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Instruments and Channels in Quantum Information Theory
Abstract
While a positive operator valued measure gives the probabilities in a quantum measurement, an instrument gives both the probabilities
and the a posteriori states. By interpreting the instrument as a quantum channel and by using the typical inequalities for
the quantum and classical relative entropies, many bounds on the classical information extracted in a quantum measurement,
of the type of the Holevo bound, are obtained in a unified manner.
... We can think the instrument to be a channel: from a quantum state (the pre-measurement state) to a quantum/classical state (a posteriori state plus probabilities). The mathematical formalization of the idea that an instrument is a channel is central in our paper and allows for a unified approach to various bounds for I c and for related quantities [11,12]. ...
... The original SWW bound [4] is inequality (39) in the case of an instrument with no sum on k in the definition (1b) of the operations O(ω). Eq. (39) is a slight generalization to the case of (1b) with sums and was already proven in [11]; a different proof, more similar to the SWW original one, was given after in [20]. Inequality (39) has been generalized to the infinite and continuous case in [12]. ...
While a positive operator valued measure gives the probabilities in a quantum measurement, an instrument gives both the probabilities and the a posteriori states. By interpreting the instrument as a quantum channel and by using the monotonicity theorem for relative entropies many bounds on the classical information extracted in a quantum measurement are obtained in a unified manner. In particular, it is shown that such bounds can all be stated as inequalities between mutual entropies. This approach based on channels gives rise to a unified picture of known and new bounds on the classical information (Holevo's, Shumacher-Westmoreland-Wootters', Hall's, Scutaru's bounds, a new upper bound and a new lower one). Some examples clarify the mutual relationships among the various bounds.
... The search for a consistent formulation of the dynamics for quantum-classical hybrid systems has a long history; the motivations include computational advantages, description of mesoscopic systems, unification of gravity and quantum theory, description of quantum measurements. . . , see [1][2][3][4][5][6][7][8][9][10][11][12] and references there in. Also the quantum measurements in continuous time can be interpreted in terms of hybrid systems; in this case the classical component is the monitored signal extracted from the quantum system [2,3,[13][14][15][16]. ...
... 10]. On the other side, measurements on a quantum system [23,24] can be interpreted as involving hybrid systems: a positive operator valued measure is a channel from a quantum system to a classical one, an instrument is a channel from a quantum system to a hybrid system [2,4,10,11]. Let us recall that an instrument gives the probabilities and the conditional state after the measurement. ...
... One motivation to the introduce the embedding representation is to provide an avenue for a continuous time measurement interpretation [81] even in the case when ρ t is non positive [43,80]. In the coming section, we analyze the application to time reversal of a completely positive evolution. ...
... The physical interpretation of the measurement record requires the statistics to be non-anticipating (i.e. the present record cannot be affected by events in the future see discussion e.g. in [9]) and to establish a correspondence with an instrument i.e. a completely positive unital map see e.g. [6,8,81]. The influence martingale unraveling framework is by construction non-anticipating. ...
We explore algebraic and dynamical consequences of unraveling general time-local master equations. We show that the ``influence martingale'', the paramount ingredient of a recently discovered unraveling framework, pairs any time-local master equation with a one parameter family of Lindblad-Gorini-Kossakowski-Sudarshan master equations. At any instant of time, the variance of the influence martingale provides an upper bound on the Hilbert-Schmidt distance between solutions of paired master equations. Finding the lowest upper bound on the variance of the influence martingale yields an explicit criterion of ``optimal pairing''. The criterion independently retrieves the measure of isotropic noise necessary for the structural physical approximation of the flow the time-local master equation with a completely positive flow. The optimal pairing also allows us to invoke a general result on linear maps on operators (the ``commutant representation'') to embed the flow of a general master equation in the off-diagonal corner of a completely positive semi-group which in turn solves a time-local master equation that we explicitly determine.We use the embedding to reverse a completely positive evolution, a quantum channel, to its initial condition thereby providing a protocol to preserve quantum memory against decoherence. We thus arrive at a model of continuous time error correction by a quantum channel.
