Information and quantum measurement

Davies, E. B.

doi:10.1109/tit.1978.1055941

Cited by 257 publications

(305 citation statements)

References 8 publications

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“…It represents an ultimate limit upon performance set by the best possible output of an N-trial instrument. The analysis proves an earlier conjecture due to Jones [80], and extends it to a general method of proof, in the style of earlier related results due to Holevo [81], Davies [82], and Bendjaballah and Charbit [83] on the optimal receiver problem in quantum communication theory. (31) [(32)] which is attained when all Aq are equal to some rank-one projector P.…”

Section: A Quantum Bayes Inversionsupporting

confidence: 80%

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Fundamental limits upon the measurement of state vectors

Jones

1994

Phys. Rev. A

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Using the Shannon information theory and the Bayesian methodology for inverting quantum data [K.R.W. Jones, Ann. Phys. (N.Y. ) 207, 140 (1991)]we prove a fundamental bound upon the measurability of 6nite-dimensional quantum states. To do so we imagine a thought experiment for the quantum communication of a pure state vP, known to one experimenter, to his colleague via the transmission of N identical copies of it in the limit of zero temperature. Initial information available to the second experimenter is merely that of the allowed manifold of superpositions upon which the chosen @ may lie. Her efForts to determine it, in an optimal way, subject to the fundamental constraints imposed by quantum noise, de6ne a statistical uncertainty principle. This limits the accuracy with which Q can be measured according to the number N of transmitted copies. [19].Here we employ the Bayesian formalism of quantum inference [19 -21], which unifies the methods of communication theory [22] with quantum measurement [23,24]. This approach treats the limits on state measurement within the paradigm of Shannon's analysis of physical communication channels, just as in the early quantum communication theory studies of Helstrom [13] The first experimenter is thus using quantized information carriers to communicate the setting of his preparing

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Section: A Quantum Bayes Inversionsupporting

confidence: 80%

“…Using the optimal configuration of a binary symmetric channel [36], see Fig. 4 To find the resulting channel performance we substitute the matrices (84), for the optimal instrument, into Holevo's bound (82) and get…”

Section: Illustration For Two-state Systemsmentioning

confidence: 99%

Fundamental limits upon the measurement of state vectors

Jones

1994

Phys. Rev. A

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“…The problem of maximization of I (E, Π) consists of two dual aspects [7,63,65]: maximization over all possible measurements, providing the ensemble E is given, see, e.g., [39,60,101,114], and (less explored) maximization over all ensembles, when the POVM Π is fixed [6,90]. In the former case, the maximum is called accessible information.…”

Section: Relation To Informational Powermentioning

confidence: 99%

Highly symmetric POVMs and their informational power

Słomczyński

Szymusiak

2015

Quantum Inf Process

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We discuss the dependence of the Shannon entropy of normalized finite rank-1 POVMs on the choice of the input state, looking for the states that minimize this quantity. To distinguish the class of measurements where the problem can be solved analytically, we introduce the notion of highly symmetric POVMs and classify them in dimension 2 (for qubits). In this case, we prove that the entropy is minimal, and hence, the relative entropy (informational power) is maximal, if and only if the input state is orthogonal to one of the states constituting a POVM. The method used in the proof, employing the Michel theory of critical points for group action, the Hermite interpolation, and the structure of invariant polynomials for unitary-antiunitary groups, can also be applied in higher dimensions and for other entropy-like functions. The links between entropy minimization and entropic uncertainty relations, the Wehrl entropy, and the quantum dynamical entropy are described.

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“…We will sometimes use subscripts to denote the supremum of (1) restricted to a particular class of ensembles; in particular, we write C n (Φ) to denote the restriction to ensembles of n states. It is well-known [1] that for qubit channels the maximum can be achieved with an ensemble containing at most four states.…”

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confidence: 99%

“…The corresponding circle of density matrices lies within a 3-dimensional subspace. By a straightforward adaptation of the general arguments in [1], the maximum capacity on a circle is achieved with at most three states. Numerical studies of examples of this type have confirmed that the 3-state capacity is strictly greater than the (unrestricted) 2-state capacity, and that C 2 < C 4 = C 3 .…”

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confidence: 99%

Qubit Channels Can Require More Than Two Inputs to Achieve Capacity

2002

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We give examples of qubit channels that require three input states in order to achieve the Holevo capacity.The Holevo capacity C(Φ) of a channel Φ is defined as the supremum over all possible ensembles E = {π j , ρ j } (consisting of a probability distribution π j and set of density matrices ρ j ), of the quantitywhere ρ = j π j ρ j is the average input, and S(γ) = −Trγ log γ denotes the von Neumann entropy. Thus, C(Φ) = sup E χ(E). It has been shown [4,10] that C(Φ) is the maximum information carrying capacity of a channel restricted to product inputs, but permitting entangled measurements. C(Φ) ≥ C Shan (Φ), where the classical Shannon capacity C Shan (Φ) describes the information carrying capacity of a channel when output measurements (as well as input ensembles) are restricted to products. (See, e.g., [2,5,7] for precise definitions.) We will sometimes use subscripts to denote the supremum of (1) restricted to a particular class of ensembles; in particular, we write C n (Φ) to denote the restriction to ensembles of n states. It is well-known [1] that for qubit channels the maximum can be achieved with an ensemble containing at most four states.In general, a qubit channel maps the Bloch sphere to an ellipsoid [6,9]. If the channel is unital, i.e, if Φ(I) = I, then the ellipsoid is centered at the origin, and the capacity is achieved with a pair of orthogonal inputs whose images, which are also states of minimal entropy [6], are the endpoints of the major axis of the ellipsoid. For nonunital channels the ellipsoid is displaced from the origin, and examples are known [2,11] for which the capacity is achieved with two non-orthogonal inputs which are not mapped onto states of minimal entropy. Here we present * king@neu.edu † ruskai@mediaone.net examples of non-unital channels which require three input states to achieve capacity. Furthermore these channels are non-extreme points in the set of all qubit channels, and this property is essential to our construction. To motivate our strategy, we consider the capacity of two well-known channels which have rotational symmetry about an axis of the Bloch sphere, and we maximize (1) over ensembles consisting of a pair of states on a line either parallel or orthogonal to this axis.First, let Φ D denote the shifted depolarizing channel which contracts the Bloch sphere to a sphere of radius µ and shifts it up until it touches the unit sphere, i.e, if ρ is written in the formWhen µ = 0.5, the image of the Bloch sphere satisfiesas shown in Fig. 1, and the poles ρ = 1 2 I ± σ 3 are mapped to the states 1 2 I + σ 3 and 1 2 I which have entropy 0 and 1 respectively (using base 2 for logarithms). If we restrict the input ensemble to a vertical line, the χ-quantity (1) is maximized by a convex combination of the poles, so that the average output state Φ D (ρ) = 1 2 I + zσ 3 lies on the z-axis. One finds that this vertical capacity, which we denote C V , is achieved when the output average is at z = 0.6 and its value is(This is also the capacity of a quantum-classical channel Φ QC , ...

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Information and quantum measurement

Cited by 257 publications

References 8 publications

Fundamental limits upon the measurement of state vectors

Fundamental limits upon the measurement of state vectors

Highly symmetric POVMs and their informational power

Qubit Channels Can Require More Than Two Inputs to Achieve Capacity

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