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Tight informationally complete quantum measurements

Scott, A J

Published Apr 7, 2006• 168 citations
OpenAlex

Abstract

We introduce a class of informationally complete positive-operator-valued measures which are, in analogy with a tight frame, "as close as possible" to orthonormal bases for the space of general quantum states. Complete sets of mutually unbiased bases and symmetric informationally complete positive-operator-valued measures are both members of this class, the latter being the unique minimal members. Such measures share a particularly simple state reconstruction formula, allowing painless quantum state tomography, and are shown to be optimal for quantum cloning.

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Tight informationally complete quantum measurements

A J Scott

We introduce a class of informationally complete positive-operator-valued measures which are, in analogy with a tight frame, ‘as close as possible’ to orthonormal bases for the space of quantum states. These measures are distinguished by an exceptionally simple state-reconstruction formula which allows ‘painless’ quantum state tomography. Complete sets of mutually unbiased bases and symmetric informationally complete positive-operator-valued measures are both members of this class, the latter being the unique minimal rank-one members. Recast as ensembles of pure quantum states, the rank-one members are in fact equivalent to weighted 2-designs in complex projective space. These measures are shown to be optimal for quantum cloning and linear quantum state tomography.

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Abstract Informationally overcomplete measurements find important applications in quantum tomography and quantum state estimation. The most popular are maximal sets of mutually unbiased bases, for which trace relations between measurement operators are well known. In this paper, we introduce a more general class of informationally overcomplete positive, operator-valued measure (POVMs) that are generated by equiangular tight frames of arbitrary rank. This class provides a generalization of equiangular measurements to non-projective POVMs, which include rescaled mutually unbiased measurements and bases. We provide a method of their construction, analyze their symmetry properties, and provide examples for highly symmetric cases. In particular, we find a wide class of generalized equiangular measurements that are conical two-designs, which allows us to derive the index of coincidence. Our results show benefits of considering a single informationally overcomplete measurement over informationally complete collections of POVMs.

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Experimental Realization of Quantum Tomography of Photonic Qudits via Symmetric Informationally Complete Positive Operator-Valued Measures

Nicolas Bent, Hammam Qassim, A. A. Tahir +5 more

Symmetric informationally complete positive operator-valued measures provide efficient quantum state tomography in any finite dimension.In this work, we implement state tomography using symmetric informationally complete positive operator-valued measures for both pure and mixed photonic qudit states in Hilbert spaces of orbital angular momentum, including spaces whose dimension is not power of a prime.Fidelities of reconstruction within the range of 0.81-0.96are obtained for both pure and mixed states.These results are relevant to high-dimensional quantum information and computation experiments, especially to those where a complete set of mutually unbiased bases is unknown.

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Discrete Wigner functions from informationally complete quantum measurements

John B. DeBrota, Blake C. Stacey

Wigner functions provide a way to do quantum physics using quasiprobabilities, that is, ``probability'' distributions that can go negative. Informationally complete positive operator-valued measures, a much younger subject than phase-space formulations of quantum mechanics, are less familiar but provide wholly probabilistic representations of quantum theory. In this paper, we show that the Born rule links these two classes of structure and discuss the art of interconverting between them. In particular, we demonstrate that the operator bases corresponding to minimal discrete Wigner functions (Wigner bases) are orthogonalizations of minimal informationally complete measurements (MICs). By not imposing a particular discrete phase-space structure at the outset, we push Wigner functions to their limits in a suitably quantified sense, revealing an extremality condition satisfied by the symmetric informationally complete measurements (SICs). This contributes another member to the set of optimality criteria that SICs are known to satisfy. Finally, we speculate that astute choices of MICs from the orthogonalization preimages of Wigner bases may in general give quantum measurements that are conceptually illuminating for the quasiprobability representations defined by those Wigner bases.

