Minimal sufficient positive-operator valued measure on a separable Hilbert space

Kuramochi, Yui

doi:10.1063/1.4934235

Cited by 19 publications

(32 citation statements)

References 20 publications

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“…9 Recently, the author has introduced the concept of the minimal sufficient POVM, which is the least redundant POVM among the POVMs that bring us the same information about the measured quantum system. 10 In Ref. 10 it is shown that any POVM on a separable Hilbert space is postprocessing equivalent to a minimal sufficient POVM unique up to almost isomorphism.…”

Section: Introductionmentioning

confidence: 99%

“…10 In Ref. 10 it is shown that any POVM on a separable Hilbert space is postprocessing equivalent to a minimal sufficient POVM unique up to almost isomorphism. Then it is natural to ask whether we can generalize the notion of minimal sufficiency to noncommutative statistical experiments and whether we can establish existence and uniqueness up to isomorphism for such general statistical experiments as in the case of POVM.…”

Section: Introductionmentioning

confidence: 99%

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Minimal sufficient statistical experiments on von Neumann algebras

Kuramochi

2017

Journal of Mathematical Physics

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A statistical experiment on a von Neumann algebra is a parametrized family of normal states on the algebra. This paper introduces the concept of minimal sufficiency for statistical experiments in such operator algebraic situations. We define equivalence relations of statistical experiments indexed by a common parameter set by completely positive or Schwarz coarse-graining and show that any statistical experiment is equivalent to a minimal sufficient statistical experiment unique up to normal isomorphism of outcome algebras. We also establish the relationship between the minimal sufficiency condition for statistical experiment in this paper and those for subalgebra. These concepts and results are applied to the concatenation relation for completely positive channels with general input and outcome von Neumann algebras.In the case of the quantum-classical channel corresponding to the positive-operator valued measure (POVM), we prove the equivalence of the minimal sufficient condition previously proposed by the author and that in this paper. We also give a characterization of the discreteness of a POVM up to postprocessing equivalence in terms of the corresponding quantum-classical channel.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Minimal sufficient statistical experiments on von Neumann algebras

Kuramochi

2017

Journal of Mathematical Physics

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“…An observable A with an outcome set Ω is said to be pairwise linearly independent if any pair (A(x 1 ), A(x 2 )), x 1 , x 2 ∈ Ω; x 1 = x 2 , is linearly independent. Every observable is post-processing equivalent to a pairwise linearly independent observable unique up to the permutation of the outcome set [11]. An observable A is pairwise linearly independent if and only if A is minimal sufficient, that is, for any Markov kernel p ∈ Markov(Ω, Ω) the condition p * A = A implies p(x, x ) = δ x,x [11].…”

Section: Appendix B Pairwise Linearly Independent Observablesmentioning

confidence: 99%

“…Every observable is post-processing equivalent to a pairwise linearly independent observable unique up to the permutation of the outcome set [11]. An observable A is pairwise linearly independent if and only if A is minimal sufficient, that is, for any Markov kernel p ∈ Markov(Ω, Ω) the condition p * A = A implies p(x, x ) = δ x,x [11]. We recall that two Markov kernels p ∈ Markov(Ω 1 , Ω 2 ) and q ∈ Markov(Ω 2 , Ω 3 ) can be combined into a new Markov kernel as follows:…”

Section: Appendix B Pairwise Linearly Independent Observablesmentioning

confidence: 99%

Post-processing minimal joint observables

Heinosaari¹,

Kuramochi²

2019

J. Phys. A: Math. Theor.

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A finite set of quantum observables (positive operator valued measures) is called compatible if these observables are marginals of a some observable, called a joint observable of them. For a given set of compatible observables, their joint observable is in general not unique and it is desirable to take a minimal joint observable in the post-processing order since a less informative observable disturbs less the system. We address the question of the minimality of finite-outcome joint observables and prove that any joint observable is lower bounded by a minimal joint observable in the post-processing order. We also give characterizations of the minimality of a joint observable that can be checked by finitestep algorithms and apply them to the case of non-commuting dichotomic qubit observables.

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“…i.e., the relabeling mediated by f takes B into A B. In statistical terms, this means that this coarse-graining (statistic) is sufficient for B and, according to [14,Prop. 5], there is a function h :…”

Section: Define the Observable B : ωmentioning

confidence: 99%

The unavoidable information flow to environment in quantum measurements

Haapasalo

Heinosaari

Miyadera

2018

Journal of Mathematical Physics

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One of the basic lessons of quantum theory is that one cannot obtain information on an unknown quantum state without disturbing it. Hence, by performing a certain measurement, we limit the other possible measurements that can be effectively implemented on the original input state. It has been recently shown that one can implement sequentially any device, either channel or observable, which is compatible with the first measurement [8]. In this work we prove that this can be done, apart from some special cases, only when the succeeding device is implemented on a larger system than just the input system. This means that some part of the still available quantum information has been flown to the environment and cannot be gathered by accessing the input system only. We characterize the size of the post-measurement system by determining the class of measurements for the observable in question that allow the subsequent realization of any measurement process compatible with the said observable. We also study the class of measurements that allow the subsequent realization of any observable jointly measurable with the first one and show that these two classes coincide when the first observable is extreme. arXiv:1710.08681v1 [quant-ph]

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Minimal sufficient positive-operator valued measure on a separable Hilbert space

Cited by 19 publications

References 20 publications

Minimal sufficient statistical experiments on von Neumann algebras

Minimal sufficient statistical experiments on von Neumann algebras

Post-processing minimal joint observables

The unavoidable information flow to environment in quantum measurements

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