Quantum Approximate Markov Chains are Thermal
Quantum Approximate Markov Chains are Thermal
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DOI：
10.1007/s00220-019-03485-6
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发表时间：
2016-09
影响因子：
2.4
通讯作者：
Kohtaro Kato;F. Brandão
中科院分区：
文献类型：
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作者：
Kohtaro Kato;F. Brandão
我们证明了在系统的所有特定的三体分裂中，任何具有小量子条件互信息的一维量子态，我们称之为量子近似马尔可夫链，都可以被短程量子哈密顿量的Gibbs态很好地近似。反之，我们还得到了一维短程量子哈密顿量的Gibbs态的(量子)条件互信息的上界。我们证明了以中间区域B为条件的两个区域A和C之间的条件互信息以B的长度的平方根为指数衰减。这两个结果构成了一维量子系统的Hammersley-Clifford定理的变体(该定理刻画了马尔可夫网络，即条件互信息为零的概率分布，作为经典短程哈密顿量的Gibbs态)。这一结果可以被视为对一维系统的热态互信息区域定律的加强。它直接意味着通过恒定深度量子电路在有限温度下有效地制备任意一维Gibbs态。
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 aquantum 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 regionsAandCconditioned on the middle regionBdecays exponentially with the square root of the length ofB. 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.