We prove an area law for the entanglement entropy in gapped one dimensional quantum systems. The bound on the entropy grows surprisingly rapidly with the correlation length; we discuss this in terms of properties of quantum expanders and present a conjecture on completely positive maps which may provide an alternate way of arriving at an area law. We also show that, for gapped, local systems, the bound on Von Neumann entropy implies a bound on Rényi entropy for sufficiently large α < 1 and implies the ability to approximate the ground state by a matrix product state.There are many reasons to believe that the entanglement entropy of a quantum system with a gap obeys an area law: that the entanglement entropy of a given region scales as the boundary area, rather than as the volume. In one dimension, conformal field theory calculations show that away from the critical point the entanglement entropy is bounded, diverging proportionally to the correlation length as a critical point is approached [1]. In higher dimensions, systems represented by matrix product states [2,3] obey an area law.However, despite this, there is no general proof of an area law. This is somewhat surprising, since it has been proven that correlation functions in a gapped system decay exponentially [4], and one might guess that the decay of correlation functions implies that only degrees of freedom near the boundary of the region may entangle with those outside. However, the existence of data hiding states [5] shows that one can have states on bipartite systems with small correlations and large entanglement. Further, the existence of quantum expanders [6,7] shows that one may have matrix product states in one dimension that have all correlation functions decaying exponentially in the distance between the operators, but still have large entanglement. This indicates some of the difficulty in proving an area law.At the same time, for a gapped system to violate an area law would require some very strange properties. For one thing [8], the thermal density matrix can be well approximated by a matrix product operator. This implies that unless a plausible assumption[6] on the density of low energy states is violated, the ground state can be well approximated by a matrix product state.In this paper, we succeed in providing a proof of an area law for one dimensional systems under the assumption of a gap. The result, however, bounds the entanglement entropy by a quantity that grows exponentially in the correlation length. This is much faster than the linear growth one might have expected. We will comment later on why this bound might in fact be reasonably tight. In the process of deriving this result, we will derive bounds for gapped local systems which inter-relate three quantities: the Von Neumann entropy, the Rényi entropy, and the error involved in approximating the ground state by a matrix product state.We begin by defining the lattice and Hamiltonian. We consider finite range Hamiltonians for simplicity. It is likely that the results can be extended ...

We study the relation between the spectral gap above the ground state and the decay of the correlations in the ground state in quantum spin and fermion systems with short-range interactions on a wide class of lattices. We prove that, if two observables anticommute with each other at large distance, then the nonvanishing spectral gap implies exponential decay of the corresponding correlation. When two observables commute with each other at large distance, the connected correlation function decays exponentially under the gap assumption. If the observables behave as a vector under the U(1) rotation of a global symmetry of the system, we use previous results on the large distance decay of the correlation function to show the stronger statement that the correlation function itself, rather than just the connected correlation function, decays exponentially under the gap assumption on a lattice with a certain self-similarity in (fractal) dimensions D < 2. In particular, if the system is translationally invariant in one of the spatial directions, then this self-similarity condition is automatically satisfied. We also treat systems with long-range, power-law decaying interactions.

The Lieb-Robinson bound states that local Hamiltonian evolution in nonrelativistic quantum mechanical theories gives rise to the notion of an effective light cone with exponentially decaying tails. We discuss several consequences of this result in the context of quantum information theory. First, we show that the information that leaks out to spacelike separated regions is negligible and that there is a finite speed at which correlations and entanglement can be distributed. Second, we discuss how these ideas can be used to prove lower bounds on the time it takes to convert states without topological quantum order to states with that property. Finally, we show that the rate at which entropy can be created in a block of spins scales like the boundary of that block. The principle of causality forms one of the pillars of modern physics. It dictates that there is a finite speed at which information can propagate. Because of the existence of a light cone, relativistic quantum field theories automatically exhibit that property. The situation is, however, not so clear in nonrelativistic quantum mechanics, where a strict notion of a light cone does not exist. It has indeed been noticed that local operations can in principle be used to send information over arbitrary distances in arbitrary small times [1]. The seminal work of Lieb and Robinson [2] and recent generalizations due to Hastings [3] and Nachtergaele and Sims [4], however, show that the situation in not so bad: if evolution is governed by local Hamiltonians, then nonrelativistic quantum mechanics gives rise to an effective light cone with exponentially decaying tails. Because of this exponential attenuation, we will show how a quantitative version of causality emerges where the amount of information that can be exchanged is exponentially small within space-time regions not connected by a light cone.A related question is how fast correlations can be created between two widely separated regions in space. Note that in this case, the principle of causality does not prohibit the buildup of correlations faster than the speed of light, as correlations as such cannot be used to signal information; this is precisely the argument used to show that the existence of entanglement does not violate causality. Again using the Lieb-Robinson bound, we will show that there is a finite velocity at which correlations can be distributed. This automatically implies that the time it takes to distribute entanglement between two nodes in a spin-network scales as the distance between the nodes, solving an open question raised in [5]. Note that we assume that all classical communication is also described by local Hamiltonian evolution, as otherwise it is possible to distribute entanglement over arbitrary distances in a single unit of time by making use of the concept of quantum teleportation [6,7]. Similar techniques can be used to prove lower bounds on the time it takes to create exotic quantum states exhibiting topological quantum order [8,9] by local Hamiltonian evolution: a time propo...

