Dynamics of Renyi entanglement entropy in local quantum circuits with charge conservation

TL;DR

This paper investigates the growth of Renyi entanglement entropy in charge-conserving quantum circuits, proving an upper bound under diffusive charge transport and confirming it with recent numerical results.

Contribution

The paper provides a rigorous proof of the upper bound on Renyi entanglement entropy growth and connects it with recent numerical findings in charge-conserving quantum circuits.

Findings

Renyi entanglement entropy with index > 1 grows at most as O(√t ln t) under diffusive charge transport.

Recent numerical simulations show this bound is nearly saturated in random local quantum circuits.

The work links entanglement dynamics with charge transport properties in quantum circuits.

Abstract

In local quantum circuits with charge conservation, we initialize the system in random product states and study the dynamics of the Renyi entanglement entropy $R_\alpha$. We rigorously prove that $R_\alpha$ with Renyi index $\alpha>1$ at time $t$ is $\le O(\sqrt{t\ln t})$ if the transport of charges is diffusive. Very recent numerical results of Rakovszky et al. show that this upper bound is saturated (up to the sub-logarithmic correction) in random local quantum circuits with charge conservation.

Equations

$$R_{\alpha }({\rho }_A)=\frac{1}{1-\alpha }\ln \mathrm{tr}{\rho }_A^{\alpha }=\frac{1}{1-\alpha }\ln \sum _{i\ge 1}{\Lambda }_i^{\alpha },$$

$$R_{\infty }({\rho }_A):=\underset{\alpha \to +\infty }{\lim }{R}_{\alpha }({\rho }_A)=-\ln {\Lambda }_1.$$

$$\underset{\alpha \to 1}{\lim }{R}_{\alpha }({\rho }_A)=-\mathrm{tr}({\rho }_A\ln {\rho }_A).$$

$$R_{\infty }({\rho }_A)\le R_{\alpha }({\rho }_A)\le \frac{\alpha }{\alpha -1}{R}_{\infty }({\rho }_A).$$

$$R_{\alpha }({\rho }_A)=\frac{1}{1-\alpha }\ln \sum _{i\ge 1}{\Lambda }_i^{\alpha }\le \frac{1}{1-\alpha }\ln {\Lambda }_1^{\alpha }=\frac{\alpha }{\alpha -1}{R}_{\infty }({\rho }_A).$$

$$U(t,0)=U(t,t-1)U(t-1,t-2)\cdots U(1,0),$$

$$U(t,t-1)=\prod _{i=1}^{n-1}{U}_t^{2i,2i+1}\prod _{i=1}^n{U}_t^{2i-1,2i}.$$

$$U_t^{i,i+1}=\left(\begin{array}{cccc}\ast & 0& 0& 0\\ 0& \ast & \ast & 0\\ 0& \ast & \ast & 0\\ 0& 0& 0& \ast \end{array}\right),$$

$$R_{\alpha }({\rho }_A)\le \frac{\alpha }{\alpha -1}O(\sqrt{t\ln t})$$

$$|{\psi }_{\mathrm{init}}⟩=|{\psi }_{\mathrm{init}}{⟩}_{\mathrm{in}}\otimes |{\psi }_{\mathrm{init}}{⟩}_{\mathrm{out}},$$

$$|{\psi }_0⟩=|0{⟩}_{\mathrm{in}}^{\otimes 2m}\otimes |{\psi }_{\mathrm{init}}{⟩}_{\mathrm{out}}$$

$$|⟨U(t,0){\psi }_0,U(t,0){\psi }_{\mathrm{init}}⟩|=2^{-m}.$$

$$\Vert (1-P)U(t,0)|{\psi }_0⟩\Vert \le e^{-\Omega (m^2/t)}.$$

$$V(t,0)=V(t,t-1)V(t-1,t-2)\cdots V(1,0),$$

$$V(t,t-1)=\prod _{i=1}^{n-1}{U}_t^{2i,2i+1}\prod _{i=1}^{(n-1)/2}{U}_t^{2i-1,2i}{u}_t^{n,n+1}\prod _{i=(n+3)/2}^n{U}_t^{2i-1,2i},$$

$$U(t,t-1)P=V(t,t-1)P.$$

$$U(t,0)|{\psi }_0⟩$$

$$=V(t,t-1)PU(t-1,0)|{\psi }_0⟩\approx V(t,t-1)U(t-1,0)|{\psi }_0⟩,$$

$$\Vert |{\Delta }_t⟩\Vert \le t{e}^{-\Omega (m^2/t)},|{\Delta }_t⟩:=U(t,0)|{\psi }_0⟩-V(t,0)|{\psi }_0⟩.$$

