Joint probability distributions for projection probabilities of random orthonormal states

Alonso, Lazaro; Gorin, T.

doi:10.1088/1751-8113/49/14/145004

Cited by 5 publications

(8 citation statements)

References 44 publications

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“…For general unitary transformations from SU (N ) we showed that the overlap distribution can be computed, in principle, following a method introduced in Ref. [14]. The solution is based on the calculation of the joint probability distribution (JPD) of partial overlaps, for which we could obtain a surprisingly simple analytical expression in the qutrit, i.e., SU (3), case.…”

Section: Discussionmentioning

confidence: 99%

“…We will closely follow the derivation for the unitary case in Ref. [14]. According to this work, Eq.…”

Section: Jpd For General Casementioning

confidence: 97%

“…The JPD P λ (t ) can be calculated by generalizing an earlier work on the projection probabilities of random orthonormal states [14].…”

Section: General Definitions and Notationmentioning

confidence: 99%

“…In Ref. [14], the distribution of sums of K such absolute values squared have been calculated as averages over U (N ), denoted by P N K (t…”

Section: General Definitions and Notationmentioning

confidence: 99%

“…For that, it is crucial to compute P (q) analytically, as this allows us to find convenient strategies to solve the inverse problem of estimating the eigenvalues. In order to compute P (q), we generalize previous results on the joint probability distribution of projection probabilities for random orthonormal quantum states [14]. We then concentrate on the qutrit case, where we obtain P (q) in closed form.…”

Section: Introductionmentioning

confidence: 99%

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Eigenvalue Determination for Mixed Quantum States using Overlap Statistics

Alonso,

Bermudez,

Gorin

2018

Preprint

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We consider the statistics of overlaps between a mixed state and its image under random unitary transformations. Choosing the transformations from the unitary group with its invariant (Haar) measure, the distribution of overlaps depends only on the eigenvalues of the mixed state. This allows one to estimate these eigenvalues from the overlap statistics. In the first part of this work, we present explicit results for qutrits, including a discussion of the expected uncertainties in the eigenvalue estimation. In the second part, we assume that the set of available unitary transformations is restricted to SO(3), realized as Wigner D-matrices. In that case, the overlap statistics does not depend only on the eigenvalues, but also on the eigenstates of the mixed state under scrutiny. The overlap distribution then shows a complicated pattern, which may be considered as a fingerprint of the mixed state. When using random transformations from the unitary group, the eigenvalues can be determined quite simply from the lower and the upper limit of the overlap statistics. This may still be possible in the SO(3) case, but only at the expense of a finite systematic uncertainty.

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Section: Discussionmentioning

confidence: 99%

“…We will closely follow the derivation for the unitary case in Ref. [14]. According to this work, Eq.…”

Section: Jpd For General Casementioning

confidence: 97%

“…The JPD P λ (t ) can be calculated by generalizing an earlier work on the projection probabilities of random orthonormal states [14].…”

Section: General Definitions and Notationmentioning

confidence: 99%

“…In Ref. [14], the distribution of sums of K such absolute values squared have been calculated as averages over U (N ), denoted by P N K (t…”

Section: General Definitions and Notationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Eigenvalue Determination for Mixed Quantum States using Overlap Statistics

Alonso,

Bermudez,

Gorin

2018

Preprint

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The statistics of spectral shifts due to finite rank perturbations

Dietz

Schanz²,

Smilansky³

et al. 2020

J. Phys. A: Math. Theor.

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This article is dedicated to the following class of problems. Start with an N × N Hermitian matrix randomly picked from a matrix ensemble—the reference matrix. Applying a rank-t perturbation to it, with t taking the values 1 ⩽ t ⩽ N, we study the difference between the spectra of the perturbed and the reference matrices as a function of t and its dependence on the underlying universality class of the random matrix ensemble. We consider both, the weaker kind of perturbation which either permutes or randomizes t diagonal elements and a stronger perturbation randomizing successively t rows and columns. In the first case we derive universal expressions in the scaled parameter τ = t/N for the expectation of the variance of the spectral shift functions, choosing as random-matrix ensembles Dyson’s three Gaussian ensembles. In the second case we find an additional dependence on the matrix size N.

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Beyond the Swap Test: Optimal Estimation of Quantum State Overlap

et al. 2020

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We study the estimation of the overlap between two Haar-random pure quantum states in a finite-dimensional Hilbert space, given M and N copies of them. We compute the statistics of the optimal measurement, which is a projection onto permutation-invariant subspaces, and provide lower bounds for the mean square error for both local and Bayesian estimation. In the former case, the bound is asymptotically saturable by a maximum likelihood estimator, whereas in the latter we give a simple exact formula for the optimal value. Furthermore, we introduce two LOCC strategies, relying on the estimation of one or both the states, and show that, although they are suboptimal, they outperform the commonly-used swap test. In particular, the swap test is extremely inefficient for small values of the overlap, which become exponentially more likely as the dimension increases. Finally, we show that the optimal measurement is less invasive than the swap test and study the robustness to depolarizing noise for qubit states.

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Joint probability distributions for projection probabilities of random orthonormal states

Cited by 5 publications

References 44 publications

Eigenvalue Determination for Mixed Quantum States using Overlap Statistics

Eigenvalue Determination for Mixed Quantum States using Overlap Statistics

The statistics of spectral shifts due to finite rank perturbations

Beyond the Swap Test: Optimal Estimation of Quantum State Overlap

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