Quantum measurements and entropic bounds

2006 ◽  
Vol 6 (1) ◽  
pp. 16-45
Author(s):  
A. Barchielli ◽  
G. Lupieri
Keyword(s):  
Quantum Measurement ◽  
Positive Operator ◽  
Upper Bound ◽  
Quantum Channel ◽  
Quantum Measurements ◽  
A Posteriori ◽  
Classical Information ◽  
Positive Operator Valued Measure

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 (the bounds by Holevo, by Shumacher, Westmoreland and Wootters, by Hall, by Scutaru, a new upper bound and a new lower one). Some examples clarify the mutual relationships among the various bounds.

scholarly journals Self-testing nonprojective quantum measurements in prepare-and-measure experiments

2020 ◽  
Vol 6 (16) ◽  
pp. eaaw6664 ◽  
Author(s):  
Armin Tavakoli ◽  
Massimiliano Smania ◽  
Tamás Vértesi ◽  
Nicolas Brunner ◽  
Mohamed Bourennane
Keyword(s):  
Experimental Data ◽  
Hilbert Space ◽  
Quantum System ◽  
Positive Operator ◽  
Space Dimension ◽  
Quantum Measurements ◽  
Quantum Devices ◽  
Self Testing ◽  
Robust To Noise ◽  
Positive Operator Valued Measure

Self-testing represents the strongest form of certification of a quantum system. Here, we theoretically and experimentally investigate self-testing of nonprojective quantum measurements. That is, how can one certify, from observed data only, that an uncharacterized measurement device implements a desired nonprojective positive-operator valued measure (POVM). We consider a prepare-and-measure scenario with a bound on the Hilbert space dimension and develop methods for (i) robustly self-testing extremal qubit POVMs and (ii) certifying that an uncharacterized qubit measurement is nonprojective. Our methods are robust to noise and thus applicable in practice, as we demonstrate in a photonic experiment. Specifically, we show that our experimental data imply that the implemented measurements are very close to certain ideal three- and four-outcome qubit POVMs and hence non-projective. In the latter case, the data certify a genuine four-outcome qubit POVM. Our results open interesting perspective for semi–device-independent certification of quantum devices.

REMOTE INFORMATION CONCENTRATION VIA FOUR-PARTICLE CLUSTER STATE AND BY POSITIVE OPERATOR-VALUE MEASUREMENT

2013 ◽  
Vol 27 (18) ◽  
pp. 1350091 ◽  
Author(s):  
JIA-YIN PENG ◽  
MING-XING LUO ◽  
ZHI-WEN MO
Keyword(s):  
Quantum Information ◽  
Positive Operator ◽  
Quantum Channel ◽  
Cluster State ◽  
Projective Measurement ◽  
Particle Cluster ◽  
Single Qubit ◽  
Global Operations ◽  
Value Measurement ◽  
Positive Operator Valued Measure

By using a proper positive operator-valued measure (POVM), we present two new schemes for remote information concentration via four-particle cluster state. It is demonstrated that by employing a maximally entangled (respectively, nonmaximally entangled) four-particle cluster state as the quantum channel, quantum information initially and generally (respectively, specially) distributed in three spatially separated qubits can be remotely and probabilistically concentrated back to a single qubit without performing any global operations. Moreover, the total success probabilities of these two protocols are also worked out. It is more easier to realize experimentally by POVM than by projective measurement.

scholarly journals Uncertainty Relation for Errors Focusing on General POVM Measurements with an Example of Two-State Quantum Systems

Entropy ◽  
2020 ◽  
Vol 22 (11) ◽  
pp. 1222
Author(s):  
Jaeha Lee ◽  
Izumi Tsutsui
Keyword(s):  
Quantum Measurement ◽  
Uncertainty Principle ◽  
Positive Operator ◽  
Pure State ◽  
Uncertainty Relation ◽  
Measure Theory ◽  
Quantum Systems ◽  
Quantum Measurements ◽  
Positive Operator Valued Measures ◽  
Informative Case

A novel uncertainty relation for errors of general quantum measurement is presented. The new relation, which is presented in geometric terms for maps representing measurement, is completely operational and can be related directly to tangible measurement outcomes. The relation violates the naïve bound ℏ/2 for the position-momentum measurement, whilst nevertheless respecting Heisenberg’s philosophy of the uncertainty principle. The standard Kennard–Robertson uncertainty relation for state preparations expressed by standard deviations arises as a corollary to its special non-informative case. For the measurement on two-state quantum systems, the relation is found to offer virtually the tightest bound possible; the equality of the relation holds for the measurement performed over every pure state. The Ozawa relation for errors of quantum measurements will also be examined in this regard. In this paper, the Kolmogorovian measure-theoretic formalism of probability—which allows for the representation of quantum measurements by positive-operator valued measures (POVMs)—is given special attention, in regard to which some of the measure-theory specific facts are remarked along the exposition as appropriate.

scholarly journals Decoherence and its role in the modern measurement problem

2012 ◽  
Vol 370 (1975) ◽  
pp. 4576-4593 ◽  
Author(s):  
David Wallace
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Quantum Measurement ◽  
Measurement Problem ◽  
Scientific Practice ◽  
Measurement Theory ◽  
Quantum Measurements ◽  
Quantum Measurement Problem ◽  
Modern Physics ◽  
Conceptual Problems

Decoherence is widely felt to have something to do with the quantum measurement problem, but getting clear on just what is made difficult by the fact that the ‘measurement problem’, as traditionally presented in foundational and philosophical discussions, has become somewhat disconnected from the conceptual problems posed by real physics. This, in turn, is because quantum mechanics as discussed in textbooks and in foundational discussions has become somewhat removed from scientific practice, especially where the analysis of measurement is concerned. This paper has two goals: firstly (§§1–2), to present an account of how quantum measurements are actually dealt with in modern physics (hint: it does not involve a collapse of the wave function) and to state the measurement problem from the perspective of that account; and secondly (§§3–4), to clarify what role decoherence plays in modern measurement theory and what effect it has on the various strategies that have been proposed to solve the measurement problem.

Optimal quantum measurements for spin-1 and spin-3/2 particles

2001 ◽  
Vol 1 (3) ◽  
pp. 52-61
Author(s):  
P Aravind
Keyword(s):  
Positive Operator ◽  
Pure State ◽  
Quantum Measurements ◽  
Positive Operator Valued Measures ◽  
Unknown State

Positive operator valued measures (POVMs) are presented that allow an unknown pure state of a spin-1 particle to be determined with optimal fidelity when 2 to 5 copies of that state are available. Optimal POVMs are also presented for a spin-3/2 particle when 2 or 3 copies of the unknown state are available. Although these POVMs are optimal they are not always minimal, indicating that there is room for improvement.

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