Notes on Joint Measurability of Quantum Observables

Heinosaari, Teiko; Reitzner, Daniel; Stano, Peter

doi:10.1007/s10701-008-9256-7

Cited by 92 publications

(163 citation statements)

References 15 publications

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“…For more observables or outcomes the set of Bell inequalities becomes fairly monstrous and is largely unexplored. Unfortunately, a similar reduction to the CHSH-case is not always possible for POVMs: incompatibility of observables can in this case be 'overlooked' if one only considers pairwise incompatibility of several observables or if one groups measurement outcomes together [17]. Hence, for the cases of POVMs (with more than two outcomes or settings) where this happens the question is still open.…”

Section: Discussionmentioning

confidence: 99%

Measurements Incompatible in Quantum Theory Cannot Be Measured Jointly in Any Other No-Signaling Theory

Wolf¹,

2009

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It is well known that jointly measurable observables cannot lead to a violation of any Bell inequalityindependent of the state and the measurements chosen at the other site. In this letter we prove the converse: every pair of incompatible quantum observables enables the violation of a Bell inequality and therefore must remain incompatible within any other no-signaling theory. While in the case of von Neumann measurements it is sufficient to use the same pair of observables at both sites, general measurements can require different choices. The main result is obtained by showing that for arbitrary dimension the CHSH inequality provides the Lagrangian dual of the characterization of joint measurability. This leads to a simple criterion for joint measurability beyond the known qubit case."While [...] More than seventy years after Einstein, Podolsky and Rosen (EPR) raised this puzzle we know, as a consequence of Bell's argument [2], that a complete theory in the sense of EPR would force us to pay a high price-such as giving up Einstein locality. Could there, however, be a theory which provides more information than quantum mechanics but still is 'incomplete enough' to circumvent such fundamental conflicts? In this work we address a particular instance of this question, in the context of which the answer is clearly negative.We consider observables which are not jointly measurable, i.e., incompatible within quantum mechanics and show that they all enable the violation of a Bell inequality. That is, there exists a bipartite quantum state and a set of observables for an added site together with which the given observables violate a Bell inequality. As a consequence the observed probabilities do not admit a joint distribution [3] unless this depends on the observable chosen at the added site, which conflicts with Einstein locality, i.e., the no-signaling condition (see appendix). So, if a hypothetical no-signaling theory is a refinement of quantum mechanics (but otherwise consistent with it [19]), it can not render possible the joint measurability of observables which are incompatible within quantum mechanics-even if these observables are already almost jointly measurable in quantum theory.An enormous amount of work has been done in related directions: Bell inequalities [4] and no-signaling theories [5] are lively fields of research. It is well known (and used in constructing quantum states admitting a local hidden variable description [6,7]) that jointly measurable quantum observables can never lead to a violation of a Bell inequality [3]. The converse, however, has hardly been addressed. For generalized measurements (POVMs) this might partly be due to the fact that no criterion for joint measurability is known beyond twolevel systems, for which it was derived only recently [8]. A first indication of the present result can be found in [9] where it has been observed that for particular two-level observables the border of joint measurability [10] coincides with the one for the violation of the CHSH Bell inequality [11]. ...

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Section: Discussionmentioning

confidence: 99%

Measurements Incompatible in Quantum Theory Cannot Be Measured Jointly in Any Other No-Signaling Theory

Wolf¹,

2009

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“…We call the maps Ψ (1) and Ψ (2) as the marginals of Ψ. Suppose that Φ 1 ∈ CP P (A; H) and Φ 2 ∈ CP P (B; H).…”

Section: Marginal Maps and Joint Mapsmentioning

confidence: 99%

“…Suppose that Ψ ∈ CP P (A ⊗ B; H) and (M, π, J) is a minimal Stinespring dilation of the first marginal Ψ (1) . There is a unique map E ∈ CP ½ (B; M) such that [π(a), E(b)] = 0 for all a ∈ A, b ∈ B and…”

Section: Marginal Maps and Joint Mapsmentioning

confidence: 99%

When do pieces determine the whole? Extreme marginals of a completely positive map

2014

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Abstract. We will consider completely positive maps defined on tensor products of von Neumann algebras and taking values in the algebra of bounded operators on a Hilbert space and particularly certain convex subsets of the set of such maps. We show that when one of the marginal maps of such a map is an extremal point, then the marginals uniquely determine the map. We will further prove that when both of the marginals are extremal, then the whole map is extremal. We show that this general result is the common source of several well-known results dealing with, e.g., jointly measurable observables. We also obtain new insight especially in the realm of quantum instruments and their marginal observables and channels.

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“…A finite set of measurements is called jointly measurable or compatible if there exists a single measurement whose various coarse-grainings recover the original measurements. The problem of characterizing the joint measurability of observables has been studied in the litera-ture [7,8], and at least the joint measurability of binary qubit observables has been completely characterized [9,10]. The connection between Bell inequality violations and the joint measurability of observables has also been quantitatively studied [11,12].…”

Section: Introductionmentioning

confidence: 99%

Quantum realization of arbitrary joint measurability structures

2014

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In many a traditional physics textbook, a quantum measurement is defined as a projective measurement represented by a Hermitian operator. In quantum information theory, however, the concept of a measurement is dealt with in complete generality and we are therefore forced to confront the more general notion of positive-operator valued measures (POVMs) which suffice to describe all measurements that can be implemented in quantum experiments. We study the (in)compatibility of such POVMs and show that quantum theory realizes all possible (in)compatibility relations among sets of POVMs. This is in contrast to the restricted case of projective measurements for which commutativity is essentially equivalent to compatibility. Our result therefore points out a fundamental feature with respect to the (in)compatibility of quantum observables that has no analog in the case of projective measurements.

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Notes on Joint Measurability of Quantum Observables

Cited by 92 publications

References 15 publications

Measurements Incompatible in Quantum Theory Cannot Be Measured Jointly in Any Other No-Signaling Theory

Measurements Incompatible in Quantum Theory Cannot Be Measured Jointly in Any Other No-Signaling Theory

When do pieces determine the whole? Extreme marginals of a completely positive map

Quantum realization of arbitrary joint measurability structures

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