^{1,a)}and Piotr Śniady

^{1,2,3,b)}

### Abstract

We investigate the asymptotic behavior of the empirical eigenvalues distribution of the partial transpose of a random quantum state. The limiting distribution was previously investigated via Wishart random matrices indirectly (by approximating the matrix of trace 1 by the Wishart matrix of random trace) and shown to be the semicircular distribution or the free difference of two free Poisson distributions, depending on how dimensions of the concerned spaces grow. Our use of Wishart matrices gives exact combinatorial formulas for the moments of the partial transpose of the random state. We find three natural asymptotic regimes in terms of geodesics on the permutation groups. Two of them correspond to the above two cases; the third one turns out to be a new matrix model for the meander polynomials. Moreover, we prove the convergence to the semicircular distribution together with its extreme eigenvalues under weaker assumptions, and show large deviation bound for the latter.

We thank the anonymous referee who not only suggested use Wick formula (instead of Weingarten calculus which was used in the preliminary version of the current paper) but also gave other useful comments.

The research of M.F. was financially supported by the CHIST-ERA/BMBF project CQC.

In the initial phase of research, P.Ś. was a holder of a fellowship of *Alexander von Humboldt-Stiftung*. P.Ś.'s research has been supported by a grant of *Deutsche Forschungsgemeinschaft* (DFG) (SN 101/1-1).

I. INTRODUCTION

II. PRELIMINARIES

A. Free probability, noncrossing partitions, and permutations

1. Noncrossing partitions

2. Permutations

3. Noncrossing partition and permutations

4. Genus functions

5. Free cumulants

6. Semicircular distribution

7. Free Poisson distribution

B. Meanders

C. Our model

D. Moments of

III. THE CASE WHEN DIMENSIONS OF ENVIRONMENT AND BOTH SYSTEM PARTS ARE LARGE

A. Limiting eigenvalues

B. Behavior of extreme eigenvalues

1. Convergence of extreme eigenvalues

2. Speed of convergence

3. Implications for quantum informationtheory and some remarks

IV. THE CASE WHEN THE ENVIRONMENT SPACE IS FIXED AND ITS CONNECTION TO MEANDERS

A. Our model

B. Random pure states

C. Meander polynomials with our model

V. THE CASE WHEN ONE OF THE SYSTEM PARTS IS FIXED

VI. CONCLUDING REMARK

### Key Topics

- Eigenvalues
- 37.0
- Probability theory
- 18.0
- Polynomials
- 14.0
- Inequalities
- 6.0
- Graphical methods
- 3.0

## Figures

A graphical representation of a noncrossing partition τ = {{1}, {2, 3, 6}, {4, 5}}.

A graphical representation of a noncrossing partition τ = {{1}, {2, 3, 6}, {4, 5}}.

(a) Graphical representation of the permutation *t*(τ) = (1)(236)(45) from the geodesic corresponding to the noncrossing partition τ from Figure 1 . (b) Graphical representation of the permutation (*t*(τ))^{−1} = (1)(632)(54) from the geodesic corresponding to the noncrossing partition τ from Figure 1 .

(a) Graphical representation of the permutation *t*(τ) = (1)(236)(45) from the geodesic corresponding to the noncrossing partition τ from Figure 1 . (b) Graphical representation of the permutation (*t*(τ))^{−1} = (1)(632)(54) from the geodesic corresponding to the noncrossing partition τ from Figure 1 .

Meander of order *p* = 4 with *k* = 2 connected components corresponding to σ_{1} = {{1, 2}, {3, 4}, {5, 8}, {6, 7}} and σ_{2} = {{1, 6}, {2, 5}, {3, 4}, {7, 8}}.

Meander of order *p* = 4 with *k* = 2 connected components corresponding to σ_{1} = {{1, 2}, {3, 4}, {5, 8}, {6, 7}} and σ_{2} = {{1, 6}, {2, 5}, {3, 4}, {7, 8}}.

(a) Graphical representation of a noncrossing partition , (b) the corresponding noncrossing pair-partition from .

(a) Graphical representation of a noncrossing partition , (b) the corresponding noncrossing pair-partition from .

Graphical representation of noncrossing partitions τ_{1} = (1, 3, 5)(7) and τ_{2} = (2, 8)(4, 6).

Graphical representation of noncrossing partitions τ_{1} = (1, 3, 5)(7) and τ_{2} = (2, 8)(4, 6).

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