... One motivation to the introduce the embedding representation is to provide an avenue for a continuous time measurement interpretation [81] even in the case when ρ t is non positive [45,80]. In the coming section, we analyze the application to time reversal of a completely positive evolution. ...
... The physical interpretation of the measurement record requires the statistics to be non-anticipating (i.e. the present record cannot be affected by events in the future see discussion e.g. in [9]) and to establish a correspondence with an instrument i.e. a completely positive unital map see e.g. [6,8,81]. The influence martingale unraveling framework is by construction nonanticipating. ...
We explore algebraic and dynamical consequences of unraveling general time-local master equations. We show that the "influence martingale", the paramount ingredient of a recently discovered unraveling framework, pairs any time-local master equation with a one parameter family of Lindblad-Gorini-Kossakowski-Sudarshan master equations. At any instant of time, the variance of the influence martingale provides an upper bound on the Hilbert-Schmidt distance between solutions of paired master equations. Finding the lowest upper bound on the variance of the influence martingale yields an explicit criterion of "optimal pairing". The criterion independently retrieves the measure of isotropic noise necessary for the structural physical approximation of the flow the time-local master equation with a completely positive flow. The optimal pairing also allows us to invoke a general result on linear maps on operators (the "commutant representation") to embed the flow of a general master equation in the off-diagonal corner of a completely positive map which in turn solves a time-local master equation that we explicitly determine. We use the embedding to reverse a completely positive evolution, a quantum channel, to its initial condition thereby providing a protocol to preserve quantum memory against decoherence. We thus arrive at a model of continuous time error correction by a quantum channel.
... [35][36][37]. It should be noticed that relative entropy, of classical or quantum type, has already been used in quantum measurement theory to give proper measures of information gains and losses in various scenarios [37][38][39][40][41]. ...
We introduce a new information-theoretic formulation of quantum measurement uncertainty relations, based on the notion of relative entropy between measurement probabilities. In the case of a finite-dimensional system and for any approximate joint measurement of two target discrete observables, we define the entropic divergence as the maximal total loss of information occurring in the approximation at hand. For fixed target observables, we study the joint measurements minimizing the entropic divergence, and we prove the general properties of its minimum value. Such a minimum is our uncertainty lower bound: the total information lost by replacing the target observables with their optimal approximations, evaluated at the worst possible state. The bound turns out to be also an entropic incompatibility degree, that is, a good information-theoretic measure of incompatibility: indeed, it vanishes if and only if the target observables are compatible, it is state-independent, and it enjoys all the invariance properties which are desirable for such a measure. In this context, we point out the difference between general approximate joint measurements and sequential approximate joint measurements; to do this, we introduce a separate index for the tradeoff between the error of the first measurement and the disturbance of the second one. By exploiting the symmetry properties of the target observables, exact values, lower bounds and optimal approximations are evaluated in two different concrete examples: (1) a couple of spin-1/2 components (not necessarily orthogonal); (2) two Fourier conjugate mutually unbiased bases in prime power dimension. Finally, the entropic incompatibility degree straightforwardly generalizes to the case of many observables, still maintaining all its relevant properties; we explicitly compute it for three orthogonal spin-1/2 components.
... In order to treat the two cases altogether, we consider POVMs with outcomes in R m × R m ≡ R 2m , which we call bi-observables; they correspond to a measurement of m position components and m momentum components. The specific covariance requirements will be given in the Definitions 5,6,7. In studying the properties of probability measures on R k , a very useful notion is that of the characteristic function, that is, the Fourier cotransform of the measure at hand; the analogous quantity for POVMs turns out to have the same relevance. ...
