JMathPhys_50_062108

Quantum error correction on infinite-dimensional Hilbert spaces Cédric Bény,1,2 Achim Kempf,2,3,4 and David W. Kribs4,5,a� 1Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore 2Department of Applied Mathematics and Department of Physics, University of Waterloo, Ontario N2L 3G1, Canada 3Department of Physics, University of Queensland, St. Lucia, Brisbane, Queensland 4072, Australia 4Institute for Quantum Computing, University of Waterloo, Ontario N2L 3G1, Canada 5Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario N1G 2W1, Canada �Received 12 February 2009; accepted 28 April 2009; published online 26 June 2009� We present a generalization of quantum error correction to infinite-dimensional Hilbert spaces. We find that, under relatively mild conditions, much of the structure known from systems in finite-dimensional Hilbert spaces carries straightforwardly over to infinite dimensions. We also find that, at least in principle, there exist qualitatively new classes of quantum error correcting codes that have no finite- dimensional counterparts. We begin with a shift of focus from states to algebras of observables. Standard subspace codes and subsystem codes are seen as the special case of algebras of observables given by finite-dimensional von Neumann factors of type I. The new classes of codes that arise in infinite dimensions are shown to be characterized by von Neumann algebras of types II and III, for which we give in-principle physical examples. © 2009 American Institute of Physics. �DOI: 10.1063/1.3155783� I. INTRODUCTION A common challenge in the numerous fields of quantum information science is to devise techniques that protect the evolution of quantum systems from external perturbations. Since inter- actions with the environment are generally unavoidable, a variety of so-called quantum error correction procedures have been developed in order to correct the effect of environmental noise on quantum systems. The basic idea underlying the correction procedures is to exploit any knowledge that one may possess about the nature of the noise. To this end, the quantum information is encoded into a larger system in such a way that any effect of the noise on the data can be undone. In this way, the original qubits remain retrievable; that is, the errors are correctable. Most work in quantum error correction has assumed a finite number of qubits embedded in a finite-dimensional Hilbert space. Real systems are, however, always ultimately described in an infinite-dimensional Hilbert space. This implies two levels of generalization. A finite-dimensional code in an infinite- dimensional Hilbert space can undergo an infinite number of discrete errors �elements of the noise channel�. In addition, one may consider infinite-dimensional codes. Algorithms for the cases of a finite or an infinite number of qubits encoded in an infinite-dimensional Hilbert space have been proposed, see Refs. 10, 26, 17, and 12, respectively. However, no general theory �for instance, in the spirit of Ref. 20� exists. Here, we lay the foundations for quantum error correction in infinite-dimensional Hilbert spaces in full generality. We find that many of the basic results for quantum error correction extend a�Electronic mail: dkribs@uoguelph.ca. JOURNAL OF MATHEMATICAL PHYSICS 50, 062108 �2009� 50, 062108-10022-2488/2009/50�6�/062108/24/$25.00 © 2009 American Institute of Physics Downloaded 13 Apr 2011 to 202.115.51.3. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions to the infinite-dimensional setting and that there are new phenomena which appear only in the infinite-dimensional setting. The new phenomena extend beyond obvious features such as the possibility of an infinite number of individual errors. In particular, we uncover types of infinite- dimensional codes that have no finite-dimensional counterparts or approximations. Historically, following the realization that quantum error correction is possible and the dis- covery of seminal examples,4,30,31,16 Knill and Laflamme found a general mathematical condition characterizing the codes that are correctable for a given arbitrary noise model.20 In this framework, the state of the information to be corrected is encoded in a subspace of the �finite-dimensional� physical Hilbert space. It was later realized21,23,24 that there is a more subtle way of encoding the relevant information, namely, in a subsystem. This amounts to assuming that we ignore the effect of the noise on the complementary subsystem. In finite dimensions, this does not lead to larger quantum codes but to more efficient correction procedures.29,3,1,2 It was shown in Refs. 6 and 7 that this idea can be further generalized if, instead of a subsystem, one focuses on a restricted set of observables, which, for the purpose of quantum error correction, can always be assumed to span a finite-dimensional