The preparation of quantum states using short quantum circuits is one of the most promising near-term applications of small quantum computers, especially if the circuit is short enough and the fidelity of gates high enough that it can be executed without quantum error correction. Such quantum state preparation can be used in variational approaches, optimizing parameters in the circuit to minimize the energy of the constructed quantum state for a given problem Hamiltonian. For this purpose we propose a simple-to-implement class of quantum states motivated by adiabatic state preparation. We test its accuracy and determine the required circuit depth for a Hubbard model on ladders with up to 12 sites (24 spin-orbitals), and for small molecules. We find that this ansatz converges faster than previously proposed schemes based on unitary coupled clusters. While the required number of measurements is astronomically large for quantum chemistry applications to molecules, applying the variational approach to the Hubbard model (and related models) is found to be far less demanding and potentially practical on small quantum computers. We also discuss another application of quantum state preparation using short quantum circuits, to prepare trial ground states of models faster than using adiabatic state preparation.

A generalization of the Lieb-Schultz-Mattis theorem to higher dimensional spin systems is shown. The physical motivation for the result is that such spin systems typically either have long-range order, in which case there are gapless modes, or have only short-range correlations, in which case there are topological excitations. The result uses a set of loop operators, analogous to those used in gauge theories, defined in terms of the spin operators of the theory. We also obtain various cluster bounds on expectation values for gapped systems. These bounds are used, under the assumption of a gap, to rule out the first case of long-range order, after which we show the existence of a topological excitation. Compared to the ground state, the topologically excited state has, up to a small error, the same expectation values for all operators acting within any local region, but it has a different momentum.

We present designs for scalable quantum computers composed of qubits encoded in aggregates of four or more Majorana zero modes, realized at the ends of topological superconducting wire segments that are assembled into superconducting islands with significant charging energy. Quantum information can be manipulated according to a measurement-only protocol, which is facilitated by tunable couplings between Majorana zero modes and nearby semiconductor quantum dots. Our proposed architecture designs have the following principal virtues:(1) the magnetic field can be aligned in the direction of all of the topological superconducting wires since they are all parallel; (2) topological T junctions are not used, obviating possible difficulties in their fabrication and utilization; (3) quasiparticle poisoning is abated by the charging energy; (4) Clifford operations are executed by a relatively standard measurement: detection of corrections to quantum dot energy, charge, or differential capacitance induced by quantum fluctuations; (5) it is compatible with strategies for producing good approximate magic states.

The holographic principle states that on a fundamental level the information content of a region should depend on its surface area rather than on its volume. In this Letter we show that this phenomenon not only emerges in the search for new Planck-scale laws but also in lattice models of classical and quantum physics: the information contained in part of a system in thermal equilibrium obeys an area law. While the maximal information per unit area depends classically only on the number of degrees of freedom, it may diverge as the inverse temperature in quantum systems. It is shown that an area law is generally implied by a finite correlation length when measured in terms of the mutual information.

The design of error-correcting codes used in modern communications relies on information theory to quantify the capacity of a noisy channel to send information 1 . This capacity can be expressed using the mutual information between input and output for a single use of the channel; although correlations between subsequent input bits are used to correct errors, they cannot increase the capacity. For quantum channels, it has been an open question whether entangled input states can increase the capacity to send classical information 2 . The additivity conjecture 3,4 states that entanglement does not help, making practical computations of the capacity possible. Although additivity is widely believed to be true, there is no proof. Here, we show that additivity is false, by constructing a random counter-example. Our results show that the most basic question of classical capacity of a quantum channel remains open, with further work needed to determine in which other situations entanglement can boost capacity.In the classical setting, Shannon presented a formal definition of a noisy channel E as a probabilistic map from input states to output states. In the quantum setting, the channel becomes a linear, completely positive, trace-preserving map from density matrices to density matrices, modelling noise in the system due to interaction with an environment. Such a channel can be used to send either quantum or classical information. In the first case, a marked violation of operational additivity was recently shown, in that there exist two channels, both having zero capacity to send quantum information no matter how many times it is used, which can be used in tandem to send quantum information 5 .Here, we address the classical capacity of a quantum channel. To specify how information is encoded in the channel, we must pick a set of states ρ i which we use as input signals with probabilities p i . Then the Holevo formula 2 for the capacity is:where H (ρ) = −Tr(ρ ln(ρ)) is the von Neumann entropy. The maximum capacity of a channel is the maximum over all input ensembles:Suppose we have two different channels, E 1 and E 2 . To compute this capacity, it seems necessary to consider entangled input states between the two channels. Similarly, when using the same channel multiple times, it may be useful to use input states that are entangled across multiple uses of the same channel. The additivity conjectureCenter for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA. *e-mail: hastings@lanl.gov. a, A set of states ρ i are used with probabilities p i as signal states on the channel E. The inputs are unentangled between channels E and E. The capacity of E is equal to that of E. b, A set of entangled input states ρ i are used on the channel E ⊗ E. The question addressed is whether entangling can increase capacity.(see Fig. 1) is the conjecture that this does not help and that insteadThe additivity conjecture makes it possible to compute the classical capacity of a quantum chan...