Entanglement scaling of operators: a conformal field theory approach, with a glimpse of simulability of long-time dynamics in 1+1d

TL;DR

This work investigates the operator space entanglement entropy (OSEE) as the keystone for understanding the simulability of quantum dynamics in 1D systems via MPOs. Using two-dimensional conformal field theory (CFT) methods, it establishes an operator area law for thermal states and generalized Gibbs ensembles, showing OSEE remains bounded at finite temperature and grows only logarithmically with inverse temperature. It demonstrates that after a global quench the reduced density matrix’s OSEE exhibits linear growth in time before saturating, implying MPO-based methods struggle primarily in the transient regime, while the evolution operator’s OSEE grows linearly in time, signaling exponential bond-dimension growth for long-time evolution. For free fermions, the OSEE of JW strings and local operators grows at most logarithmically, with exact MPO representations for fermionic operators; however, extending these results to interacting systems via CFT faces breakdowns of conformal invariance in certain setups, leaving as an open problem the general OSEE behavior of Heisenberg-picture operators in interacting 1D systems. Overall, the paper clarifies when MPOs can efficiently simulate long-time dynamics and highlights key open questions about operator dynamics in interacting conformal field theories.

Abstract

In one dimension, the area law and its implications for the approximability by Matrix Product States are the key to efficient numerical simulations involving quantum states. Similarly, in simulations involving quantum operators, the approximability by Matrix Product Operators (in Hilbert-Schmidt norm) is tied to an operator area law, namely the fact that the Operator Space Entanglement Entropy (OSEE)---the natural analog of entanglement entropy for operators, investigated by Zanardi [Phys. Rev. A 63, 040304(R) (2001)] and by Prosen and Pizorn [Phys. Rev. A 76, 032316 (2007)]---, is bounded. In the present paper, it is shown that the OSEE can be calculated in two-dimensional conformal field theory, in a number of situations that are relevant to questions of simulability of long-time dynamics in one spatial dimension. It is argued that: (i) thermal density matrices $ρ\propto e^{-βH}$ and Generalized Gibbs Ensemble density matrices $ρ\propto e^{- H_{\rm GGE}}$ with local $H_{\rm GGE}$ generically obey the operator area law; (ii) after a global quench, the OSEE first grows linearly with time, then decreases back to its thermal or GGE saturation value, implying that, while the operator area law is satisfied both in the initial state and in the asymptotic stationary state at large time, it is strongly violated in the transient regime; (iii) the OSEE of the evolution operator $U(t) = e^{-i H t}$ increases linearly with $t$, unless the Hamiltonian is in a localized phase; (iv) local operators in Heisenberg picture, $φ(t) = e^{i H t} φe^{-i H t}$, have an OSEE that grows sublinearly in time (perhaps logarithmically), however it is unclear whether this effect can be captured in a traditional CFT framework, as the free fermion case hints at an unexpected breakdown of conformal invariance.

Figures (10)

• Figure 1: Cartoon of the area law/operator area law and approximability by MPS/MPOs: (a) the states whose entanglement entropy remains bounded when $L_A,L_B \rightarrow \infty$ can be well approximated by MPSs with small (finite) bond dimension (b) the operators whose operator space entanglement entropy (OSEE) remains bounded can be well approximated (in Hilbert-Schmidt norm) by MPOs with small (finite) bond dimension. (c) Of course, by viewing $O \in {\rm End}(\mathcal{H})$ as a state $\left| O \right> \in \mathcal{H} \otimes \overline{\mathcal{H}}$, the two things are exactly the same. This 'operator-folding' trick typically does not simplify analytic calculations, as it merely leads to rewritings of equivalent formulas. But it is sometimes helpful because it allows to use results that are well-established about the entanglement entropy to make analogous claims about the OSEE; it is also useful numerically, to turn MPS-algorithms into MPO-algorithms and vice versa.
• Figure 2: Cartoon of an infinitely long one-dimensional quantum system at finite temperature $1/\beta$, bipartitioned as $A \cup B$ where $A$ is a finite interval. The entanglement entropy famously obeys a volume law, while the OSEE of the thermal density matrix $\rho_\beta \, \propto \, e^{-\beta H}$ saturates as $L_A \gg \beta v$: the thermal density matrix obeys an operator area law. In a CFT, $v$ is the velocity of gapless excitations; in more general systems, $v$ would be some natural velocity, for instance a Lieb-Robinson velocity. In the opposite limit of low temperatures, $\beta v \gg L_A$, the thermal density matrix is close to the projector onto the ground-state, so relation (\ref{['eq:OSEE_pure']}) holds.
• Figure 3: (a) Drawing of the thermal density matrix; taking the trace corresponds to identifying the two dashed lines; this gives the partition function $Z = {\rm tr} [e^{-\beta H}]$ of the CFT on a cyclinder of circumference $\beta v$. In order for $Z$ to be finite, one needs an IR cutoff $\Lambda$, namely some truncation of the system at the two ends. (b) The replicated surface that is used to calculate the entanglement entropy, in the setup of Cardy and Calabrese calabrese2004entanglement. The swap operator $\mathcal{S}^A_\sigma$, with a cyclic permutation $\sigma = (1,2\dots, \alpha)$, can be equivalently represented by two twist field $\mathcal{T}_{\sigma}$ and $\mathcal{T}_{\sigma^{-1}}$ located at the two ends of the interval $A$. (c) For the OSEE, there are two swap operators $\mathcal{S}^A_{\sigma}$ and $\mathcal{S}^A_{\sigma^{-1}}$, that are equivalent to a four-point function of twist fields.
• Figure 4: Numerical check of formula (\ref{['eq:exact_thermal_free']}) for free fermions on the lattice with dispersion $\varepsilon(k) = -\cos k$ ( a.k.a the XX chain after a Jordan-Wigner transformation); the Fermi velocity is $v=1$, and the constant $a_0$ in (\ref{['eq:exact_thermal_free']}) is obtained by fitting. We take a chain of $1000$ sites at inverse temperature $\beta = 20$, and cut an interval of length $L_A= 1,2,\dots ,100$ in the middle. Notice that formula (\ref{['eq:exact_thermal_free']}) works for non-integer $\alpha$, and, in particular, it works for $\alpha < 1$.
• Figure 5: Cartoon of an algorithm that would try to exploit the fact that $\rho_A$ can be approximated by an MPO at short and large times. Top: starting from an initial state written as an MPS, one would construct the reduced density matrix $\rho_A$ in the form of an MPO, with left and right 'environments' $E_L$ and $E_R$. One would then design a protocol for updating both the matrices $V$ and the two vectors $E_L$ and $E_R$ that encode the environment, at each time step. Designing such a protocol for the update is of course the non-trivial part. However, what we argue in this section is that, no matter how smart this protocol is, such an algorithm will face an exponential growth of the bond dimension at intermediate times $t \sim L_A/{2v}$.