Heisenberg's uncertainty principle has recently led to general measurement uncertainty relations for quantum systems: incompatible observables can be measured jointly or in sequence only with some unavoidable approximation, which can be quantified in various ways. The relative entropy is the natural theoretical quantifier of the information loss when a `true' probability distribution is replaced by an approximating one. In this paper, we provide a lower bound for the amount of information that is lost by replacing the distributions of the sharp position and momentum observables, as they could be obtained with two separate experiments, by the marginals of any smeared joint measurement. The bound is obtained by introducing an entropic error function, and optimizing it over a suitable class of covariant approximate joint measurements. We fully exploit two cases of target observables: (1) n-dimensional position and momentum vectors; (2) two components of position and momentum along different directions. In (1), we connect the quantum bound to the dimension n; in (2), going from parallel to orthogonal directions, we show the transition from highly incompatible observables to compatible ones. For simplicity, we develop the theory only for Gaussian states and measurements.
... The maps in this set must sum to a quantum channel, namely to a completely positive and trace-preserving (CPTP) linear map [8][9][10]. This situation is described by a quantum instrument [10][11][12][13]: a quantum channel that takes a quantum system as input, and outputs a classical-quantum system, where the classical system represents the 'meter' read by the experimenter. From the classical outcome read on the meter, one can infer which CPTNI map occurred during the experiment. ...
Quantum supermaps are a higher-order genera- lization of quantum maps, taking quantum maps to quantum maps. It is known that any completely positive and trace non-increasing (CPTNI) map can be performed as part of a quantum measurement. By providing an explicit counterexample we show that, instead, not every quantum supermap sending a quantum channel to a CPTNI map can be realized in a measurement on quantum channels. We find that the supermaps that can be implemented in this way are exactly those transforming quantum channels into CPTNI maps even when tensored with the identity supermap. We link this result to the fact that the principle of causality fails in the theory of quantum supermaps.
... The maps in this set must sum to a quantum channel, namely to a completely positive and trace-preserving (CPTP) linear map [8][9][10]. This situation is described by a quantum instrument [10][11][12][13], a quantum channel that takes a quantum system as input, and outputs a a classical-quantum system, where the classical system represents the "meter" read by the experimenter. From the classical outcome read on the meter, one can infer which CPTNI map occurred during the experiment. ...
Quantum supermaps are a higher-order generalization of quantum maps, taking quantum maps to quantum maps. It is known that any completely positive, trace non-increasing (CPTNI) map can be performed as part of a quantum measurement. By providing an explicit counterexample we show that, instead, not every quantum supermap sending a quantum channel to a CPTNI map can be realized in a measurement on quantum channels. We find that the supermaps that can be implemented in this way are exactly those transforming quantum channels into CPTNI maps even when tensored with the identity supermap.
It is well known that the state operator of an open quantum system can be generically represented as the solution of a time-local equation — a quantum master equation. Unraveling in quantum trajectories offers a picture of open system dynamics dual to solving master equations. In the unraveling picture, physical indicators are computed as Monte Carlo averages over a stochastic process valued in the Hilbert space of the system. This approach is particularly adapted to simulate systems in large Hilbert spaces. We show that the dynamics of an open quantum system generically admits an unraveling in the Hilbert space of the system described by a Markov process generated by ordinary stochastic differential equations for which rigorous concentration estimates are available. The unraveling can be equivalently formulated in terms of norm-preserving state vectors or in terms of linear “ostensible” processes trace preserving only on average. We illustrate the results in the case of a two level system in a simple boson environment. Next, we derive the state-of-the-art form of the Diósi-Gisin-Strunz Gaussian random ostensible state equation in the context of a model problem. This equation provides an exact unraveling of open systems in Gaussian environments. We compare and contrast the two unravelings and their potential for applications to quantum error mitigation.
