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arXiv:2303.13587 · Quantum

Entanglement Trajectory and its Boundary

Ruge Lin

Published Mar 14, 2024• 0 citations
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Abstract

In this article, we present a novel approach to investigating entanglement in the context of quantum computing. Our methodology involves analyzing reduced density matrices at different stages of a quantum algorithm's execution and representing the dominant eigenvalue and von Neumann entropy on a graph, creating an "entanglement trajectory." To establish the trajectory's boundaries, we employ random matrix theory. Through the examination of examples such as quantum adiabatic computation, the Grover algorithm, and the Shor algorithm, we demonstrate that the entanglement trajectory remains within the established boundaries, exhibiting unique characteristics for each example. Moreover, we show that these boundaries and features can be extended to trajectories defined by alternative entropy measures. The entanglement trajectory serves as an invariant property of a quantum system, maintaining consistency across varying situations and definitions of entanglement. Numerical simulations accompanying this research are available via open access.

DOI
10.22331/q-2024-03-14-1282

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In this article, we present a novel approach to investigating entanglement in the context of quantum computing. Our methodology involves analyzing reduced density matrices at different stages of a quantum algorithm's execution and representing the dominant eigenvalue and von Neumann entropy on a graph, creating an "entanglement trajectory." To establish the trajectory's boundaries, we employ random matrix theory. Through the examination of examples such as quantum adiabatic computation, the Grover algorithm, and the Shor algorithm, we demonstrate that the entanglement trajectory remains within the established boundaries, exhibiting unique characteristics for each example. Moreover, we show that these boundaries and features can be extended to trajectories defined by alternative entropy measures. The entanglement trajectory serves as an invariant property of a quantum system, maintaining consistency across varying situations and definitions of entanglement. Numerical simulations accompanying this research are available via open access.
In this paper, we show how the entropy (including the von Neumann entropy obtained by tracing across various sizes of subsystems, the entanglement gap, as well as different degrees of Rényi entropy) of the random reduced density matrices are related to their dominant eigenvalue. Analytical results are deduced from Random Matrix Theory (RMT) for decentralized Wishart matrices and backed up by computer simulations. The correlation between our study and entanglement generated by quantum computing is provided with various examples.
The cost of classical simulations of quantum many-body dynamics is often determined by the amount of entanglement in the system. In this paper, we study entanglement in stochastic quantum trajectory approaches that solve master equations describing open quantum system dynamics. First, we introduce and compare adaptive trajectory unravelings of master equations. Specifically, building on Ref. [Phys. Rev. Lett. 128, 243601 (2022)], we study several greedy algorithms that generate trajectories with a low average entanglement entropy. Second, we consider various conventional unravelings of a one-dimensional open random Brownian circuit and locate the transition points from area- to volume-law-entangled trajectories. Third, we compare various trajectory unravelings using matrix product states with a direct integration of the master equation using matrix product operators. We provide concrete examples of dynamics, for which the simulation cost of stochastic trajectories is exponentially smaller than the one of matrix product operators.
We compute, for massive particles, the explicit Wigner rotations of one-particle states for arbitrary Lorentz transformations; and the explicit Hermitian generators of the infinite-dimensional unitary representation. For a pair of spin 1/2 particles, Einstein–Podolsky–Rosen–ell entangled states and their behaviour under the Lorentz group are analyzed in the context of quantum field theory. Group theoretical considerations suggest a convenient definition of the Bell states which is slightly different from the conventional assignment. The behaviour of Bell states under arbitrary Lorentz transformations can then be described succinctly. Reduced density matrices applicable to systems of identical particles are defined through Yang's prescription. The von Neumann entropy of each of the reduced density matrix is Lorentz invariant; and its relevance as a measure of entanglement is discussed, and illustrated with an explicit example. A regularization of the entropy in terms of generalized zeta functions is also suggested.
The cost of classical simulations of quantum many-body dynamics is often determined by the amount of entanglement in the system. In this paper, we study entanglement in stochastic quantum trajectory approaches that solve master equations describing open quantum system dynamics. First, we introduce and compare adaptive trajectory unravelings of master equations. Specifically, building on [Phys. Rev. Lett. 128, 243601 (2022)], we study several greedy algorithms that generate trajectories with a low average entanglement entropy. Second, we consider various conventional unravelings of a one-dimensional open random Brownian circuit and locate the transition points from area- to volume-law-entangled trajectories. Third, we compare various trajectory unravelings using matrix product states with a direct integration of the master equation using matrix product operators. We provide concrete examples of dynamics, for which the simulation cost of stochastic trajectories is exponentially smaller than the one of matrix product operators.
The definition of entanglement entropy in quantum systems relies on the factorization of the total Hilbert space into subsystems, which allows the construction of reduced density matrices through the partial trace operation. However, there exist important physical systems for which such a factorization is not available. In these cases, reduced density matrices and entanglement entropy are not well defined in the conventional sense. In this thesis, we develop an operator-algebraic framework for describing entanglement in such systems using the theory of von Neumann algebras. As a primary example, we consider eternal Schwarzschild black holes in Anti-de Sitter (AdS) spacetime. Starting from the dual boundary theory, we show how a suitable notion of trace can be defined, leading to a well-defined von Neumann entropy. We then reproduce the same entropy from the bulk gravitational description and demonstrate the agreement between the two calculations. Finally, we show that this quantity can be naturally interpreted as the generalized entropy.
We propose a measure of entanglement through the quantum trajectory approach. The advantage of this measure is that it can avoid the problem of finding all possible decompositions for mixed states, which is usually complicated. This method can be applied to any physical system whose time evolution is governed by the master equation. As an example, we calculate the entanglement of the following cases: (A) entangled system composed of a pair of two-level atoms coupling to two separate baths and, (B) two entangled modes subjected to two separate baths.