Highly symmetric POVMs and their informational power

Słomczyński, Wojciech; Szymusiak, Anna

doi:10.1007/s11128-015-1157-z

Cited by 21 publications

(36 citation statements)

References 123 publications

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“…Notice that, as expected, the capacities in Theorem 2 reduce to those given in Refs. [23,[27][28][29][30][31][32]45] for projective t designs when one takes the limit λ → 1.…”

Section: Resultsmentioning

confidence: 99%

“…For the completeness of our reasoning we add a brief description of the optimization method based on Hermite interpolation, first introduced in this context and discussed in detail in Ref. [27]. Let us first recall the well known formula for the Hermite interpolation error.…”

Section: Appendix A: Hermite Interpolationmentioning

confidence: 99%

“…Therein, the simple case of SIC measurement of a qubit was solved. Subsequently, considerable interest arose for computing the capacity of several symmetric measurements [24][25][26][27][28][29][30][31][32]. With very limited exceptions (e.g.…”

Section: Introductionmentioning

confidence: 99%

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Communication capacity of mixed quantumt-designs

2016

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We operationally introduce mixed quantum t designs as the most general arbitrary-rank extension of projective quantum t designs which preserves indistinguishability from the uniform distribution for t copies. First, we derive upper bounds on the classical communication capacity of any mixed t design measurement, for t ∈ [1, 5]. Second, we explicitly compute the classical communication capacity of several mixed t design measurements, including the depolarized version of: any qubit and qutrit symmetric, informationally complete (SIC) measurement and complete mutually unbiased bases (MUB), the qubit icosahedral measurement, the Hoggar SIC measurement, any anti-SIC (where each element is proportional to the projector on the subspace orthogonal to one of the elements of the original SIC), and the uniform distribution over pure effects.

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“…Notice that, as expected, the capacities in Theorem 2 reduce to those given in Refs. [23,[27][28][29][30][31][32]45] for projective t designs when one takes the limit λ → 1.…”

Section: Resultsmentioning

confidence: 99%

Section: Appendix A: Hermite Interpolationmentioning

confidence: 99%

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Communication capacity of mixed quantumt-designs

2016

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“…for an input state ρ ∈ S C d ; see [45], [56] for the history and information-theoretic interpretation of this notion. If ρ = |ψ ψ| ∈ P C d , we put H(|ψ , Π) := H(ρ, Π).…”

Section: Entropy Of Measurement and Povm-entropymentioning

confidence: 99%

Quantum Dynamical Entropy, Chaotic Unitaries and Complex Hadamard Matrices

Słomczyński

Szczepanek

2017

IEEE Trans. Inform. Theory

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We introduce two information-theoretical invariants for the projective unitary group acting on a finite-dimensional complex Hilbert space: PVM-and POVM-dynamical (quantum) entropies, which are analogues of the classical Kolmogorov-Sinai entropy rate. They quantify the maximal randomness of the successive quantum measurement results in the case where the evolution of the system between each two consecutive measurements is described by a given unitary operator. We study the class of chaotic unitaries, i.e., the ones of maximal entropy, or equivalently, such that they can be represented by suitably rescaled complex Hadamard matrices in some orthonormal bases. We provide necessary conditions for a unitary operator to be chaotic, which become also sufficient for qubits and qutrits. These conditions are expressed in terms of the relation between the trace and the determinant of the operator. We also compute the volume of the set of chaotic unitaries in dimensions two and three, and the average PVM-dynamical entropy over the unitary group in dimension two. We prove that this mean value behaves as the logarithm of the dimension of the Hilbert space, which implies that the probability that the dynamical entropy of a unitary is almost as large as possible approaches unity as the dimension tends to infinity.

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“…symmetric informationally complete (SIC) measurements [9] and mutually unbiased bases [10] (MUB), both in the real and complex cases, and in the presence of isotropic noise. SIC measurements play a fundamental role in quantum tomography [4][5][6], quantum communication [11][12][13][14][15][16][17], and foundations of quantum theory [18][19][20][21][22], while MUBs are pivotal elements in quantum cryptography [23], entropic uncertainty relations [24][25][26], and locking of classical information in quantum states [27].…”

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confidence: 99%

Device-Independent Tests of Quantum Measurements

et al. 2017

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We consider the problem of characterizing the set of input-output correlations that can be generated by an arbitrarily given quantum measurement. Our main result is to provide a closed-form, full characterization of such a set for any qubit measurement, and to discuss its geometrical interpretation. As applications, we further specify our results to the cases of real and complex symmetric, informationally complete measurements and mutually unbiased bases of a qubit, in the presence of isotropic noise. Our results provide the optimal device-independent tests of quantum measurements. DOI: 10.1103/PhysRevLett.118.250501 In operational quantum theory, it is a natural question to ask whether a given data sample, provided in the form of a conditional probability distribution representing the measured input-output correlation, is compatible with a particular hypothesis about the theoretical model underlying the experiment. A theoretical model can be more or less specific: for example, it could consist only of a general hypothesis about the theory describing the physics, as in the case in Bell tests [1][2][3], or it could be extremely detailed, as in the case of a tomographic reconstruction [4][5][6]. More generally, a hypothesis could be specific about a portion of the underlying model, while leaving the remaining elements completely uncharacterized.Here, we address hypotheses about the measurement producing the final outcomes of the experiment. This is the problem of characterizing the set SðπÞ of all input-output correlations p yjx ¼ Tr½ρ x π y compatible with an arbitrarily given quantum measurement π ≔ fπ y g (i.e., the hypothesis) and any family of input quantum states fρ x g, namely,We first note that a correlation p is compatible with measurement π if and only if for any fixed x there exists a state ρ x such that p yjx ¼ Tr½ρ x π y . Hence, SðπÞ is fully characterized by the range of π, namely, the set S 1 ðπÞ of output distributions q y ¼ Tr½ρπ y generated by π for varying input state ρ. This is in stark contrast with the analogous problem of characterizing the set of correlations compatible with a given quantum channel, which in general requires more than one input [7].Our main result is a closed-form characterization of the range S 1 ðπÞ, and hence of the set SðπÞ of all compatible correlations, when the hypothesis π is a qubit measurement. It turns out that an output distribution q y belongs to S 1 ðπÞ if and only ifwhere 1 is the identity matrix, t is the vector t y ≔ . Imposing the system of equalities in Eq. (2) causes linear dependencies-if any-among measurement elements π y to emerge as linear constraints on the probabilities. Such linear dependencies are present, for example, in any overcomplete measurement. Provided that these constraints are satisfied, the inequality in Eq. (2) recasts-through the transformation Q þ -the set of distributions compatible with π as an ellipsoid centered on distribution t. This in particular provides a simple and clear geometrical representation for the range of any q...

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Highly symmetric POVMs and their informational power

Cited by 21 publications

References 123 publications

Communication capacity of mixed quantumt-designs

Communication capacity of mixed quantumt-designs

Quantum Dynamical Entropy, Chaotic Unitaries and Complex Hadamard Matrices

Device-Independent Tests of Quantum Measurements

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