Matrix product states form a powerful ansatz for the simulation of a wide range of one-dimensional quantum systems that are in a pure state.Their power stems from the fact that they faithfully approximate states with a low amount of entanglement, the "area law."However, in order to accurately capture the physics of realistic systems, one generally needs to apply a mixed-state description.In this work, we establish the mixed-state analog of this characterization.We show that one-dimensional mixed states with a low amount of entanglement, quantified by the entanglement of purification, can be efficiently approximated by matrix product density operators.
Matrix product states form a powerful ansatz for the simulation of a wide range of one-dimensional quantum systems that are in a pure state.Their power stems from the fact that they faithfully approximate states with a low amount of entanglement, the "area law."However, in order to accurately capture the physics of realistic systems, one generally needs to apply a mixed-state description.In this work, we establish the mixed-state analog of this characterization.We show that one-dimensional mixed states with a low amount of entanglement, quantified by the entanglement of purification, can be efficiently approximated by matrix product density operators.
We analyze the description of quantum many-body mixed states using matrix product states and operators. We consider two such descriptions: (i) as a matrix product density operator of bond dimension D ; and (ii) as a purification that is written as a matrix product state of bond dimension D '. We show that these descriptions are inequivalent in the sense that D ' cannot be upper bounded by D only. Then we provide two constructive methods to obtain (ii) out of (i). The sum of squares (sos) polynomial method scales exponentially in the number of different eigenvalues, and its approximate version is formulated as a semidefinite program, which gives efficient approximate purifications whose D ' only depends on D . The eigenbasis method scales quadratically in the number of eigenvalues, and its approximate version is very efficient for rapidly decaying distributions of eigenvalues. Our results imply that a description of mixed states which is both efficient and locally positive semidefinite does not exist, but that good approximations do.
Classical simulation of quantum many-body systems is in general a challenging problem for the simple reason that the dimension of the Hilbert space grows exponentially with the system size. In particular, merely encoding a generic quantum many-body state requires an exponential number of bits. However, condensed matter physicists are mostly interested in local Hamiltonians and especially their ground states, which are highly non-generic. Thus, we might hope that at least some physical systems allow efficient classical simulation.Starting with one-dimensional (1D) quantum systems (i.e., the simplest nontrivial case), the first basic question is: Which classes of states have efficient classical representations? It turns out that this question is quantitatively related to the amount of entanglement in the state, for states with ``little entanglement'' are well approximated by matrix product states (a data structure that can be manipulated efficiently on a classical computer). At a technical level, the mathematical notion for ``little entanglement'' is area law, which has been proved for unique ground states in 1D gapped systems. We establish an area law for constant-fold degenerate ground states in 1D gapped systems and thus explain the effectiveness of matrix-product-state methods in (e.g.) symmetry breaking phases. This result might not be intuitively trivial as degenerate ground states in gapped systems can be long-range correlated.Suppose an efficient classical representation exists. How can one find it efficiently? The density matrix renormalization group is the leading numerical method for computing ground states in 1D quantum systems. However, it is a heuristic algorithm and the possibility that it may fail in some cases cannot be completely ruled out. Recently, a provably efficient variant of the density matrix renormalization group has been developed for frustration-free 1D gapped systems. We generalize this algorithm to all (i.e., possibly frustrated) 1D gapped systems. Note that the ground-state energy of 1D gapless Hamiltonians is computationally intractable even in the presence of translational invariance.It is tempting to extend methods and tools in 1D to two and higher dimensions (2+D), e.g., matrix product states are generalized to tensor network states. Since an area law for entanglement (if formulated properly) implies efficient matrix product state representations in 1D, an interesting question is whether a similar implication holds in 2+D. Roughly speaking, we show that an area law for entanglement (in any reasonable formulation) does not always imply efficient tensor network representations of the ground states of 2+D local Hamiltonians even in the presence of translational invariance. It should be emphasized that this result does not contradict with the common sense that in practice quantum states with more entanglement usually require more space to be stored classically; rather, it demonstrates that the relationship between entanglement and efficient classical representations is still far from being well understood.Excited eigenstates participate in the dynamics of quantum systems and are particularly relevant to the phenomenon of many-body localization (absence of transport at finite temperature in strongly correlated systems). We study the entanglement of excited eigenstates in random spin chains and expect that its singularities coincide with dynamical quantum phase transitions. This expectation is confirmed in the disordered quantum Ising chain using both analytical and numerical methods.Finally, we study the problem of generating ground states (possibly with topological order) in 1D gapped systems using quantum circuits. This is an interesting problem both in theory and in practice. It not only characterizes the essential difference between the entanglement patterns that give rise to trivial and nontrivial topological order, but also quantifies the difficulty of preparing quantum states with a quantum computer (in experiments).
