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Entanglement thresholds for random induced states

Guillaume Aubrun, Stanis Law, J. Szarek, Deping Ye

Published Jan 1, 2014• 51 citations
OpenAlex

Abstract

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.

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For a random quantum state on H=Cd⊗CdH=C^d \otimes C^dH=Cd⊗Cd obtained by partial tracing a random pure state on H⊗CsH \otimes C^sH⊗Cs, we consider the whether it is typically separable or typically entangled. For this problem, we show the existence of a sharp threshold s0=s0(d)s_0=s_0(d)s0​=s0​(d) of order roughly d3d^3d3. More precisely, for any a>0a > 0a>0 and for d large enough, such a random state is entangled with very large probability when s<(1−a)s0s < (1-a)s_0s<(1−a)s0​, and separable with very large probability when s>(1+a)s0s > (1+a)s_0s>(1+a)s0​. One consequence of this result is as follows: for a system of N identical particles in a random pure state, there is a threshold k0=k0(N)∼N/5k_0 = k_0(N) \sim N/5k0​=k0​(N)∼N/5 such that two subsystems of k particles each typically share entanglement if k>k0k > k_0k>k0​, and typically do not share entanglement if k<k0k < k_0k<k0​. Our methods work also for multipartite systems and for "unbalanced" systems such as Cd⊗Cd′C^{d} \otimes C^{d'}Cd⊗Cd′, d≠d′d \neq d'd=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.
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.
The entanglement in a pure state of N qudits (d-dimensional distinguishable quantum particles) can be characterized by specifying how entangled its subsystems are. A generally mixed subsystem of m qudits is obtained by tracing over the other $N\ensuremath{-}m$ qudits. We examine the entanglement in the space of mixed states of m qudits. We show that for a typical pure state of N qudits, its subsystems smaller than N/3N/3N/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/3N/3N/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.
For a system of NNN 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 kkk particles each typically share entanglement if k>k0k>{k}_{0}k>k0​, and typically do not share entanglement if k<k0k<{k}_{0}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\text{PPT}PPT) property can be described even more precisely. For example, for NNN qubits the two subsystems of size kkk are typically in a PPT\text{PPT}PPT state if $k<{k}_{1}:=N/4\ensuremath{-}1/2$ and typically in a non-PPT\text{PPT}PPT state if k>k1k>{k}_{1}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.
In the presence of a globally conserved charge NNN, 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 NNN. 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 NNN. We also touch upon the question of the complexity of SSP, and multiparty entanglement.
We consider random bipartite quantum states obtained by tracing out one subsystem from a random, uniformly distributed, tripartite pure quantum state. We compute thresholds for the dimension of the system being traced out, so that the resulting bipartite quantum state satisfies the reduction criterion in different asymptotic regimes. We consider as well the basis-independent version of the reduction criterion (the absolute reduction criterion), computing thresholds for the corresponding eigenvalue sets. We do the same for other sets relevant in the study of absolute separability, using techniques from random matrix theory. Finally, we gather and compare the known values for the thresholds corresponding to different entanglement criteria, and conclude with a list of open questions.
For a given Hamiltonian HHH on a multipartite quantum system, one is interested in finding the energy E0{E}_{0}E0​ of its ground state. In the separability approximation, arising as a natural consequence of measurement in a separable basis, one looks for the minimal expectation value ${\ensuremath{\lambda}}_{\mathrm{min}}^{\ensuremath{\bigotimes}}$ of HHH among all product states. For several concrete model Hamiltonians, we investigate the difference ${\ensuremath{\lambda}}_{\mathrm{min}}^{\ensuremath{\bigotimes}}\ensuremath{-}{E}_{0}$, called the separability gap, which vanishes if the ground state has a product structure. In the generic case of a random Hermitian matrix of the Gaussian orthogonal ensemble, we find explicit bounds for the size of the gap which depend on the number of subsystems and hold with probability one. This implies an effective entanglement criterion applicable for any multipartite quantum system: If an expectation value of a typical observable of a given state is sufficiently distant from the average value, the state is almost surely entangled.