algebra. We will show that this idea naturally generalizes to infinite- dimensional Hilbert spaces, where the finite-dimensional algebras are replaced by arbitrary von Neumann algebras. In this context, a subsystem, as defined by a tensor-product structure on the underlying Hilbert space, corresponds to a von Neumann factor of type I. Other types of factors, however, also correspond to full-fledged logical quantum systems. This paper is organized as follows. In Sec. II we give relevant background. We then begin our investigation, which includes an analysis of the sharp fixed and correctable observables, a deriva- tion of the simultaneously correctable observables and operator systems, a brief review of von Neumann algebras and discussion of the special cases captured by the theory, presentation of explicit type I and type II examples, and finally an outlook section. II. BACKGROUND A. Quantum error correction We begin with a general review of quantum error correction motivated by the presentation of Ref. 21, which focused on the finite-dimensional case but applies equally well to the case of bounded interaction operators �as described below�. Full details on our formalism in the infinite- dimensional setting will be provided below. Let us suppose that some information is sent through a quantum channel E. The aim of quantum error correction is to find certain degrees of freedom, the error correcting code �whose exact nature we deliberately keep imprecise for the moment�, on which the effect of the channel can be inverted. Since the inversion must be implemented physi- cally, it must be a valid physical transformation, i.e., a channel. The inverse channel R is called the correction channel. Under reasonable assumptions �see Sec. II C� a channel E can always be written as E��� = � i Ei�Ei † , which means that we can assume that the noise is given by a discrete family of individual error operators Ei on Hilbert space. In fact we will see that if R corrects this channel on some code, then it will correct also any channel whose elements span the same operator space as the elements of E. This is important because often one does not know the precise channel elements Ei. Indeed, suppose that the system interacts continuously with its environment via a general Hamiltonian H=H† of the form H = � i Ji � Ki, �1� where the interaction operators Ji act on the system Hilbert space and the operators Ki act on the environment. We assume here that none of the operators Ki are equal to zero. 062108-2 Bény, Kempf, and Kribs J. Math. Phys. 50, 062108 �2009� Downloaded 13 Apr 2011 to 202.115.51.3. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions In order for the state of the system at a given time to depend unambiguously on its initial state, we must assume that the initial state of the environment is uncorrelated with that of the system. For simplicity, we will further assume that it is a pure state ���. It is unlikely that we know much about Ki or ��� given that the environment may be very large and complex. However, it is generally conceivable that we have good knowledge of what the operators Ji, or rather their span, can be �the set of interaction operators can be replaced by any other set spanning the same vector space by changing the definition of the operators Ki�. If Et is the channel describing the evolution of the system alone up to time t and Et� is the corresponding dual map, then we have Et��A� = �1 � ����eitH�A � 1�e−itH�1 � ���� = � k �1 � ����eitH�1 � ����A�1 � ����e−itH�1 � ���� = � k Ek †�t�AEk�t� , �2� where the channel elements are Ek�t� = �1 � �k��e−itH�1 � ���� = � n �− it�n n! �1 � �k��Hn�1 � ���� = � n � j1. . .jn �− it�n n! �k�Kj1 ¯ Kjn���Jj1 ¯ Jjn. �3� Hence, we know that no matter what the environment operators and initial state are, the span of the channel elements Ek�t� belongs to the algebra generated by the interaction operators Ji. One can look for correctable codes for all channels with this property. Such codes would be a form of infinite-distance code �in fact an example of noiseless subsystems�.21 While there are important instances in which these codes do exist, they form a rather restrictive class of quantum error correcting code. On the other hand, if the time t at which we aim to perform the correction is small enough, we can do better. Indeed, suppose that � is some interaction parameter with unit of energy; then the above series is expressed in powers of t�. We see that to the nth order in t�, the elements of the channel are in the span of the nth order products Jj1¯Jjn. Hence, if we correct often enough �in order to limit the value of t��, then we only need to find a correction channel and a code for channel elements in the span of the operators Jj1¯Jjn, n�N, for a fixed N. In this context, the operators Ji are seen as representing individual errors, and our code corrects up to N independent errors. 