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Quantum measurements and entropic bounds on information transmission
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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 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.
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General quantum measurements are represented by instruments. In this paper the mathematical formalization is given of the idea that an instrument is a channel which accepts a quantum state as input and produces a probability and an a posteriori state as output. Then, by using mutual entropies on von Neumann algebras and the identification of instruments and channels, many old and new informational inequalities are obtained in a unified manner. Such inequalities involve various quantities which characterize the performances of the instrument under study; in particular, these inequalities include and generalize the famous Holevo's bound.
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.
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 concerned with the limits imposed by quantum mechanics on correlations between statistical sources. Typical examples are the correlations between the transmitter and receiver of a quantum communication channel, and between the measurement results of two detectors that make simultaneous measurements.KeywordsMutual InformationQuantum CorrelationJoint Probability DistributionCommunication ContextBell InequalityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
A measuring process is described by an interaction between an object to be measured and a measuring apparatus and by a subsequent reading of the outcome. By a measurement statistics we shall mean the following two elements determined by a measuring process: the probability distribution dP(x|ρ) of the outcome x and the state reduction from prior states ρ to the posterior states ρ
x conditioned by the outcome x. The problem of mathematical characterizations of possible measurement statistics has a long history initiated by von Neumann’s pioneering work [1] and up to now it is known as a general solution that every normalized operation valued measures dX(x), called an operational measure in brief, represents a realizable measurement statistics by the relation dX(x) ρ = ρxdP(x|ρ)[2], where an operation is a contraction on the space of trace class operators such that its dual is completely positive.
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.
In order to provide a mathmatical framework for the process of making repeated measurements on continuous observables in a statistical system we make a mathematical definition of an instrument, a concept which generalises that of an observable and that of an operation. It is then possible to develop such notions as joint and conditional probabilities without any of the commutation conditions needed in the approach via observables. One of the crucial notions is that of repeatability which we show is implicitly assumed in most of the axiomatic treatments of quantum mechanics, but whose abandonment leads to a much more flexible approach to measurement theory.
It has long been known that the von Neumann entropy S is an upper bound on the information one can extract from a quantum system in an unknown pure state. In this paper we define the ‘‘subentropy’’ Q, which we prove to be a lower bound on this information. Moreover, just as the von Neumann entropy is the best upper bound that depends only on the density matrix, we show that Q is the best lower bound that depends only on the density matrix. Other parallels between S and Q are also demonstrated.
The possible amount of information transfer between any source and any user via a quantum system is bounded through the quantum entropy function. In contrast to the classical case, this shows that infinite information transfer implies infinite entropy. The entropy bound is also applied to obtain the ultimate quantum information transmission capacity of the free electromagnetic field under a power and a bandwidth constraint.
We obtain a lower bound for the mutual information of a quantum transmission channel, which is perfectly analogous with Holevo's upper bound. The Uhlmann inequality for relative entropies is used, in a reverted order, for a completely positive mapping related with the mixed coherent states introduced by us seventeen years ago. Possible applications to quantum cryptography are discussed.
We prove a new result limiting the amount of accessible information in a quantum channel. This generalizes Kholevo's theorem and implies it as a simple corollary. Our proof uses the strong subadditivity of the von Neumann entropy functional S(ae) and a specific physical analysis of the measurement process. The result presented here has application in information obtained from "weak" measurements, such as those sometimes considered in quantum cryptography. PACS numbers: 03.65.Bz, 89.70.+c 1 1 Introduction When a measurement is performed on a quantum system, information is acquired about the preparation of the system. In a quantum communication channel, for example, the receiver uses a "decoding observable" to infer something about the "signal state" of the channel [1]. One of the central problems of quantum information theory is the extent to which the laws of quantum mechanics provide limitations and opportunities for information acquisition in various contexts. A theorem stated b...
The Holevo bound is a bound on the mutual information for a given quantum encoding. In 1996 Schumacher, Westmoreland and Wootters [Schumacher, Westmoreland and Wootters, Phys. Rev. Lett. 76, 3452 (1996)] derived a bound which reduces to the Holevo bound for complete measurements, but which is tighter for incomplete measurements. The most general quantum operations may be both incomplete and inefficient. Here we show that the bound derived by SWW can be further extended to obtain one which is yet again tighter for inefficient measurements. This allows us in addition to obtain a generalization of a bound derived by Hall, and to show that the average reduction in the von Neumann entropy during a quantum operation is concave in the initial state, for all quantum operations. This is a quantum version of the concavity of the mutual information. We also show that both this average entropy reduction, and the mutual information for pure state ensembles, are Schur-concave for unitarily covariant measurements; that is, for these measurements, information gain increases with initial uncertainty.
When a measurement is made on a quantum system in which classical information is encoded, the measurement reduces the observers average Shannon entropy for the encoding ensemble. This reduction, being the {\em mutual information}, is always non-negative. For efficient measurements the state is also purified; that is, on average, the observers von Neumann entropy for the state of the system is also reduced by a non-negative amount. Here we point out that by re-writing a bound derived by Hall [Phys. Rev. A {\bf 55}, 100 (1997)], which is dual to the Holevo bound, one finds that for efficient measurements, the mutual information is bounded by the reduction in the von Neumann entropy. We also show that this result, which provides a physical interpretation for Hall's bound, may be derived directly from the Schumacher-Westmoreland-Wootters theorem [Phys. Rev. Lett. {\bf 76}, 3452 (1996)]. We discuss these bounds, and their relationship to another bound, valid for efficient measurements on pure state ensembles, which involves the subentropy. Comment: 4 pages, Revtex4. v3: rewritten and reinterpreted somewhat
This paper presents self-contained proofs of the strong subadditivity inequality for quantum entropy and some related inequalities for the quantum relative entropy, most notably its convexity and its monotonicity under stochastic maps. Moreover, the approach presented here, which is based on Klein's inequality and one of Lieb's less well-known concave trace functions, allows one to obtain conditions for equality. Using the fact that the Holevo bound on the accessible information in a quantum ensemble can be obtained as a consequence of the monotonicity of relative entropy, we show that equality can be attained for that bound only when the states in the ensemble commute. The paper concludes with an Appendix giving a short description of Epstein's elegant proof of the relevant concavity theorem of Lieb. Comment: 28 pages, latex Added reference to M.J.W. Hall, "Quantum Information and Correlation Bounds" Phys. Rev. A, 55, pp 100--112 (1997)
The capacity of a quantum channel for transmission of classical information
depends in principle on whether product states or entangled states are used at
the input, and whether product or entangled measurements are used at the
output. We show that when product measurements are used, the capacity of the
channel is achieved with product input states, so that entangled inputs do not
increase capacity. We show that this result continues to hold if sequential
measurements are allowed, whereby the choice of successive measurements may
depend on the results of previous measurements.
We also present a new simplified expression which gives an upper bound for
the Shannon capacity of a channel, and which bears a striking resemblance to
the well-known Holevo bound.
On the Heisenberg principle, namely on the information-disturbance tradeoff in a quantum measurement
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