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Register a new userCoexistence and Joint Measurability in Quantum Mechanics
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Pekka Lahti
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This talk is a survey of the question of joint measurability of coexistent observables and its is based on the monograph Operational Quantum Physics [1] and on the papers [2,3,4].
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Occupying a position between entanglement and Bell nonlocality, Einstein-Podolsky-Rosen (EPR) steering has attracted increasing attention in recent years. Many criteria have been proposed and experimentally implemented to characterize EPR-steering. Nevertheless, only a few results are available to quantify steerability using analytical results. In this work, we propose a method for quantifying the steerability in two-qubit quantum states in the two-setting EPR-steering scenario, using the connection between joint measurability and steerability. We derive an analytical formula for the steerability of a class of X-states. The sufficient and necessary conditions for two-setting EPRsteering are presented. Based on these results, a class of asymmetric states, namely, one-way steerable states, are obtained.
In this work, we investigate the joint measurability of quantum effects and connect it to the study of free spectrahedra. Free spectrahedra typically arise as matricial relaxations of linear matrix inequalities. An example of a free spectrahedron is the matrix diamond, which is a matricial relaxation of the $ell_1$-ball. We find that joint measurability of binary POVMs is equivalent to the inclusion of the matrix diamond into the free spectrahedron defined by the effects under study. This connection allows us to use results about inclusion constants from free spectrahedra to quantify the degree of incompatibility of quantum measurements. In particular, we completely characterize the case in which the dimension is exponential in the number of measurements. Conversely, we use techniques from quantum information theory to obtain new results on spectrahedral inclusion for the matrix diamond.
In order to analyze joint measurability of given measurements, we introduce a Hermitian operator-valued measure, called $W$-measure, such that it has marginals of positive operator-valued measures (POVMs). We prove that ${W}$-measure is a POVM {em if and only if} its marginal POVMs are jointly measurable. The proof suggests to employ the negatives of ${W}$-measure as an indicator for non-joint measurability. By applying triangle inequalities to the negativity, we derive joint measurability criteria for dichotomic and trichotomic variables. Also, we propose an operational test for the joint measurability in sequential measurement scenario.
The notion coexistence of quantum observables was introduced to describe the possibility of measuring two or more observables together. Here we survey the various different formalisations of this notion and their connections. We review examples illustrating the necessary degrees of unsharpness for two noncommuting observables to be jointly measurable (in one sense of the phrase). We demonstrate the possibility of measuring together (in another sense of the phrase) noncoexistent observables. This leads us to a reconsideration of the connection between joint measurability and noncommutativity of observables and of the statistical and individual aspects of quantum measurements.
Quantum measurements can be interpreted as a generalisation of probability vectors, in which non-negative real numbers are replaced by positive semi-definite operators. We extrapolate this analogy to define a generalisation of doubly stochastic matrices that we call doubly normalised tensors (DNTs), and formulate a corresponding version of Birkhoff-von Neumanns theorem, which states that permutations are the extremal points of the set of doubly stochastic matrices. We prove that joint measurability arises as a mathematical feature of DNTs in this context, needed to establish a characterisation similar to Birkhoff-von Neumanns. Conversely, we also show that DNTs emerge naturally from a particular instance of a joint measurability problem, remarking its relevance in general operator theory.
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