21 For clarity of the presentation we will stick to the simple picture where the channel E is given. However, we will keep the more realistic situation in mind and check that the correction procedure we devised works not just for the given channel but also for any channel whose elements span the same space. B. Stochastic Heisenberg picture Traditionally, one attempts to correct states, namely, to simultaneously find a state � and a channel R such that R�E����=�. In Refs. 6 and 7 it was shown that it is convenient to consider instead the correction of observables. Most generally, an observable is specified by a positive operator-valued measure �POVM�. For instance, if the measure is discrete, the POVM X is speci- fied by a family of positive operators Xk called effects. In general an effect can be any positive operator smaller than the identity 0�Xk�1. In order to form a discrete POVM, these effects must sum to the identity �kXk=1. More generally, a POVM X sends measurable subsets of a measure space � to positive operators. It is defined by the effects X���, ���. For a discrete POVM, X���=�k��Xk. 062108-3 Quantum error correction J. Math. Phys. 50, 062108 �2009� Downloaded 13 Apr 2011 to 202.115.51.3. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions What matters physically are the expectation values of the form Tr��A� , where A is an effect. Indeed, for a POVM X, Tr��X���� is the probability that the outcome of a measurement of X falls inside ���. If the channel E describes the evolution of our system, a measurement of an observable X after the evolution will yield probabilities of the form Tr�E���X����. Alternatively, one can define a dual map E� which satisfies Tr�E���X���� = Tr��E��X����� for all ���. This dual channel is the stochastic form of the usual Heisenberg picture of quantum mechanics. It specifies the evolution of observables instead of states. In this picture, an observable X evolves to the observable Y defined by Y���=E��X����. An important conceptual point to note here, which is not apparent in the usual case of unitary evolutions, is that this evolution goes “backward” in time. Indeed, if the channel E2 follows the channel E1 in time, the overall trans- formation in the Schrödinger picture is given by the channel E2 �E1. But the dual channels com- pose in the reverse order; we have �E2 �E1��=E1� �E2�. This means that the latest transformation must be applied first. Instead of correcting states, we may thus attempt to correct observables, i.e., to pair a channel R with an observable X such that �R � E���X���� = X��� for all ���. This expression means that measuring X before or after the action of the map R �E would yield the same outcomes with the same probabilities no matter what the initial state was. Hence we can say that the observable X is corrected by R for the noise defined by E. In general, if the channel R exists such that this equation is satisfied, we say that the observable X is correctable. The sets of simultaneously correctable sharp observables, that is, correctable by a single common channel R, were characterized in Ref. 6 and shown to generalize all previous known types of quantum error correcting codes. We will show here that this approach leads naturally to an infinite-dimensional generalization. C. States and channels in infinite dimensions We consider a quantum system characterized by a Hilbert space H that may be infinite dimensional. For effects, it is natural to consider the bounded linear operators on H. Indeed, the condition 0�A�1 guarantees that A is a bounded operator on H. We denote the set of bounded operators on H by B�H� and the set of effects �or positive contractions on H� by E�H�. The set B�H� is naturally a von Neumann algebra. In general we could assume that states are positive linear functionals on B�H�. In the finite-dimensional case, this guarantees that they are of the form A�Tr��A� for some operator � and all effects A. However, this does not carry through to the infinite-dimensional case. Instead, we postulate that a state is represented by a positive operator � such that Tr��A� is well defined for any effect A and also such that Tr��A�=1. This means that states are trace-class operators. Formally, the trace-class operators B�B�H� are those for which the expression � i �i� B†B�i� converges for �one and hence� any basis �i�. We will let Bt�H� denote the set of trace-class operators on H. For self-adjoint elements ��Bt�H�, we have a trace defined by 062108-4 Bény, Kempf, and Kribs J. Math. Phys. 50, 062108 �2009� Downloaded 13 Apr 2011 to 202.115.51.3. