Emergent Irreversibility and Entanglement Spectrum Statistics

Claudio Chamon, Alioscia Hamma, and Eduardo R. Mucciolo

Phys. Rev. Lett. 112, 240501 – Published 16 June 2014

Abstract

We study the problem of irreversibility when the dynamical evolution of a many-body system is described by a stochastic quantum circuit. Such evolution is more general than a Hamiltonian one, and since energy levels are not well defined, the well-established connection between the statistical fluctuations of the energy spectrum and irreversibility cannot be made. We show that the entanglement spectrum provides a more general connection. Irreversibility is marked by a failure of a disentangling algorithm and is preceded by the appearance of Wigner-Dyson statistical fluctuations in the entanglement spectrum. This analysis can be done at the wave-function level and offers an alternative route to study quantum chaos and quantum integrability.

Received 13 November 2013

DOI:https://doi.org/10.1103/PhysRevLett.112.240501

Authors & Affiliations

Claudio Chamon¹, Alioscia Hamma², and Eduardo R. Mucciolo³

¹Department of Physics, Boston University, Boston, Massachusetts 02215, USA
²Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, People’s Republic of China
³Department of Physics, University of Central Florida, Orlando, Florida 32816, USA

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Vol. 112, Iss. 24 — 20 June 2014

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Images

Figure 1
Reversible gates used in the quantum stochastic evolutions. The output values of the gates are defined when $a$ , $b$ , and $c$ take values 0 and 1. Top row: The two-qubit gates swap and cnot. Bottom row: The different variations of the three-qubit Toffoli gates. We note that different variations of the cnot and Toffoli gates can be obtained from one fixed variation plus not gates.
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Figure 2
Evolution of the entanglement entropy $S_{1}$ obtained from the bipartition of the qubit string at the middle ( $n_{A} = n_{B} = 8$ ) as a function of the number of applied gates. First, $M_{h}$ reversible gates are randomly applied (heating period); second, a Metropolis algorithm is used to reverse the evolution and restore zero entropy (cooldown period). The shorter solid black line ( $M_{h} = 512$ ) and the longer red line ( $M_{h} = 4096$ ) result from averaging $S_{1}$ over 128 initial-state realizations. The faded brown lines show the evolution of $S_{1}$ for two typical realizations. (a) Only two-qubit gates are used. Upper inset: The heating period. Lower inset: Shifted curves, showing that the cooling, on average, is independent of the duration of the heating period. (b) A mixture of two-qubit and Toffoli gates is used. Inset: Detail of the transition between heating and cooling periods. The dashed line indicates the transition point.
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Figure 3
(a) The distribution of the spacing between consecutive unfolded singular values obtained from a bipartition at the middle of a $n = 16$ qubit string at the end of the heating period ( $M_{h} = 512$ ). Solid black circles are for two-qubit gate heating and open red squares are for heating with a mixture of two-qubit and Toffoli gates. Solid green line: GOE prediction. Dashed blue line: Poisson distribution. Dotted-dashed maroon line: Semi-Poisson distribution. Inset: The tail of the distributions. (b) The average spectral rigidity $Δ_{3} (L)$ obtained from the same spectra (a linear fit is used for the semi-Poisson line). A total of 5000 realizations were used to compute the averages.
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Figure 4
The statistical fluctuations of the entanglement spectrum for initial product states $| χ^{(1)} ⟩$ (orange empty squares) and $| χ^{(2)} ⟩$ (blue full circles) entangled with a mixture of two-qubit and Toffoli gates ( $n = 16$ , $M_{h} = 512$ ). The solid green line is the GOE prediction. (a) Distribution of unfolded singular value spacings. Inset: Distribution tails. (b) Average spectral rigidity $Δ_{3} (L)$ . Statistical averages were performed over 5000 realizations.
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