Matrix-element distributions as a signature of entanglement generation

Yaakov S. Weinstein and C. Stephen Hellberg

Phys. Rev. A 72, 022331 – Published 23 August 2005

Abstract

We explore connections between an operator’s matrix-element distribution and its entanglement generation. Operators with matrix-element distributions similar to those of random matrices generate states of high multipartite entanglement. This occurs even when other statistical properties of the operators do not coincide with random matrices. Similarly, operators with some statistical properties of random matrices may not exhibit random matrix element distributions and will not produce states with high levels of multipartite entanglement. Finally, we show that operators with similar matrix-element distributions generate similar amounts of entanglement.

Received 17 February 2005

DOI:https://doi.org/10.1103/PhysRevA.72.022331

Authors & Affiliations

Yaakov S. Weinstein^* and C. Stephen Hellberg^†

Center for Computational Materials Science, Naval Research Laboratory, Washington, DC 20375, USA

^*Author to whom correspondence should be addressed. Present address: Quantum Information Science Group, MITRE, Eatontown, NJ 07724. Electronic address: weinstein@mitre.org
^†Electronic address: hellberg@dave.nrl.navy.mil

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Vol. 72, Iss. 2 — August 2005

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Images

Figure 1
(Color online) Distributions of nearest-neighbor eigenvalue spacings (upper left) and eigenvector-element amplitudes (upper right) for matrices of the interpolating ensembles with $δ = 0.1$ (dashed), 0.5 (dotted), and 0.9 (chained). The $N = 256$ matrix element plot (lower left) includes the distribution for $δ = 0.98$ (light solid line). For this $δ$ the eigenvalue and eigenvector distributions are indistinguishable from random (solid line) for the resolution of the figure. The matrix element distribution appears to converge more slowly which may be manifest in the entanglement generated by operating with 100 eight-qubit $δ = 0.9$ (엯), and 0.98 (◇) matrices on all computational basis states (lower right).Reuse & Permissions
Figure 2
(Color online) Distribution of nearest-neighbor eigenvalue spacings (top left), eigenvector elements (top right), matrix elements (bottom left), and $Q$ (bottom right) for $N = 256$ pseudorandom maps of $m = 2 (+)$ , 4 $(\times)$ , 8 (◻), and 16 (엯). The eigenvalue and eigenvector distributions appear to converge to that of the CUE (solid lines) more quickly than the matrix-element distribution and entanglement distribution. To approach $P_{C U E} (Q)$ with one iteration of a map requires $m ≃ 40$ [6, 9].Reuse & Permissions
Figure 3
(Color online) Matrix-element distribution (left) and eigenvector-element distribution (right) for interpolating ensemble matrices with $δ = 0.9$ and $N = 256 (\times)$ , 128 (엯), 64 $(+)$ , 32 (◻), 16 (◇), and 8 (▵). The matrix-element distribution gets further and further from that expected of the CUE (dashed line for the $N \to \infty$ limit and chained line for $N = 8$ ) as $N$ increases. This is in contrast with the eigenvector-element distribution, which is remarkably stable as a function of $N$ .Reuse & Permissions
Figure 4
(Color online) Entanglement spectra for CUE operators (lines) and $δ = 0.9$ interpolating ensemble operators (shapes), for $N = 8$ (solid line and $\times$ ), 16 (dotted line and 엯), 32 (chained line and $+$ ), and 64 (dashed line and ◻). As $N$ increases the distributions of the interpolating ensemble operators diverge further from the CUE distribution. The inset shows the difference between average entanglement generation for CUE operators and the $δ = 0.9$ interpolating ensemble operators, $Δ Q$ , as a function of number of qubits $n = \log_{2} (N)$ . As $N$ increases the average entanglement production gets further from the CUE average.Reuse & Permissions
Figure 5
(Color online) Average entanglement $⟨ Q ⟩$ over all eight-qubit initial computational basis states as a function of time for quantum sawtooth maps, $k = 1.5$ (chained line) and $- 1.5$ (dotted line), and Harper maps, $γ = 1$ (solid line) and 0.1 (dashed line), compared to the random matrix average (horizontal dashed line). The chaotic maps quickly approach the random matrix average while the regular maps do not. The insets show matrix-element distributions for the regular (light) and chaotic (dark) sawtooth maps at $t = 50$ (a), the regular (light) and chaotic (dark) Harper maps at $t = 50$ (b), and $t = 1$ (light) and 100 (dark) of the baker’s map (c).Reuse & Permissions
Figure 6
(Color online) Average entanglement generation as a function of time for 50 interpolating ensemble operators with $δ = 0.9 (\times)$ and 0.98 (엯) operators on initial computational basis states. For both sets of operators the average entanglement produced approaches the CUE average (dashed line) as an exponential. The insets show the matrix-element distributions for the same set of operators as a function of time. The left inset shows the distribution for $δ = 0.9$ operators for $t = 1$ (엯), 5 (◻), 10 $(\times)$ , 15 $(+)$ , 20 (▵), and 30 (◇). The right inset shows the matrix element distributions for $δ = 0.98$ operators and $t = 1$ (엯), 2 (◻), 3 $(\times)$ , 4 $(+)$ , 5 (▵), and 6 (◇). Both sets of operator matrix-element distributions approach the CUE distribution as time increases.Reuse & Permissions

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