Devising schemes for testing the amount of entanglement in quantum systems has played a crucial role in quantum computing and information theory. Here, we study the problem of testing whether an unknown state $|\psi\rangle$ is a matrix product state (MPS) in the property testing model. MPS are a class of physically-relevant quantum states which arise in the study of quantum many-body systems. A quantum state $|\psi_{1,...,n}\rangle$ comprised of $n$ qudits is said to be an MPS of bond dimension $r$ if the reduced density matrix $\psi_{1,...,k}$ has rank $r$ for each $k \in \{1,...,n\}$. When $r=1$, this corresponds to the set of product states. For larger values of $r$, this yields a more expressive class of quantum states, which are allowed to possess limited amounts of entanglement. In the property testing model, one is given $m$ identical copies of $|\psi\rangle$, and the goal is to determine whether $|\psi\rangle$ is an MPS of bond dimension $r$ or whether $|\psi\rangle$ is far from all such states. For the case of product states, we study the product test, a simple two-copy test previously analyzed by Harrow and Montanaro (FOCS 2010), and a key ingredient in their proof that $\mathsf{QMA(2)}=\mathsf{QMA}(k)$ for $k \geq 2$. We give a new and simpler analysis of the product test which achieves an optimal bound for a wide range of parameters, answering open problems of Harrow and Montanaro (FOCS 2010) and Montanaro and de Wolf (2016). For the case of $r\geq 2$, we give an efficient algorithm for testing whether $|\psi\rangle$ is an MPS of bond dimension $r$ using $m = O(n r^2)$ copies, independent of the dimensions of the qudits, and we show that $\Omega(n^{1/2})$ copies are necessary for this task. This lower bound shows that a dependence on the number of qudits $n$ is necessary, in sharp contrast to the case of product states where a constant number of copies suffices.
翻译:用于测试量子系统中的纠缠量的计划在量子计算和信息理论中扮演了关键角色。 在这里, 我们研究测试一个未知状态$%psi\rangle$是否在属性测试模型中的矩阵产品状态( MPS) 的问题。 MPS是一个在量子多体系统研究中产生的与物理相关的量状态类别。 量子状态 $%psi1,..., 由 $nqrangle 构成的量子量值在量子计算和信息理论中扮演了关键值的 $( $%%2x美元 美元) 。 当一个未知状态$%psi\rangle$1, 美元是2美元 美元 美元 。 当量子体质测试模型中, 量子体质状态的量级比较明确, 量质子状态的量级数量有限。 在财产测试模型中, 以美元 美元 和 量基质子 美元 的值, 一个是相同的 美元 美元 美元,, 而这个目标是要确定 美元或 美元 美元 数据 。