated with a property of the system after the interaction. ThereforeC represents the only information that the apparatus acquires about the system and which stays pertinent through the in- teraction.

This analysis has implications for the theory of decoher- ence关30,31兴as well as for the theory of measurements. We have shown that any interaction between a system and its environment共which took the role of the apparatus兲automati- cally selects a unique observableCas being the onlypredic- tiveinformation about the system acquired by the environ- ment. Even though an observer who has access to the environment could learn about any observable contained in the algebraASA, only the information encoded by C bears any information about the future state of the system. This suggests that the pointer states, which characterize decoher- ence, should not be selected only for their stability under the interaction with the environment: One should also add the requirement that they encode information that the environ- ment learns about the system. Indeed, any one of those re- quirements taken separately does not select a single observ-

able unambiguously, but together they do. This is a new way of solving thebasis ambiguityproblem关32兴.

V. ERROR CORRECTION OF OPERATOR SPACES In this section we discuss an extension of OAQEC theory to the setting of operator spaces generated by observables.

We shall leave a deeper analysis of this extension for inves- tigation elsewhere. Anoperator space 关33兴is a linear mani- fold共a subspace兲of operators insideL共H兲. Operator spaces, and their Hermitian closed counterpart “operator systems,”

have arisen recently in the study of channel capacity prob- lems in quantum information关34兴. Observe that共by design兲 Definitions 3 and 8 include the case of operator spaces and systems generated by observables, and hence these cases fit into the mathematical framework for error correction intro- duced here. Let us describe how operator spaces physically arise in the present setting.

In Sec. III A we showed how to build the correction chan- nel for active error correction. We were free to choose what to do to the system in the case that the syndrome measure- ment revealed the state had not initially been in the code prior to the action of the error channel. In fact, there is an advantage in choosing to send that state back in the code, meaning that we choose the correction channel such that R^†共X兲=R^†共PXP兲 for any operator X, which is indeed the case for the correction channel defined in Theorem 11. Con- sider the set of operators defined by

VªE^†„R^†共A兲….

This set is not an algebra in general. Nevertheless, it is an operator system by the linearity and positivity of channels. If X苸Vthen there existsY苸PAPsuch thatX=E^†(R^†共Y兲)and alsoPXP=Y. Therefore for allX苸V,

E^†„R^†共X兲…=E^†„R^†共Y兲…=X.

Hence the observables inVare exactly corrected, and this is independentof what the initial state was. For instance, if we

“forgot” to make sure that the initial state was in the code, we can still recover all the information, provided that we measure the observable with elementsXk=E^†(R^†共Yk兲)when- ever we would have measuredYk苸A. Typically this would involve measuring general共unsharp兲POVMs instead of just sharp projective observables. This shows that it could be useful to consider the correction of general POVMs. Since POVM elements do not always span an algebra, this suggests that we should consider the correctability共passive or active兲 codes associated with operator systems in this way.

Consider the following simple example of a conserved operator space that is not an algebra. LetHbe a single-qutrit Hilbert space with computational basis兵兩0典,兩1典,兩2典其. Consider the channel E on H defined by its action on observables represented in this basis as follows:

E^†共关aij兴3⫻3兲=

冤

^{aa01121 aa01222 a1100+2a22}

冥

^{. 共12兲}

Observe that 共E^†兲²=E^†ⴰE^†=E^† and that the range V of E^† coincides with its fixed point set; 兵Y:Y=E^†共X兲for someX其

=兵X:E^†共X兲=X其. Thus, the operator systemVis conserved by E. Moreover,Vis not an algebra since it is not closed under multiplication.

Given results from other settings for quantum error cor- rection, it is of course desirable to find a characterization of correction for operator spaces independent of any particular recovery operation. Here we derive a necessary condition, and we leave the general question as an open problem. Ob- serve that if there exists a channel R such that PE^†(R^†共X兲)P=PXP for all X苸V then, 0艋X艋1 implies 0 艋R^†共X兲艋1, since R^† is a contractive map. This in turn implies that there exists 0艋Y艋1 such thatPXP=PE^†共Y兲P, namely,Y=R^†共X兲.

Proposition 12. A necessary condition for an operator spaceV to be correctable on states PH for Eis that for all X苸V such that 0艋X艋1, there exists 0艋Y艋1 such that PXP=PE^†共Y兲P. This condition is also sufficient when V is an algebra containingP.

Proof. The first part of the statement has been proved. We show that the condition expressed implies correctability ofV if it is an algebra containingP.

SinceP苸V, we know thatBªPVPis a subalgebra ofV.

Note it is sufficient to prove the correctability ofB. Indeed, if Bis correctable then, in particular,Pis correctable. LetRbe a correction channel for the largest correctable algebra on PH. Then we have seen in the proof of Theorem 11 that the correctability of P implies that PE^†(R^†共X兲)P

=PE^†(R^†共PXP兲)P for any X. From this it follows that the correctability ofBonPHimplies that ofV.

Let Eª兵X兩0艋X艋1其. Suppose that P共V艚E兲P

=B艚E債PE^†共E兲P. Then for all projectorsQ苸B, there ex- ists a self-adjoint operator 0艋X艋1 such that PE^†共X兲P=Q.

Hence共P−Q兲E^†共X兲共P−Q兲= 0, which impliesXEk共P−Q兲= 0, orXE_kP=XE_kQfor allk. Also we havePE^†共1−X兲P=P−Q, so that QE^†共1−X兲Q= 0 from which 共1−X兲E_kQ= 0, or E_kQ

=XE_kQ for allk. Therefore

X共E_kP兲=XE_kQ=E_kQ=共E_kP兲Q