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions Tr��� ª � i �i���i� . The product of an element of ��Bt�H� with any operator A�B�H� is also trace class, which implies that we can define Tr��A�. The set Bt�H� itself is a Banach algebra. It is the predual of B�H�, which means that B�H� is the set of linear functionals on Bt�H�. This means that effectively we have defined the effects as linear functionals on states rather than the converse. The existence of a predual is a fundamental property of von Neumann algebras. A general von Neumann algebra is equipped with a weak-� topology, which amounts to defining the convergence of a sequence An�B�H� in terms of expectation values. In the case of B�H�, this means that the sequence converges to A if and only if the numbers Tr��An� converge to Tr��A� as n→� for all states ��Bt�H�. This implies that the states represented by elements of Bt�H�, seen as linear functionals of effects, are continuous with respect to the weak-� topology. Indeed, if the sequence An�n=1 � converges to A in this topology, then, by definition Tr��An�→Tr��A�. Hence the map A�Tr��A� is continuous. Conversely, those are all the weak-� continuous positive linear functionals. There- fore, our choice of states corresponds to restricting the natural set of all linear functionals on effects to only those which are weak-� continuous, or normal for short. A channel from a system represented by the Hilbert space H1 to a system represented by H2 can be defined by a trace-preserving completely positive linear map E:Bt�H1� → Bt�H2� on states. It has a dual E�:B�H2� → B�H1� , which describes the evolution of effects, and hence observables, in the Heisenberg picture. A channel can be represented as E��A� = � k=1 � Ek †AEk or E��� = � k=1 � Ek�Ek † , where the sum can now be infinite.22 We will call this the operator-sum form of the channel E. The elements Ek are bounded linear operators from H1 to H2. This can be understood starting from the Stinespring dilation theorem for completely positive maps between C�-algebras, which states that there is a representation of B�H2� on some Hilbert space K and an isometry V :H1→K, such that E��A� = V† �A�V for all A�B�H2�. In the case that we are considering �von Neumann factor of type I and normal map�, the representation on H is of the form �A�=A� 1. Also, if H2 is separable, then so is K.28 Therefore the subsystem on which �A� acts trivially is also separable and possesses a discrete basis �i�. This implies that E��A� = V†�A � 1�V = � i V†�A � �i��i��V = � i V†�1 � �i��A�1 � �i��V , where the operators �1� �i��V :H1→H2 are defined by the induced tensor structure on the dilation space K. Thus the elements of the channel E can be chosen to be Ei= �1� �i��V. Note from this 062108-5 Quantum error correction J. Math. Phys. 50, 062108 �2009� Downloaded 13 Apr 2011 to 202.115.51.3. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions observation that it is clear that there is a large ambiguity in the choice of the elements Ei. Indeed, any orthonormal basis �i� would potentially yield a different set of elements. III. SHARP CORRECTABLE OBSERVABLES A POVM X is correctable if the effect of the channel can be inverted on all the effects X��� in the following sense. Definition 3.1: We say that an effect 0�A�1 is correctable for the channel E if there exists a channel R such that A = E��R��A�� . In this section we will characterize the correctability of certain types of effects, namely, projectors. This is important because the effects of a sharp observable. i.e., a traditional observ- able represented by a self-adjoint operator, are always projectors �they are the spectral projectors of the corresponding self-adjoint operator�. The result obtained in this section will form the basis of our understanding of correctable observables. We begin with a “warm up,” the case of passive error correction in this setting. A. Sharp fixed observables We consider the problem of characterizing the sharp observables that are unaffected by the action of the channel, that is, which are correctable in the above sense but with the trivial correc- tion channel R���=�. This requires that we use the same source and destination Hilbert spaces, namely, H ª H1 = H2. Definition 3.2: We say that an observable X is fixed by the channel E if X��� = E��X���� for all � . A slightly more general form of this problem �see below� was addressed for channels defined on finite-dimensional Hilbert spaces in Ref. 6 and shown to yield all noiseless subsystems.21,19,13 These results may be readily generalized to the infinite-dimensional setting provided that we model the proof on the approach of Ref. 7 which does not refer to the structure theory of finite-dimensional algebras. Since we focus on sharp observables, the effects X��� are all projectors. Let us therefore first characterize the fixed projectors P, which satisfy P=E��P�. By multiplying on both sides by the orthogonal projector P�=1− P, we obtain 0= P�E��P�P�=�kP�Ek†PEkP�. The right hand side is a sum of positive operators, which must therefore all equal zero: �PEkP��†PEkP�=0 for all k, which in turn implies PEk