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.

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 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.

scholarly journals Operational link between mutually unbiased bases and symmetric informationally complete positive operator-valued measures

2013 ◽  
Vol 88 (3) ◽  
Author(s):  
Roberto Beneduci ◽  
Thomas J. Bullock ◽  
Paul Busch ◽  
Claudio Carmeli ◽  
Teiko Heinosaari ◽  
...  
Keyword(s):  
Positive Operator ◽  
Mutually Unbiased Bases ◽  
Positive Operator Valued Measures

scholarly journals Probing qubit by qubit: Properties of the POVM and the information/disturbance tradeoff

2014 ◽  
Vol 12 (02) ◽  
pp. 1461012 ◽  
Author(s):  
Carlo Sparaciari ◽  
Matteo G. A. Paris
Keyword(s):  
Positive Operator ◽  
Projective Measurement ◽  
Matrix Elements ◽  
Positive Operator Valued Measures ◽  
Unitary Coupling ◽  
Probe System ◽  
Initial Preparation ◽  
Free Parameters

We address the class of positive operator-valued measures (POVMs) for qubit systems that are obtained by coupling the signal qubit with a probe qubit and then performing a projective measurement on the sole probe system. These POVMs, which represent the simplest class of qubit POVMs, depends on 3 + 3 + 2 = 8 free parameters describing the initial preparation of the probe qubit, the Cartan representative of the unitary coupling, and the projective measurement at the output, respectively. We analyze in some detail the properties of the POVM matrix elements, and investigate their values for given ranges of the free parameters. We also analyze in detail the tradeoff between information and disturbance for different ranges of the free parameters, showing, among other things, that (i) typical values of the tradeoff are close to optimality and (ii) even using a maximally mixed probe one may achieve optimal tradeoff.

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