Freeness and The Partial Transposes of Wishart Random Matrices

Mingo, James A.; Popa, Mihai

doi:10.4153/cjm-2018-002-2

Cited by 14 publications

(26 citation statements)

References 10 publications

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“…If a random unitary matrix U is large enough, the unitarity constraints become so weak that after reshuffling the matrix U R shows statistical properties close to these of the Ginibre ensemble [45,49]-see also the recent rigorous results in Ref. [78]. Thus the corresponding positive matrix, U R (U R ) † , displays spectra in agreement with the Marčenko-Pastur law [48], P MP (x) = (2π ) −1 √ (4 − x)/x, derived to describe the spectral density of random Wishart matrices GG † .…”

Section: B Thermalization Of the Spectra Of Reshuffled And Partiallymentioning

confidence: 82%

Entanglement measures of bipartite quantum gates and their thermalization under arbitrary interaction strength

Jonnadula

Mandayam

Życzkowski

et al. 2020

Phys. Rev. Research

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Entanglement properties of bipartite unitary operators are studied via their local invariants, namely the entangling power e p and a complementary quantity, the gate typicality g t. We characterize the boundaries of the set K 2 representing all two-qubit gates projected onto the plane (e p , g t) showing that the fractional powers of the SWAP operator form a parabolic boundary of K 2 , while the other bounds are formed by two straight lines. In this way, a family of gates with extreme properties is identified and analyzed. We also show that the parabolic curve representing powers of SWAP persists in the set K N for gates of higher dimensions (N > 2). Furthermore, we study entanglement of bipartite quantum gates applied sequentially n times, and we analyze the influence of interlacing local unitary operations, which model generic Hamiltonian dynamics. An explicit formula for the entangling power of a gate applied n times averaged over random local unitary dynamics is derived for an arbitrary dimension of each subsystem. This quantity shows an exponential saturation to the value predicted by the random matrix theory, indicating "thermalization" in the entanglement properties of sequentially applied quantum gates that can have arbitrarily small, but nonzero, entanglement to begin with. The thermalization is further characterized by the spectral properties of the reshuffled and partially transposed unitary matrices.

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Section: B Thermalization Of the Spectra Of Reshuffled And Partiallymentioning

confidence: 82%

Entanglement measures of bipartite quantum gates and their thermalization under arbitrary interaction strength

Jonnadula

Mandayam

Życzkowski

et al. 2020

Phys. Rev. Research

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“…Here we improve the results from [14], giving a general sufficient condition that implies asymptotic Boolean independence between the two matrices from above. In particular, we show the asymptotic Boolean independence from matrix transposes (result somehow analogous to [7]) and classes of matrix partial transposes (see also [8]). The last part of the paper, Section 5, presents a result concerning the asymptotic distribution of self-adjoint random matrices with identically distributed and Boolean independent entries.…”

Section: Introductionmentioning

confidence: 77%

“…To formulate the following consequence of the Theorem above, we need first to define (following [8]) the notion of partial m-transpose of a square matrix. Definition 4.4.…”

Section: Permutations Of Entries and Asymptotic Boolean Independencementioning

confidence: 99%

An asymptotic property of large matrices with identically distributed Boolean independent entries

Popa

Hao

2019

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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Motivated by the recent work on asymptotic independence relations for random matrices with non-commutative entries, we investigate the limit distribution and independence relations for large matrices with identically distributed and Boolean independent entries. More precisely, we show that, under some moment conditions, such random matrices are asymptotically B-diagonal and Boolean independent from each other. The paper also gives a combinatorial condition under which such matrices are asymptotically Boolean independent from the matrix obtained by permuting the entries (thus extending a recent result in Boolean probability). In particular, we show that random matrices considered are asymptotically Boolean independent from their partial transposes. The main results of the paper are based on combinatorial techniques.Mathematics Subject Classification: Primary 60B20; Secondary 46L53, 06A07.

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“…Lemma 4.1. Suppose that there exists some k ∈ [r] such that the sequence σ k,N N satisfies the following property: (16) lim…”

Section: Resultsmentioning

confidence: 99%

“…In the recent years one can notice the emergence of a small but growing body of literature addressing questions connected to permutations of entries of various classes of matrices. For example, partial transposes (see part 5.1 for a precise definition and more details) appear in connection to Quantum Information Theory in [13], [5], [12]; furthermore, distributions of partial transposes of Wishart random matrices are described in [4], [7], also in [16], [17]; very interesting examples of entry permutations and other linear transforms on matrix entries are studied in [2], [8]. One should also mention the groundbreaking work [1] on random entry permutations on Haar unitary random matrices.…”

Section: Introductionmentioning

confidence: 99%

Answer to a question by A. Mandarino, T. Linowski and K. Życzkowski

Popa¹

2021

Preprint

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A recent work by A. Mandarino, T. Linowski and K. Życzkowski left open the following question. If µ N is a certain permutation of entries of a N 2 × N 2 matrix ("mixing map") and U N is a N 2 × N 2 Haar unitary random matrix, then is the family U N , U µN N , (U 2 N ) µN , . . . , (U m N ) µN asymptotically free? (here by A µ we understand the matrix resulted by permuting the entries of A according to the permutation µ). This paper presents some techniques for approaching such problems. In particular, one easy consequence of the main result is that the question above has an affirmative answer.

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Freeness and The Partial Transposes of Wishart Random Matrices

Cited by 14 publications

References 10 publications

Entanglement measures of bipartite quantum gates and their thermalization under arbitrary interaction strength

Entanglement measures of bipartite quantum gates and their thermalization under arbitrary interaction strength

An asymptotic property of large matrices with identically distributed Boolean independent entries

Answer to a question by A. Mandarino, T. Linowski and K. Życzkowski

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