arXiv:1912.07554•Physical Review A•Sep 28, 2020•21 citations
We introduce a class of informationally complete positive-operator-valued measures which are, in analogy with a tight frame, "as close as possible" to orthonormal bases for the space of general quantum states. Complete sets of mutually unbiased bases and symmetric informationally complete positive-operator-valued measures are both members of this class, the latter being the unique minimal members. Such measures share a particularly simple state reconstruction formula, allowing painless quantum state tomography, and are shown to be optimal for quantum cloning.
We introduce a class of informationally complete positive-operator-valued measures which are, in analogy with a tight frame, ‘as close as possible’ to orthonormal bases for the space of quantum states. These measures are distinguished by an exceptionally simple state-reconstruction formula which allows ‘painless’ quantum state tomography. Complete sets of mutually unbiased bases and symmetric informationally complete positive-operator-valued measures are both members of this class, the latter being the unique minimal rank-one members. Recast as ensembles of pure quantum states, the rank-one members are in fact equivalent to weighted 2-designs in complex projective space. These measures are shown to be optimal for quantum cloning and linear quantum state tomography.
Abstract Informationally overcomplete measurements find important applications in quantum tomography and quantum state estimation. The most popular are maximal sets of mutually unbiased bases, for which trace relations between measurement operators are well known. In this paper, we introduce a more general class of informationally overcomplete positive, operator-valued measure (POVMs) that are generated by equiangular tight frames of arbitrary rank. This class provides a generalization of equiangular measurements to non-projective POVMs, which include rescaled mutually unbiased measurements and bases. We provide a method of their construction, analyze their symmetry properties, and provide examples for highly symmetric cases. In particular, we find a wide class of generalized equiangular measurements that are conical two-designs, which allows us to derive the index of coincidence. Our results show benefits of considering a single informationally overcomplete measurement over informationally complete collections of POVMs.
A broad class of informationally complete symmetric measurements is introduced. It can be understood as a common generalization of symmetric, informationally complete positive operator-valued measures and mutually unbiased bases. Additionally, it provides a natural way to define two more families of mutually unbiased symmetric measurement operators in any finite dimension. We show a general method of their construction, together with an example of an optimal measurement. Finally, we analyze the properties of symmetric measurements and provide applications in entropic relations and entanglement detection.
In this work, the concept of mutually unbiased frames is introduced as the most general notion of unbiasedness for sets composed by linearly independent and normalized vectors. It encompasses the already existing notions of unbiasedness for orthonormal bases, regular simplices, equiangular tight frames, positive operator valued measure, and also includes symmetric informationally complete quantum measurements. After introducing the tool, its power is shown by finding the following results about the last mentioned class of constellations: (i) real fiducial states do not exist in any even dimension, and (ii) unknown d -dimensional fiducial states are parameterized, a priori, with roughly 3d/2 real variables only, without loss of generality. Furthermore, multi-parametric families of pure quantum states having minimum uncertainty with regard to several choices of d+1 orthonormal bases are shown, in every dimension d . These last families contain all existing fiducial states in every finite dimension, and the bases include maximal sets of d+1 mutually unbiased bases, when d is a prime number.
Symmetric informationally complete positive operator-valued measures provide efficient quantum state tomography in any finite dimension.In this work, we implement state tomography using symmetric informationally complete positive operator-valued measures for both pure and mixed photonic qudit states in Hilbert spaces of orbital angular momentum, including spaces whose dimension is not power of a prime.Fidelities of reconstruction within the range of 0.81-0.96are obtained for both pure and mixed states.These results are relevant to high-dimensional quantum information and computation experiments, especially to those where a complete set of mutually unbiased bases is unknown.
Wigner functions provide a way to do quantum physics using quasiprobabilities, that is, ``probability'' distributions that can go negative. Informationally complete positive operator-valued measures, a much younger subject than phase-space formulations of quantum mechanics, are less familiar but provide wholly probabilistic representations of quantum theory. In this paper, we show that the Born rule links these two classes of structure and discuss the art of interconverting between them. In particular, we demonstrate that the operator bases corresponding to minimal discrete Wigner functions (Wigner bases) are orthogonalizations of minimal informationally complete measurements (MICs). By not imposing a particular discrete phase-space structure at the outset, we push Wigner functions to their limits in a suitably quantified sense, revealing an extremality condition satisfied by the symmetric informationally complete measurements (SICs). This contributes another member to the set of optimality criteria that SICs are known to satisfy. Finally, we speculate that astute choices of MICs from the orthogonalization preimages of Wigner bases may in general give quantum measurements that are conceptually illuminating for the quasiprobability representations defined by those Wigner bases.