Entanglement plays an important role in our ability to understand, simulate, and harness quantum many-body phenomena. In this work, we investigate the entanglement spectrum for open one-dimensional (1D) systems and propose a natural quantifier for how much a 1D quantum state is entangled while being subject to decoherence. We demonstrate our method using a simple case of single-particle evolution and find that the open system entanglement spectrum is composed of generalized concurrence values, as well as quantifiers of the state's purity. Our proposed entanglement spectrum can be directly obtained using a correct scaling of a matrix product state decomposition of the system's density matrix. Our method thus offers observables that are easily acquired in the study of interacting 1D systems and sheds light on the approximations employed in matrix product state simulations of open system dynamics.
Tensor networks [1] and in particular matrix-product states [2] have proven extremely useful for the investigation of quantum many-body states, such as, e.g., ground states of local many-body Hamiltonians. They provide an efficient description of these states, avoiding the exponential increase of Hilbert space with particle number at the cost of limiting the entanglement. In particular for matrix-product states, the relation between area laws of Rényi entanglement entropies and efficient simulability of quantum states is nowadays well understood [3]. However, it is not always clear a priori whether or not the relevant states of a given quantum system (e.g., its eigenstates) fulfil such an area law and can thus efficiently be represented by matrix-product states. By investigating specific states of relevance for many-body quantum dynamics, in particular symmetric states and Fock states, we here aim to derive criteria beyond the area law to assess for which quantum systems a description by matrix-product states or more general tensor networks is beneficial. [1] S. Montangero, Introduction to Tensor Networks (Springer, Cham, 2018) [2] J. I. Cirac, D. Pérez-García, N. Schuch, and F. Verstraete, Rev. Mod. Phys 93, 045003 (2021) [3] N. Schuch, M. W. Wolf, F. Verstraete, and J. I. Cirac, Phys. Rev. Lett. 100, 030504 (2008)
We show a fundamental limitation in the description of quantum many-body mixed states with tensor networks in purification form. Namely, we show that there exist mixed states which can be represented as a translationally invariant (TI) matrix product density operator valid for all system sizes, but for which there does not exist a TI purification valid for all system sizes. The proof is based on an undecidable problem and on the uniqueness of canonical forms of matrix product states. The result also holds for classical states.
The entanglement area law is a universal principle that characterizes quantum many-body phases and underpins tensor network algorithms. Traditionally, its validity has been limited to systems with short-range interactions and bounded local energy. Achieving a complete generalization that removes both of these constraints has been a longstanding goal in quantum many-body theory, especially for interacting boson systems where unbounded energy presents intrinsic difficulties. In this work, we rigorously prove the area law for one-dimensional interacting boson systems with long-range interactions, covering broad models including the Bose-Hubbard and ϕ4 classes. Furthermore, we establish an efficiency guarantee for Matrix-Product-State approximations of the ground states, offering a practical route to numerical simulation. One of our main technical contributions is a general method for Hilbert space dimension reduction, whose applicability extends to arbitrary spatial dimensions. These results address two major challenges simultaneously and provide important foundations for simulating long-range cold atomic systems. Entanglement area law has been established in systems with short range interactions and bounded local energy. Here the authors extend the proof to 1D bosonic systems with long-range interactions and unbounded local energy and outline implications for matrix product state approximations of the ground states.