Quantum states far from the energy eigenstates of any local Hamiltonian

What quantum states are possible energy eigenstates of a many-body Hamiltonian? Suppose the Hamiltonian is non-trivial, i.e., not a multiple of the identity, and L-local, in the sense of containing interaction terms involving at most L bodies, for some fixed L. We construct quantum states \psi which are ``far away'' from all the eigenstates E of any non-trivial L-local Hamiltonian, in the sense that |\psi-E| is greater than some constant lower bound, independent of the form of the Hamiltonian.Comment: 4 page

Separable states are more disordered globally than locally

A remarkable feature of quantum entanglement is that an entangled state of two parties, Alice (A) and Bob (B), may be more disordered locally than globally. That is, S(A) > S(A,B), where S(.) is the von Neumann entropy. It is known that satisfaction of this inequality implies that a state is non-separable. In this paper we prove the stronger result that for separable states the vector of eigenvalues of the density matrix of system AB is majorized by the vector of eigenvalues of the density matrix of system A alone. This gives a strong sense in which a separable state is more disordered globally than locally and a new necessary condition for separability of bipartite states in arbitrary dimensions. We also investigate the extent to which these conditions are sufficient to characterize separability, exhibiting examples that show separability cannot be characterized solely in terms of the local and global spectra of a state. We apply our conditions to give a simple proof that non-separable states exist sufficiently close to the completely mixed state of $n$ qudits.Comment: 4 page

Fault-Tolerant Quantum Computation via Exchange interactions

Quantum computation can be performed by encoding logical qubits into the states of two or more physical qubits, and controlling a single effective exchange interaction and possibly a global magnetic field. This "encoded universality" paradigm offers potential simplifications in quantum computer design since it does away with the need to perform single-qubit rotations. Here we show that encoded universality schemes can be combined with quantum error correction. In particular, we show explicitly how to perform fault-tolerant leakage correction, thus overcoming the main obstacle to fault-tolerant encoded universality.Comment: 5 pages, including 1 figur

The Dynamics of 1D Quantum Spin Systems Can Be Approximated Efficiently

In this Letter we show that an arbitrarily good approximation to the propagator e^{itH} for a 1D lattice of n quantum spins with hamiltonian H may be obtained with polynomial computational resources in n and the error \epsilon, and exponential resources in |t|. Our proof makes use of the finitely correlated state/matrix product state formalism exploited by numerical renormalisation group algorithms like the density matrix renormalisation group. There are two immediate consequences of this result. The first is that the Vidal's time-dependent density matrix renormalisation group will require only polynomial resources to simulate 1D quantum spin systems for logarithmic |t|. The second consequence is that continuous-time 1D quantum circuits with logarithmic |t| can be simulated efficiently on a classical computer, despite the fact that, after discretisation, such circuits are of polynomial depth.Comment: 4 pages, 2 figures. Simplified argumen

Entanglement, quantum phase transitions, and density matrix renormalization

We investigate the role of entanglement in quantum phase transitions, and show that the success of the density matrix renormalization group (DMRG) in understanding such phase transitions is due to the way it preserves entanglement under renormalization. We provide a reinterpretation of the DMRG in terms of the language and tools of quantum information science which allows us to rederive the DMRG in a physically transparent way. Motivated by our reinterpretation we suggest a modification of the DMRG which manifestly takes account of the entanglement in a quantum system. This modified renormalization scheme is shown,in certain special cases, to preserve more entanglement in a quantum system than traditional numerical renormalization methods.Comment: 5 pages, 1 eps figure, revtex4; added reference and qualifying remark

Frustration, interaction strength and ground-state entanglement in complex quantum systems

Entanglement in the ground state of a many-body quantum system may arise when the local terms in the system Hamiltonian fail to commute with the interaction terms in the Hamiltonian. We quantify this phenomenon, demonstrating an analogy between ground-state entanglement and the phenomenon of frustration in spin systems. In particular, we prove that the amount of ground-state entanglement is bounded above by a measure of the extent to which interactions frustrate the local terms in the Hamiltonian. As a corollary, we show that the amount of ground-state entanglement is bounded above by a ratio between parameters characterizing the strength of interactions in the system, and the local energy scale. Finally, we prove a qualitatively similar result for other energy eigenstates of the system.Comment: 11 pages, 3 figure

A holographic proof of the strong subadditivity of entanglement entropy

When a quantum system is divided into subsystems, their entanglement entropies are subject to an inequality known as "strong subadditivity". For a field theory this inequality can be stated as follows: given any two regions of space $A$ and $B$, $S(A) + S(B) \ge S(A \cup B) + S(A \cap B)$. Recently, a method has been found for computing entanglement entropies in any field theory for which there is a holographically dual gravity theory. In this note we give a simple geometrical proof of strong subadditivity employing this holographic prescription.Comment: 9 pages, 3 figure

The spatial relation between the event horizon and trapping horizon

The relation between event horizons and trapping horizons is investigated in a number of different situations with emphasis on their role in thermodynamics. A notion of constant change is introduced that in certain situations allows the location of the event horizon to be found locally. When the black hole is accreting matter the difference in area between the two different horizons can be many orders of magnitude larger than the Planck area. When the black hole is evaporating the difference is small on the Planck scale. A model is introduced that shows how trapping horizons can be expected to appear outside the event horizon before the black hole starts to evaporate. Finally a modified definition is introduced to invariantly define the location of the trapping horizon under a conformal transformation. In this case the trapping horizon is not always a marginally outer trapped surface.Comment: 16 pages, 1 figur

Theoretical Setting of Inner Reversible Quantum Measurements

We show that any unitary transformation performed on the quantum state of a closed quantum system, describes an inner, reversible, generalized quantum measurement. We also show that under some specific conditions it is possible to perform a unitary transformation on the state of the closed quantum system by means of a collection of generalized measurement operators. In particular, given a complete set of orthogonal projectors, it is possible to implement a reversible quantum measurement that preserves the probabilities. In this context, we introduce the concept of "Truth-Observable", which is the physical counterpart of an inner logical truth.Comment: 11 pages. More concise, shortened version for submission to journal. References adde

Quantum computation with unknown parameters

We show how it is possible to realize quantum computations on a system in which most of the parameters are practically unknown. We illustrate our results with a novel implementation of a quantum computer by means of bosonic atoms in an optical lattice. In particular we show how a universal set of gates can be carried out even if the number of atoms per site is uncertain.Comment: 3 figure