We study the computational complexity of quantum discord (a measure of quantum correlation beyond entanglement), and prove that computing quantum discord is NP-complete. Therefore, quantum discord is computationally intractable: the running time of any algorithm for computing quantum discord is believed to grow exponentially with the dimension of the Hilbert space so that computing quantum discord in a quantum system of moderate size is not possible in practice. As by-products, some entanglement measures (namely entanglement cost, entanglement of formation, relative entropy of entanglement, squashed entanglement, classical squashed entanglement, conditional entanglement of mutual information, and broadcast regularization of mutual information) and constrained Holevo capacity are NP-hard/NP-complete to compute. These complexity-theoretic results are directly applicable in common randomness distillation, quantum state merging, entanglement distillation, superdense coding, and quantum teleportation; they may offer significant insights into quantum information processing. Moreover, we prove the NP-completeness of two typical problems: linear optimization over classical states and detecting classical states in a convex set, providing evidence that working with classical states is generically computationally intractable.

In many-body localized systems, propagation of information forms a light cone that grows logarithmically with time. However, local changes in energy or other conserved quantities typically spread only within a finite distance. Is it possible to detect the logarithmic light cone generated by a local perturbation from the response of a local operator at a later time? We numerically calculate various correlators in the random-field Heisenberg chain. While the equilibrium retarded correlator A(t = 0)B (t > 0) is not sensitive to the unbounded information propagation, the out-of-time-ordered corre- In the presence of disorder, localization can occur not only in single-particle systems [1], but also in interacting many-body systems [2][3][4][5][6][7][8][9][10][11][12]. The former is known as Anderson localization (AL), and the latter is called manybody localization (MBL). Neither AL nor MBL systems transfer energy, charge, or other local conserved quantities: Changes in energy or charge at position x = 0 from equilibrium can spread and lead to changes in the corresponding quantity only within a finite distance |x| < L 0 , where L 0 is the localization length.A characteristic feature that distinguishes MBL from AL lies in the dynamics of entanglement after a global quench. Initialized in a random product state at time t = 0, the half-chain entanglement entropy remains bounded in AL systems [13], but grows logarithmically with time in MBL systems [14][15][16][17][18][19][20][21]. In sharp contrast to the transport phenomena, the unbounded growth of entanglement in MBL systems suggests that information propagates throughout the system, although very slowly.The propagation of information can be formulated by adapting the Lieb-Robinson (LR) bound [22][23][24] to the present context. In particular, it is manifested as the noncommutativity of a local operator A at x = 0 and t = 0 with another local operator B at position x and evolved for some time t . In MBL systems, the operator norm of the commutator [A(0, 0), B (x, t )] is non-negligible inside a light cone whose radius is given by |x| ∼ log |t |, and decays exponentially with distance outside the light cone, i.e. [25],after averaging over disorder. Here, B (x, t ) = e i H t B (x, 0)e −i H t is the time-evolved operator; · is the operator norm (the largest singular value); C , v LR , ξ are positive constants.Is it possible to detect the logarithmic light cone (LLC) with equilibrium correlators of A(0, 0) and B (x, t )? Arguably the most straightforward approach is to measure the commutator in the LR bound (1) on equilibrium states, i.e., thermal states or eigenstates, using the Kubo formula in linear response theory: (2) where U = e −i Aτ with τ 1 is a local unitary perturbation if A is Hermitian. The first and second terms on the lefthand side of (2) are the expectation values of B in the presence and absence of the perturbation U , respectively. The difference is the effect of U observed by measuring B . Note that U and B can be, but do not have to be, chosen as the op...

Constraint satisfaction problems are a central pillar of modern computational complexity theory. This survey provides an introduction to the rapidly growing field of Quantum Hamiltonian Complexity, which includes the study of quantum constraint satisfaction problems. Over the past decade and a half, this field has witnessed fundamental breakthroughs, ranging from the establishment of a "Quantum Cook-Levin Theorem" to deep insights into the structure of 1D low-temperature quantum systems via so-called area laws. Our aim here is to provide a computer science-oriented introduction to the subject in order to help bridge the language barrier between computer scientists and physicists in the field. As such, we include the following in this survey: (1) The motivations and history of the field, (2) a glossary of condensed matter physics terms explained in computer-science friendly language, (3) overviews of central ideas from condensed matter physics, such as indistinguishable particles, mean field theory, tensor networks, and area laws, and (4) brief expositions of selected computer science-based results in the area. For example, as part of the latter, we provide a novel information theoretic presentation of Bravyi's polynomial time algorithm for Quantum 2-SAT.Comment: v4: published version, 127 pages, introduction expanded to include brief introduction to quantum information, brief list of some recent developments added, minor changes throughou

Many low-dimensional materials are well described by integrable one-dimensional models such as the Hubbard model of electrons or the Heisenberg model of spins. However, the small perturbations to these models required to describe real materials are expected to have singular effects on transport quantities: integrable models often support dissipationless transport, while weak non-integrable terms lead to finite conductivities. We use matrix-product-state methods to obtain quantitative values of spin/electrical and thermal conductivities in an almost integrable gapless (XXZ-like) spin chain. At low temperatures, we observe power laws whose exponents are solely determined by the Luttinger liquid parameter. This indicates that our results are independent of the actual model under consideration.The physics of many one-dimensional systems with idealized interactions is rather special: the quantum Hamiltonian has infinitely many independent conserved quantities that are sums of local operators. Such Hamiltonians are called "integrable" in analogy with classical Hamiltonian systems that decompose into independent action and angle variables. Examples relevant to experiments on crystalline materials include the Hubbard model of electrons and the XXZ model of spins; ultracold atomic systems can realize integrable continuum models of bosons. However, in all these cases it is expected that integrability is only an approximation to reality and that experimental systems have integrability-breaking perturbations which, while small, drastically change some of the physical properties.Transport properties provide an experimentally important example of the effects of non-integrable perturbations. In integrable systems, parts of charge, spin, or energy currents are conserved, and thus transport is dissipationless even at non-zero temperature. This corresponds to a finite "Drude weight" D in the frequency-dependent conductivity: 1-16 σ(ω) = 2πDδ(ω) + σ reg (ω) .(1)In reality, many quasi-one-dimensional systems are expected to be well described by integrable Hamiltonians plus weak non-integrable perturbations. The zerofrequency conductivity is regularized (D = 0) by these perturbations [4][5][6][7]12,14,[17][18][19][20][21] . An experimental example is the large (but not dissipationless) anisotropic thermal transport observed in Sr 14 Cu 24 O 41 attributed to a long mean free path of quasi-1D magnons 22-24 . However, computing σ reg (ω) quantitatively for a microscopic nonintegrable Hamiltonian is a challenging problem.We study a generic gapless non-integrable system (a XXZ-like spin chain) using density matrix renormalization group (DMRG) methods, which were developed in the past few years to access finite-temperature dynamics of correlated systems. Using linear prediction, we extrapolate current correlations functions to large times. This allows to quantitatively observe the destruction of the thermal and electrical Drude weights; we compute the corresponding conductivities and analyze how they depend on temperature and the strength ...