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arXiv:1112.4582 · Physical Review A

Phase transitions for random states and a semicircle law for the partial transpose

Guillaume Aubrun, Stanisław J. Szarek, Deping Ye

Published Mar 12, 2012• 28 citations
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Abstract

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.

DOI
10.1103/physreva.85.030302

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For a system of NNN identical particles in a random pure state, there is a threshold k0=k0(N)\ensuremath∼N/5{k}_{0}={k}_{0}(N)\ensuremath{\sim}N/5k0​=k0​(N)\ensuremath∼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<k1:=N/4\ensuremath−1/2k<{k}_{1}:=N/4\ensuremath{-}1/2k<k1​:=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.
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.
The entanglement between two arbitrary subsystems of random pure states is studied via properties of the density matrix's partial transpose ${\ensuremath{\rho}}_{12}^{{T}_{2}}$. The density of states of ${\ensuremath{\rho}}_{12}^{{T}_{2}}$ is close to the semicircle law when both subsystems have dimensions which are not too small and are of the same order. A simple random matrix model for the partial transpose is found to capture the entanglement properties well, including a transition across a critical dimension. Log negativity is used to quantify entanglement between subsystems and analytic formulas for this are derived based on the simple model. The skewness of the eigenvalue density of ${\ensuremath{\rho}}_{12}^{{T}_{2}}$ is derived analytically, using the average of the third moment over the ensemble of random pure states. The third moment after partial transpose is also shown to be related to a generalization of the Kempe invariant. The smallest eigenvalue after partial transpose is found to follow the extreme value statistics of random matrices; namely, the Tracy-Widom distribution. This distribution, with relevant parameters obtained from the model, is found to be useful in calculating the fraction of entangled states at critical dimensions. These results are tested in a quantum dynamical system of three coupled standard maps, where one finds that if the parameters represent a strongly chaotic system, the results are close to those of random states, although there are some systematic deviations at critical dimensions.
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.
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\mathbb C^d\otimes\mathbb C^dCd⊗Cd, obtained as the marginal over an environment Cs\mathbb C^sCs of a uniformly distributed pure state, generically satisfy or violate them. For each fixed level m∈Nm\in\mathbb Nm∈N in this hierarchy of moment-based PPT criteria, we are able to identify a threshold environment dimension s=λmd2s=λ_md^2s=λm​d2 at which the behavior of the associated random state switches from violating to satisfying it, with probability going to 111 as ddd 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 ddd is large. We additionally need tools from the theory of Hankel determinant evaluation via orthogonal polynomials.
The entanglement properties of random pure states are relevant to a variety of problems ranging from chaotic quantum dynamics to black-hole physics.The averaged bipartite entanglement entropy of such states admits a volume law and upon increasing the subregion size follows the Page curve.In this paper, we generalize this setup to random mixed states by coupling the system to a bath and use the partial transpose to study their entanglement properties.We develop a diagrammatic method to incorporate partial transpose within random matrix theory and formulate a perturbation theory in 1/L, the inverse of the Hilbert-space dimension.We compute several quantities including the spectral density of partial transpose (or entanglement negativity spectrum), two-point correlator of eigenvalues, and the logarithmic negativity.As long as the bath is smaller than the system, we find that upon sweeping the subregion size, the logarithmic negativity shows an initial increase and a final decrease similar to the Page curve, while it admits a plateau in the intermediate regime where the logarithmic negativity depends only on the size of the system and of the bath but not on how the system is partitioned.This intermediate phase has no analog in random pure states, and is separated from the two other regimes by a critical point.We further show that when the bath is larger than the system by at least two extra qubits the logarithmic negativity is identically zero, which implies that there is no distillable entanglement.Using the diagrammatic approach, we provide a simple derivation of the semicircle law of the entanglement negativity spectrum in the latter two regimes.We show that despite the appearance of a semicircle distribution, reminiscent of Gaussian unitary ensemble (GUE), the higher-order corrections to the negativity spectrum and two-point correlator deviate from those of GUE.
The entanglement properties of random pure states are relevant to a variety of problems ranging from chaotic quantum dynamics to black-hole physics. The averaged bipartite entanglement entropy of such states admits a volume law and upon increasing the subregion size follows the Page curve. In this paper, we generalize this setup to random mixed states by coupling the system to a bath and use the partial transpose to study their entanglement properties. We develop a diagrammatic method to incorporate partial transpose within random matrix theory and formulate a perturbation theory in 1/L, the inverse of the Hilbert-space dimension. We compute several quantities including the spectral density of partial transpose (or entanglement negativity spectrum), two-point correlator of eigenvalues, and the logarithmic negativity. As long as the bath is smaller than the system, we find that upon sweeping the subregion size, the logarithmic negativity shows an initial increase and a final decrease similar to the Page curve, while it admits a plateau in the intermediate regime where the logarithmic negativity depends only on the size of the system and of the bath but not on how the system is partitioned. This intermediate phase has no analog in random pure states, and is separated from the two other regimes by a critical point. We further show that when the bath is larger than the system by at least two extra qubits the logarithmic negativity is identically zero, which implies that there is no distillable entanglement. Using the diagrammatic approach, we provide a simple derivation of the semicircle law of the entanglement negativity spectrum in the latter two regimes. We show that despite the appearance of a semicircle distribution, reminiscent of Gaussian unitary ensemble (GUE), the higher-order corrections to the negativity spectrum and two-point correlator deviate from those of GUE.