Cette thèse est consacrée à l'étude de modèles stochastiques associés aux systèmes quantiques ouverts. Plus particulièrement, nous étudions les marches quantiques ouvertes qui sont les analogues quantiques des marches aléatoires classiques. La première partie consiste en une présentation générale des marches quantiques ouvertes. Nous présentons les outils mathématiques nécessaires afin d'étudier les systèmes quantiques ouverts, puis nous exposons les modèles discrets et continus des marches quantiques ouvertes. Ces marches sont respectivement régies par des canaux quantiques et des opérateurs de Lindblad. Les trajectoires quantiques associées sont quant à elles données par des chaînes de Markov et des équations différentielles stochastiques avec sauts. La première partie s'achève avec la présentation de quelques pistes de recherche qui sont le problème de Dirichlet pour les marches quantiques ouvertes et les théorèmes asymptotiques pour les mesures quantiques non destructives. La seconde partie rassemble les articles rédigés durant cette thèse. Ces articles traîtent les sujets associés à l'irréductibilité, à la dualité récurrence-transience, au théorème central limite et au principe de grandes déviations pour les marches quantiques ouvertes à temps continu.
In this paper we will give a short presentation of the quantum Lévy-Khinchin formula andof the formulation of quantum continual measurements based on stochastic differentialequations, matters which we had the pleasure to work on in collaboration with Prof.Holevo. Then we will begin the study of various entropies and relative entropies, whichseem to be promising quantities for measuring the information content of the continualmeasurement under consideration and for analysing its asymptotic behaviour.
In order to develop a statistical theory of quantum measurements including continuous observables, a concept of a posteriori states is introduced, which generalizes the notion of regular conditional probability distributions in classical probability theory. Its statistical interpretation in measuring processes is discussed and its existence is proved. As an application, we also give a complete proof of the E. B. Davies and J. T. Lewis [Commun. Math. Phys. 17, 239-260 (1970; Zbl 0194.583)] conjecture that there are no (weakly) repeatable instruments for non-discrete observables in the standard formulation of quantum mechanics, using the notion of a posteriori states.
This paper is primarily concerned with the development and application of quantum bounds on mutual information, although some of the methods developed can be applied to any figure of merit indicating degree of correlation, such as coincidence rate. Three basic techniques for obtaining bounds are described: mappings between joint-measurement and communication correlation contexts; a duality relation for quantum ensembles and quantum measurements; and an information exclusion principle [M. J. W. Hall, Phys. Rev. Lett. 74, 3307 (1995)]. Results include a proof of Holevo's communication bound from a joint-measurement inequality; a measurement-dependent dual to Holevo's bound; lower bounds for mutual information under ensemble and measurement constraints; information exclusion relations for measurements described by probability-operator measures; a proof that Glauber coherent states are optimal signal states for quantum communication based on (noisy) optical heterodyne detection; and an information inequality for quantum eavesdropping. Relations between the three techniques are used to further obtain upper bounds for quantum information, and to extend the information exclusion principle to a joint-measurement context.
A generalization of Shannon’s amount of information into quantum measurements of continuous observables is introduced. A necessary and sufficient condition for measuring processes to have a non‐negative amount of information is obtained. This resolves Groenewold’s conjecture completely including the case of measurements of continuous observables. As an application the approximate position measuring process considered by von Neumann and later by Davies is shown to have a non‐negative amount of information.
The purpose of this paper is to provide a basis of theory of measurements of continuous observables. We generalize von Neumann’s description of measuring processes of discrete quantum observables in terms of interaction between the measured system and the apparatus to continuous observables, and show how every such measuring process determines the state change caused by the measurement. We establish a one‐to‐one correspondence between completely positive instruments in the sense of Davies and Lewis and the state changes determined by the measuring processes. We also prove that there are no weakly repeatable completely positive instruments of nondiscrete observables in the standard formulation of quantum mechanics, so that there are no measuring processes of nondiscrete observables whose state changes satisfy the repeatability hypothesis. A proof of the Wigner–Araki–Yanase theorem on the nonexistence of repeatable measurements of observables not commuting conserved quantities is given in our framework. We also discuss the implication of these results for the recent results due to Srinivas and due to Mercer on measurements of continuous observables.
For the average information gain by a quantal measurement of the first kind, an expression is suggested for a lower bound, which turns out to be non-negative.
It is proved that for an ideal quantum measurement the average entropy of the reduced states after the measurement is not greater than the entropy of the original state.
