arXiv:1106.2264 · Communications on Pure and Applied Mathematics

Entanglement Thresholds for Random Induced States

Guillaume Aubrun, Stanisław J. Szarek, Deping Ye

Published May 23, 2013• 50 citations

Abstract

For a random quantum state on obtained by partial tracing a random pure state on , we consider the question whether it is typically separable or typically entangled. For this problem, we show the existence of a sharp threshold of order roughly . More precisely, for any and for d large enough, such a random state is entangled with very large probability when , and separable with very large probability when . One consequence of this result is as follows: for a system of N identical particles in a random pure state, there is a threshold such that two subsystems of k particles each typically share entanglement if k > k 0 , and typically do not share entanglement if k < k 0 . Our methods also work for multipartite systems and for “unbalanced” systems such as , . The arguments rely on random matrices, classical convexity, high‐dimensional probability, and geometry of Banach spaces; some of the auxiliary results may be of reference value. © 2013 Wiley Periodicals, Inc.

DOI: 10.1002/cpa.21460

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arXiv:1106.2264 · Communications on Pure and Applied Mathematics

Entanglement Thresholds for Random Induced States

Guillaume Aubrun, Stanisław J. Szarek, Deping Ye

Published May 23, 2013• 50 citations

arXiv HTML PDF

Abstract

DOI: 10.1002/cpa.21460

Related papers

Entanglement thresholds for random induced states

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For a random quantum state on

H=C^d \otimes C^d

obtained by partial tracing a random pure state on

H \otimes C^s

, we consider the whether it is typically separable or typically entangled. For this problem, we show the existence of a sharp threshold

s_0=s_0(d)

of order roughly

d^3

. More precisely, for any

a > 0

and for d large enough, such a random state is entangled with very large probability when

s < (1-a)s_0

, and separable with very large probability when

s > (1+a)s_0

. One consequence of this result is as follows: for a system of N identical particles in a random pure state, there is a threshold

k_0 = k_0(N) \sim N/5

such that two subsystems of k particles each typically share entanglement if

k > k_0

, and typically do not share entanglement if

k < k_0

. Our methods work also for multipartite systems and for "unbalanced" systems such as

C^{d} \otimes C^{d'}

d \neq d'

. The arguments rely on random matrices, classical convexity, high-dimensional probability and geometry of Banach spaces; some of the auxiliary results may be of reference value. A high-level non-technical overview of the results of this paper and of a related article arXiv:1011.0275 can be found in arXiv:1112.4582.

N/3

qudits will have a positive partial transpose and hence are separable or bound entangled. Additionally, our numerical results show that the probability of finding entangled subsystems smaller than

N/3

falls exponentially in the dimension of the Hilbert space. The bulk of pure state Hilbert space thus consists of highly entangled states with multipartite entanglement encompassing at least a third of the qudits in the pure state.

In the presence of a globally conserved charge

N

, a natural question is whether a given separable state can be separated into charge-conserving components. We dub this problem the Symmetric Separability Problem (SSP). On random states, the SSP is answered negatively with probability one for almost all

N

. Using a witness to the failure of symmetric separability, namely the number entanglement (NE) introduced in arXiv:2110.09388, we show that most symmetric and separable states are actually far from being symmetrically separable, with the NE featuring Gaussian concentration around a strictly positive mean value. We discuss some consequences of our results for quantum tasks in the presence of a superselection rule or in the absence of a common reference frame. Progress is made on the question of the size of the separable space constrained by

N

. We also touch upon the question of the complexity of SSP, and multiparty entanglement.

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>{k}_{0}

, and typically do not share entanglement if

k<{k}_{0}

. By ``random'' we mean here uniformly distributed on the sphere of the corresponding Hilbert space. The analogous phase transition for the positive partial transpose (

\text{PPT}

) property can be described even more precisely. For example, for

N

qubits the two subsystems of size

k

are typically in a

\text{PPT}

state if $k<{k}_{1}:=N/4\ensuremath{-}1/2$ and typically in a non-

\text{PPT}

state if

k>{k}_{1}

. 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.