$$S=\{|+⟩,|-⟩{\}}_{\mathrm{in}}^{\otimes 2m}\otimes |{\psi }_{\mathrm{init}}{⟩}_{\mathrm{out}}$$

$$\sum _{|{\psi }_{\mathrm{init}}⟩\in S}|⟨{\Delta }_t|U(t,0)|{\psi }_{\mathrm{init}}⟩{|}^2\le \Vert |{\Delta }_t⟩{\Vert }^2⟹\frac{1}{|S|}\sum _{|{\psi }_{\mathrm{init}}⟩\in S}|⟨{\Delta }_t|U(t,0)|{\psi }_{\mathrm{init}}⟩|\le 2^{-m}\Vert |{\Delta }_t⟩\Vert ,$$

$$S^{\prime }:=\{|{\psi }_{\mathrm{init}}⟩\in S:|⟨{\Delta }_t|U(t,0)|{\psi }_{\mathrm{init}}⟩|\le 2^{-m}\Vert |{\Delta }_t⟩\Vert p(t)\}.$$

$$|S^{\prime }|/|S|\ge 1-1/p(t).$$

$$|\psi ⟩=\sum _{i\ge 1}{\lambda }_i|a_i{⟩}_A\otimes |b_i{⟩}_B$$

$$|⟨\varphi |\psi ⟩|\le |⟨{\psi }^{\prime }|\psi ⟩|=\sqrt{\sum _{i=1}^D{\lambda }_i^2}$$

$$|{\psi }^{\prime }⟩:=\frac{1}{\sqrt{{\sum }_{i=1}^D{\lambda }_i^2}}\sum _{i=1}^D{\lambda }_i|a_i{⟩}_A\otimes |b_i{⟩}_B.$$

$$|⟨V(t,0){\psi }_0,U(t,0){\psi }_{\mathrm{init}}⟩|=|⟨U(t,0){\psi }_0,U(t,0){\psi }_{\mathrm{init}}⟩-⟨{\Delta }_t|U(t,0)|{\psi }_{\mathrm{init}}⟩|$$

$$\ge 2^{-m}-|⟨{\Delta }_t|U(t,0)|{\psi }_{\mathrm{init}}⟩|\ge 2^{-m}(1-\Vert |{\Delta }_t⟩\Vert p(t))\ge 2^{-m}(1-t{e}^{-\Omega (m^2/t)}p(t)).$$

$${\lambda }_1\ge 2^{-m}(1-t{e}^{-\Omega (m^2/t)}p(t)).$$

$$R_{\alpha }({\rho }_A)\le \frac{\alpha }{\alpha -1}{R}_{\infty }({\rho }_A)=-\frac{\alpha }{\alpha -1}\ln {\Lambda }_1=-\frac{2\alpha }{\alpha -1}\ln {\lambda }_1.$$

$$R_{\alpha }({\rho }_A)\le \frac{\alpha }{\alpha -1}O(t^z\mathrm{poly}\ln t)$$

Full text

1 Introduction

Entanglement, a concept of quantum information theory, has been widely used in condensed matter and statistical physics to provide insights beyond those obtained via “conventional” quantities. For example, in a non-integrable system the growth of the von Neumann entanglement entropy reveals the ballistic light cone set by the Lieb-Robinson bound [15], while the energy transport is diffusive [13].

The von Neumann entanglement entropy is the standard entanglement measure for pure states. However, it is also instructive to study the Renyi entanglement entropy, which reflects the entanglement spectrum [14] and is easier to measure experimentally [1, 3, 11]. It is known that the von Neumann and Renyi entanglement entropies may behave differently in some cases: from the scaling of eigenstate entanglement [5, 8, 17] to describing the simulability of quantum many-body states [24, 23, 9].

Quantum circuits are not only a model of quantum computation [4, 27, 20], but also useful for the study of quantum many-body systems [25, 2, 10]. Since the dynamics of local Hamiltonians can be simulated by local quantum circuits [16], one may gain insights into the former from the latter. In particular, some recent works [18, 26, 19] studied random local quantum circuits, which are minimal models of quantum chaotic dynamics. Furthermore, the evolution governed by time-independent chaotic local Hamiltonians preserves energy. Such evolution is more faithfully described by random local quantum circuits with conserved quantities, in which the diffusive transport is easily observed [12, 21].

