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Register a new userJoint measurability of quantum effects and the matrix diamond
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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.
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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].
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.
We associate with k hermitian Ntimes N matrices a probability measure on R^k. It is supported on the joint numerical range of the k-tuple of matrices. We call this measure the joint numerical shadow of these matrices. Let k=2. A pair of hermitian Ntimes N matrices defines a complex Ntimes N matrix. The joint numerical range and the joint numerical shadow of the pair of hermitian matrices coincide with the numerical range and the numerical shadow, respectively, of this complex matrix. We study relationships between the dynamics of quantum maps on the set of quantum states, on one hand, and the numerical ranges, on the other hand. In particular, we show that under the identity resolution assumption on Kraus operators defining the quantum map, the dynamics shrinks numerical ranges.
In this work, we establish the connection between the study of free spectrahedra and the compatibility of quantum measurements with an arbitrary number of outcomes. This generalizes previous results by the authors for measurements with two outcomes. Free spectrahedra arise from matricial relaxations of linear matrix inequalities. A particular free spectrahedron which we define in this work is the matrix jewel. We find that the compatibility of arbitrary measurements corresponds to the inclusion of the matrix jewel into a free spectrahedron defined by the effect operators of the measurements under study. We subsequently use this connection to bound the set of (asymmetric) inclusion constants for the matrix jewel using results from quantum information theory and symmetrization. The latter translate to new lower bounds on the compatibility of quantum measurements. Among the techniques we employ are approximate quantum cloning and mutually unbiased bases.
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.
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