Quantum Approximate Markov Chains are Thermal

Abstract

We prove that any one-dimensional (1D) quantum state with small quantum conditional mutual information in all certain tripartite splits of the system, which we call a quantum approximate Markov chain, can be well-approximated by a Gibbs state of a short-range quantum Hamiltonian. Conversely, we also derive an upper bound on the (quantum) conditional mutual information of Gibbs states of 1D short-range quantum Hamiltonians. We show that the conditional mutual information between two regions A and C conditioned on the middle region B decays exponentially with the square root of the length of B. These two results constitute a variant of the Hammersley–Clifford theorem (which characterizes Markov networks, i.e. probability distributions which have vanishing conditional mutual information, as Gibbs states of classical short-range Hamiltonians) for 1D quantum systems. The result can be seen as a strengthening—for 1D systems—of the mutual information area law for thermal states. It directly implies an efficient preparation of any 1D Gibbs state at finite temperature by a constant-depth quantum circuit.

Appendix A: Another approach to prove Theorem 4

In Sect. 4, we employ a perturbative method to obtain a local operator O determined by the Hamiltonian and a local operator V such that

$$\begin{aligned} e^{-\beta (H+V)}\approx Oe^{-\beta H}O^\dagger . \end{aligned}$$

The existence of such operator plays a central role in the proof of Theorem 4. In this appendix, we introduce another approach to obtain similar operators, which is based on the previous work by Araki [25]. The main difference between these approaches is the origins of locality of the operator O. In the perturbative approach, the locality is originated in Lieb-Robinson bounds, which restrict real time evolutions of operators. Instead, in Araki’s approach, locality of O is originated in a restriction on imaginary-time evolutions of V.

Let us consider a 1D spin chain $\Lambda =[-n,n]$, a short-range Hamiltonian H on $\Lambda$, and a local operator V. We denote the maximum strength of H by J. A simple algebra show the following relation holds.

$$\begin{aligned} e^{-\beta (H+V)}&=e^{-\frac{\beta }{2}(H+V)}e^{\frac{\beta }{2}H}e^{-\beta H}e^{\frac{\beta }{2}H}e^{-\frac{\beta }{2}(H+V)}\\&\equiv E_r(V; H)e^{-\beta H}E_r(V; H)^{\dagger }, \end{aligned}$$

where we denote $e^{-\frac{\beta }{2}(H+V)}e^{\frac{\beta }{2}H}$ by $E_r(V; H)$. By denoting $V(\beta )=e^{-\beta H}Ve^{\beta H}$, $E_r(V; H)$ has another form written as [25]

$$\begin{aligned} E_r(V; H)=\sum _{n=0}^\infty (-1)^n\int _0^{\frac{\beta }{2}}d\beta _1\int _0^{\beta _1}d\beta _2\cdots \int _0^{\beta _{n-1}}d\beta _nV(\beta _n)\cdots V(\beta _1). \end{aligned}$$

Actually, $E_r(V; H)$ can be approximated by a local operator.

Lemma 11

[25] The following statements hold for any region $X\subset [-n,n]$ and any bounded operator V with $supp(V)=[a,b]\subset [-n,n]$.

• (i) There exists a constant $C\ge 0$ depending on $\beta$, J and $\Vert V\Vert$such that

  $$\begin{aligned} \Vert E_r(V; H_X)\Vert \le C\, \end{aligned}$$

• (ii) There exist constants $C, q\ge 0$ depending on $\beta$, J and $\Vert V\Vert$ such that

  $$\begin{aligned} \left\| E_r(V; H_X)-E_r(V; H_{X\cap [a-l,b+l]})\right\| \le C\frac{q^{1+\lfloor \frac{l}{2}\rfloor }}{(1+\lfloor \frac{l}{2}\rfloor )!}. \end{aligned}$$

Since $\log x!\approx x\log x-x$, the denominator grows faster than the numerator with respect to l, and thus, the accuracy of the above approximation is exponentially good with respect to l. Note that similar properties hold for the inverse of $E_r(V; H_X)$, $E_l(V;H_X)\equiv e^{-\frac{\beta }{2}H_X}e^{\frac{\beta }{2}(H_X+V)}$. Therefore, by choosing $V=H_{B^M}$, $E_r(V; H)$ and its local approximation play the same role as $O_{ABC}$ and $O_B$ in Sect. 4.2.