In (not necessarily random) local quantum circuits with charge conservation, we initialize the system in random product states and study the dynamics of the Renyi entanglement entropy $R_{\alpha}$. Perhaps surprisingly, we rigorously prove that $R_{\alpha}$ with Renyi index $\alpha>1$ at time $t$ is $\leq O(\sqrt{t\ln t})$ if the transport of charges is diffusive. It is straightforward to extend this result to cases where the transport is sub- or super-diffusive. Indeed, the proof explicitly shows that the growth of $R_{\alpha}$ with $\alpha>1$ is a probe of transport. This is in contrast to the linear (in $t$) growth of the von Neumann entanglement entropy.

We conjecture that the upper bound $O(\sqrt{t\ln t})$ on the Renyi entanglement entropy $R_{\alpha}$ with $\alpha>1$ holds more generally for the dynamics of time-independent local Hamiltonians with diffusive energy transport. It is known to fail [7] in the integrable $XY$ chain, whose transport is ballistic.

2 Preliminaries

We start with some basic definitions. We use the natural logarithm throughout this paper.

Definition 1** (entanglement entropy).**

The Renyi entanglement entropy $R_{\alpha}$ with index $\alpha\in(0,1)\cup(1,+\infty)$ of a bipartite pure state $\rho_{AB}=|\psi\rangle\langle\psi|$ is defined as

[TABLE]

where $\Lambda_{1}\geq\Lambda_{2}\geq\cdots\geq 0$ with $\sum_{i\geq 1}\Lambda_{i}=1$ are the eigenvalues (in descending order) of the reduced density matrix $\rho_{A}=\operatorname{tr}_{B}\rho_{AB}$. The min-entropy is defined as

[TABLE]

Note that the von Neumann entanglement entropy is given by

[TABLE]

Lemma 1**.**

For $\alpha>1$, we have

[TABLE]

Proof.

The first inequality is a consequence of fact that $R_{\alpha}$ is monotonically non-increasing with respect to $\alpha$ (this is why $R_{\infty}$ is called the min-entropy). The second inequality follows from

[TABLE]

∎

Definition 2** (local quantum circuit with charge conservation).**

Consider a chain of $2n$ spin-$1/2$’s. Let the time-evolution operator be

[TABLE]

where $t$ is a positive integer. Each layer of the circuit consists of two sub-layers of local unitaries:

[TABLE]

Each unitary $U_{t}^{i,i+1}$ acts on two neighboring spins $i,i+1$ and is block diagonal in the $\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$ basis:

[TABLE]

i.e., $U_{t}^{i,i+1}$ is the direct sum of a phase factor, a unitary matrix of order $2$, and another phase factor. It should be clear that every $U_{t}^{i,i+1}$ and hence $U(t,0)$ preserve the total charge $\sum_{i=1}^{2n}\sigma_{z}^{i}$.

Of special interest is the so-called random local quantum circuit with charge conservation, where each $U_{t}^{i,i+1}$ is the direct sum of a random phase factor, a Haar-random unitary matrix of order $2$, and another random phase factor. In this model, it is straightforward to prove that the transport of charges is diffusive [21], i.e., the evolution of the distribution of charges $\{\langle\sigma_{z}^{i}\rangle\}_{i=1}^{2n}$ is (approximately) described by the diffusion equation. We emphasize that the main result of this paper only assumes diffusive transport and does not require randomness in $U_{t}^{i,i+1}$ or $U(t,0)$.

3 Results

We are ready to state and prove the main result of this paper.

Theorem 1**.**

Consider the spin chain as a bipartite quantum system $A\otimes B$. Subsystem $A$ consists of spins $1,2,\ldots,n$, i.e., we study the entanglement across the middle cut. Initialize the system in a random product state $|\psi_{\rm init}\rangle$ in the $\sigma_{x}$ basis, i.e., each spin is in either $|+\rangle:=\frac{|0\rangle+|1\rangle}{\sqrt{2}}$ or $|-\rangle:=\frac{|0\rangle-|1\rangle}{\sqrt{2}}$ with equal probability. Let $\alpha>1$ and $\rho_{A}:=\operatorname{tr}_{B}(U(t,0)|\psi_{\rm init}\rangle\langle\psi_{\rm init}|U^{\dagger}(t,0))$ be the reduced density matrix of subsystem $A$ at time $t$. If the transport of charges under the dynamics $U(t,0)$ is diffusive, then

[TABLE]

holds with probability $\geq 1-1/p(t)$, where $p$ is a polynomial of arbitrarily high degree.

Proof.

