Let W be a Wishart random matrix of size d 2 × d 2 , considered as a block matrix with d × d blocks. Let Y be the matrix obtained by transposing each block of W. We prove that the empirical eigenvalue distribution of Y approaches a non-centered semicircular distribution when d → ∞. We also show the convergence of extreme eigenvalues towards the edge of the expected spectrum. The proofs are based on the moments method. This matrix model is relevant to Quantum Information Theory and corresponds to the partial transposition of a random induced state. A natural question is: "When does a random state have a positive partial transpose (PPT)?". We answer this question and exhibit a strong threshold when the parameter from the Wishart distribution equals 4. When d gets large, a random state on C d ⊗ C d obtained after partial tracing a random pure state over some ancilla of dimension αd 2 is typically PPT when α > 4 and typically non-PPT when α < 4.
Let W be a Wishart random matrix of size d 2 × d 2 , considered as a block matrix with d × d blocks. Let Y be the matrix obtained by transposing each block of W. We prove that the empirical eigenvalue distribution of Y approaches a non-centered semicircular distribution when d → ∞. We also show the convergence of extreme eigenvalues towards the edge of the expected spectrum. The proofs are based on the moments method. This matrix model is relevant to Quantum Information Theory and corresponds to the partial transposition of a random induced state. A natural question is: "When does a random state have a positive partial transpose (PPT)?". We answer this question and exhibit a strong threshold when the parameter from the Wishart distribution equals 4. When d gets large, a random state on C d ⊗ C d obtained after partial tracing a random pure state over some ancilla of dimension αd 2 is typically PPT when α > 4 and typically non-PPT when α < 4.
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.
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, de-pending 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 asymp-totic 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. 1.
In this paper, we first obtain an algebraic formula for the moments of a centered Wishart matrix, and apply it to obtain new convergence results in the large dimension limit when both parameters of the distribution tend to infinity at different speeds. We use this result to investigate APPT (absolute positive partial transpose) quantum states. We show that the threshold for a bipartite random induced state on Cd = Cd1 ⊗ Cd2, obtained by partial tracing a random pure state on Cd ⊗ Cs, being APPT occurs if the environmental dimension s is of order s0 = min (d1, d2)3 max (d1, d2). That is, when s ≥ Cs0, such a random induced state is APPT with large probability, while such a random states is not APPT with large probability when s ≤ cs0. Besides, we compute effectively C and c and show that it is possible to replace them by the same sharp transition constant when min (d1, d2)2 ≪ d.
Moment-based relaxations of the positive partial transpose (PPT) criterion have been recently introduced, as a hierarchy of entanglement criteria involving only experimentally accessible quantities of a given bipartite state. The goal of this work is to study their typical detection performance on high-dimensional bipartite systems. Concretely, we investigate whether random bipartite mixed states on Cd⊗Cd, obtained as the marginal over an environment Cs of a uniformly distributed pure state, generically satisfy or violate them. For each fixed level m∈N in this hierarchy of moment-based PPT criteria, we are able to identify a threshold environment dimension s=λmd2 at which the behavior of the associated random state switches from violating to satisfying it, with probability going to 1 as d grows. The proof combines combinatorics of permutations techniques to estimate the average value of moments of partially transposed random states and concentration of measure arguments to bound the probability of deviating from such average, when the underlying local dimension d is large. We additionally need tools from the theory of Hankel determinant evaluation via orthogonal polynomials.
For a system of N identical particles in a random pure state, there is a threshold ${k}_{0}={k}_{0}(N)\ensuremath{\sim}N/5$ such that two subsystems of k particles each typically share entanglement if k>k0, and typically do not share entanglement if k<k0. By ``random'' we mean here uniformly distributed on the sphere of the corresponding Hilbert space. The analogous phase transition for the positive partial transpose (PPT) property can be described even more precisely. For example, for N qubits the two subsystems of size k are typically in a PPT state if $k<{k}_{1}:=N/4\ensuremath{-}1/2$ and typically in a non-PPT state if k>k1. Since, for a given state of the entire system, the induced state of a subsystem is given by the partial trace, the above facts can be rephrased as properties of random induced states. An important step in the analysis depends on identifying the asymptotic spectral density of the partial transposes of such random induced states, a result which is interesting in its own right.
We develop a method for the random sampling of (multimode) Gaussian states in terms of their covariance matrix, which we refer to as a random quantum covariance matrix (RQCM). We analyze the distribution of marginals and demonstrate that the eigenvalues of an RQCM converge to a shifted semicircular distribution in the limit of a large number of modes. We provide insights into the entanglement of such states based on the positive partial transpose criteria. Additionally, we show that the symplectic eigenvalues of an RQCM converge to a probability distribution that can be characterized using free probability. We present numerical estimates for the probability of a RQCM being separable and, if not, its extendibility degree, for various parameter values and mode bipartitions.