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We introduce the notion of compatibility dimension for a set of quantum measurements: it is the largest dimension of a Hilbert space on which the given measurements are compatible. In the Schrodinger picture, this notion corresponds to testing compatibility with ensembles of quantum states supported on a subspace, using the incompatibility witnesses of Carmeli, Heinosaari, and Toigo. We provide several bounds for the compatibility dimension, using approximate quantum cloning or algebraic techniques inspired by quantum error correction. We analyze in detail the case of two orthonormal bases, and, in particular, that of mutually unbiased bases.
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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.
We connect two key concepts in quantum information: compatibility and divisibility of quantum channels. Two channels are compatible if they can be both obtained via marginalization from a third channel. A channel divides another channel if it reproduces its action by sequential composition with a third channel. (In)compatibility is of central importance for studying the difference between classical and quantum dynamics. The relevance of divisibility stands in its close relationship with the onset of Markovianity. We emphasize the simulability character of compatibility and divisibility, and, despite their structural difference, we find a set of channels -- self-degradable channels -- for which the two notions coincide. We also show that, for degradable channels, compatibility implies divisibility, and that, for anti-degradable channels, divisibility implies compatibility. These results motivate further research on these classes of channels and shed new light on the meaning of these two largely studied notions.
Causality is a seminal concept in science: Any research discipline, from sociology and medicine to physics and chemistry, aims at understanding the causes that could explain the correlations observed among some measured variables. While several methods exist to characterize classical causal models, no general construction is known for the quantum case. In this work, we present quantum inflation, a systematic technique to falsify if a given quantum causal model is compatible with some observed correlations. We demonstrate the power of the technique by reproducing known results and solving open problems for some paradigmatic examples of causal networks. Our results may find applications in many fields: from the characterization of correlations in quantum networks to the study of quantum effects in thermodynamic and biological processes.
We consider an environment for an open quantum system described by a Quantum Network Geometry with Flavor (QNGF) in which the nodes are coupled quantum oscillators. The geometrical nature of QNGF is reflected in the spectral properties of the Laplacian matrix of the network which display a finite spectral dimension, determining also the frequencies of the normal modes of QNGFs. We show that an a priori unknown spectral dimension can be indirectly estimated by coupling an auxiliary open quantum system to the network and probing the normal mode frequencies in the low frequency regime. We find that the network parameters do not affect the estimate; in this sense it is a property of the network geometry, rather than the values of, e.g., oscillator bare frequencies or the constant coupling strength. Numerical evidence suggests that the estimate is also robust both to small changes in the high frequency cutoff and noisy or missing normal mode frequencies. We propose to couple the auxiliary system to a subset of network nodes with random coupling strengths to reveal and resolve a sufficiently large subset of normal mode frequencies.
The distinguishability of quantum states is important in quantum information theory and has been considered by authors. However, there were no general results considering whether a set of indistinguishable states become distinguishable by viewing them in a larger system without employing extra resources. In this paper, we consider this question for LOCC$_{1}$, PPT and SEP distinguishabilities of states. We use mathematical methods to show that if a set of states is indistinguishable in $otimes _{k=1}^{K} C^{d _{k}}$, then it is indistinguishable even being viewed in $otimes _{k=1}^{K} C^{d _{k}+h _{k}}$, where $K, d _{k}geqslant2$, $h _{k}geqslant0$ are integers. This shows that LOCC$_{1}$, PPT and SEP distinguishabilities of states are properties of states themselves and independent of the dimension of quantum system. With these results, we can give the maximal number of states which can be distinguished via LOCC$_{1}$ and construct a LOCC indistinguishable basis of product states in a general system. Note that our results are also suitable for unambiguous discriminations. Also, we give a conjecture for other distinguishabilities and a framework by defining the Local-global indistinguishable property. Instead of considering such problems for special sets or special systems, we consider the problems for general states in general systems, which have not been considered yet, for our knowledge.
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