We divide the spin chain into two parts. One of them, labeled by “in,” consists of $2m$ spins (with indices $n-m+1,n-m+2,\ldots,n+m$) near the cut, where $m$ is some positive integer to be determined later. The other part, labeled by “out,” is the rest of the system. The initial state can be factored into

[TABLE]

where $|\psi_{\rm init}\rangle_{\rm in}$ and $|\psi_{\rm init}\rangle_{\rm out}$ are random product states in the “in” and “out” parts of the system, respectively. Define

[TABLE]

so that $|\langle\psi_{0}|\psi_{\rm init}\rangle|=2^{-m}$. Since $U(t,0)$ is unitary, we have

[TABLE]

The left-hand side of this equation is the absolute value of the inner product between $U(t,0)|\psi_{0}\rangle$ and $U(t,0)|\psi_{\rm init}\rangle$. Occasionally we do not use the standard Dirac notation because it is cumbersome.

Let $Z$ with $|Z|=2^{2n-2}$ be the set of all computational basis states (i.e., product states in the $\sigma_{z}$ basis) obeying the constraint that spins $n$ and $n+1$ are in the state $|00\rangle$. Let $P$ be the projection onto the subspace $\operatorname{span}Z$. The state $|\psi_{0}\rangle$ has an extended region of $|0\rangle$’s in the middle of the chain. Since the transport of charges is described by the diffusion equation, we have

[TABLE]

Assume without loss of generality that $n$ is odd. The only local unitary in $U(t,t-1)$ acting on both subsystems $A$ and $B$ is in the second product on the right-hand of Eq. (7). Define a modified local quantum circuit

[TABLE]

where $u_{t}^{n,n+1}:=\langle 00|U_{t}^{n,n+1}|00\rangle$ is a complex number. It is easy to see that

[TABLE]

Therefore,

[TABLE]

where each approximation step generates an additive error upper bounded by the right-hand side of (13). Iterating this process, we have

[TABLE]

Recall that both $|\psi_{\rm init}\rangle_{\rm in}$ and $|\psi_{\rm init}\rangle_{\rm out}$ are random product states in the $\sigma_{x}$ basis. We now fix the latter but not the former. Then, $|\psi_{0}\rangle$ is fixed but $|\psi_{\rm init}\rangle$ is not. Let

[TABLE]

be the set of all possible initial states consistent with $|\psi_{\rm init}\rangle_{\rm out}$ so that $|S|=2^{2m}$. Since the states in $S$ are pairwise orthogonal,

[TABLE]

where we used the inequality of arithmetic and geometric means. Define a subset of $S$ as

[TABLE]

Markov’s inequality implies that

[TABLE]

It suffices to prove (9) for all states in $S^{\prime}$. To this end, we make use of

Lemma 2** (Eckart-Young theorem [6]).**

Let

[TABLE]

be the Schmidt decomposition of the state $|\psi\rangle$, where $\lambda_{1}\geq\lambda_{2}\geq\cdots>0$ with $\sum_{i\geq 1}\lambda_{i}^{2}=1$ are the Schmidt coefficients in descending order. Any state $|\phi\rangle$ of Schmidt rank $D$ satisfies

[TABLE]

where

[TABLE]

For any particular state $|\psi_{\rm init}\rangle\in S^{\prime}$, we have

[TABLE]

Let $\lambda_{1}$ be the largest Schmidt coefficient of $U(t,0)|\psi_{\rm init}\rangle$, and $\Lambda_{1}=\lambda_{1}^{2}$ be the largest eigenvalue of the reduced density matrix $\rho_{A}=\operatorname{tr}_{B}(U(t,0)|\psi_{\rm init}\rangle\langle\psi_{\rm init}|U^{\dagger}(t,0))$. Since none of the local unitaries in $V(t,t-1)$ or $V(t,0)$ act on both subsystems $A$ and $B$, $V(t,0)$ does not generate any entanglement so that $V(t,0)|\psi_{0}\rangle$ is a product state between $A$ and $B$ (i.e., a state of Schmidt rank $1$). Combining this with (3) and Lemma 2, we have

[TABLE]

Therefore,

[TABLE]

We complete the proof by choosing $m=O(\sqrt{t\ln t})$ with a sufficiently large pre-factor hided in the Big-O notation. ∎

It is straightforward to extend Theorem 1 to cases where the transport is sub- or super-diffusive.

Corollary 1**.**

In the setting of Theorem 1, suppose that the transport of charges under $U(t,0)$ has the scaling: distance $\sim t^{z}$ for $0<z<1$. Then,

[TABLE]

holds with probability $\geq 1-1/p(t)$.

It is also straightforward to extend Theorem 1 to higher spatial dimensions.

Notes

Very recently, we became aware of a relate work [22], which also studied the growth of the Renyi entanglement entropy in diffusive systems. The numerical results there show that the upper bound in Theorem 1 is saturated (up to the sub-logarithmic correction) in random local quantum circuits with charge conservation.