Tensor Network States and Geometry

Tensor network states are used to approximate ground states of local Hamiltonians on a lattice in D spatial dimensions. Different types of tensor network states can be seen to generate different geometries. Matrix product states (MPS) in D=1 dimensions, as well as projected entangled pair states (PEPS) in D>1 dimensions, reproduce the D-dimensional physical geometry of the lattice model; in contrast, the multi-scale entanglement renormalization ansatz (MERA) generates a (D+1)-dimensional holographic geometry. Here we focus on homogeneous tensor networks, where all the tensors in the network are copies of the same tensor, and argue that certain structural properties of the resulting many-body states are preconditioned by the geometry of the tensor network and are therefore largely independent of the choice of variational parameters. Indeed, the asymptotic decay of correlations in homogeneous MPS and MERA for D=1 systems is seen to be determined by the structure of geodesics in the physical and holographic geometries, respectively; whereas the asymptotic scaling of entanglement entropy is seen to always obey a simple boundary law—that is, again in the relevant geometry. This geometrical interpretation offers a simple and unifying framework to understand the structural properties of, and helps clarify the relation between, different tensor network states. In addition, it has recently motivated the branching MERA, a generalization of the MERA capable of reproducing violations of the entropic boundary law in D>1 dimensions.     

Entanglement-Structured LSTM Boosts Chaotic Time Series Forecasting

Traditional machine-learning methods are inefficient in capturing chaos in nonlinear dynamical systems, especially when the time difference Δt between consecutive steps is so large that the extracted time series looks apparently random. Here, we introduce a new long-short-term-memory (LSTM)-based recurrent architecture by tensorizing the cell-state-to-state propagation therein, maintaining the long-term memory feature of LSTM, while simultaneously enhancing the learning of short-term nonlinear complexity. We stress that the global minima of training can be most efficiently reached by our tensor structure where all nonlinear terms, up to some polynomial order, are treated explicitly and weighted equally. The efficiency and generality of our architecture are systematically investigated and tested through theoretical analysis and experimental examinations. In our design, we have explicitly used two different many-body entanglement structures—matrix product states (MPS) and the multiscale entanglement renormalization ansatz (MERA)—as physics-inspired tensor decomposition techniques, from which we find that MERA generally performs better than MPS, hence conjecturing that the learnability of chaos is determined not only by the number of free parameters but also the tensor complexity—recognized as how entanglement entropy scales with varying matricization of the tensor.     

Unifying variational methods for simulating quantum many-body systems

We introduce a unified formulation of variational methods for simulating ground state properties of quantum many-body systems. The key feature is a novel variational method over quantum circuits via infinitesimal unitary transformations, inspired by flow equation methods. Variational classes are represented as efficiently contractible unitary networks, including the matrix-product states of density matrix renormalization, multiscale entanglement renormalization (MERA) states, weighted graph states, and quantum cellular automata. In particular, this provides a tool for varying over classes of states, such as MERA, for which so far no efficient way of variation has been known. The scheme is flexible when it comes to hybridizing methods or formulating new ones. We demonstrate the functioning by numerical implementations of MERA, matrix-product states, and a new variational set on benchmarks.     

Meta-Entanglement

We give a meta-logical interpretation of the entanglement mechanism of quantum space-time in terms of the sequent calculus of a quantum sub-structural logic. This meta-logical picture is based mainly on the two meta-rules cut and EPR, and on the new meta-theorem “teleportation” (TEL), built by the use of the above meta-rules, both performed in parallel. The proof of (TEL)-theorem fairly reproduces the protocol of quantum teleportation. In the framework of space-time entanglement, the conclusion of the (TEL)-theorem is that the entangled space-time can convey the quantum teleportation of an unknown quantum state. We also introduce two new structural rules: the Hadamard (H)-rule and the CNOT-rule, the latter being used, together with the cut, in the proof of the new theorem “Entanglement” (ENT).

     

Information Theories with Adversaries,Intrinsic Information,and Entanglement

There are aspects of privacy theory that are analogous to quantum theory. In particular one can define distillable key and key cost in parallel to distillable entanglement and entanglement cost. We present here classical privacy theory as a particular case of information theory with adversaries, where similar general laws hold as in entanglement theory. We place the result of Renner and Wolf—that intrinsic information is lower bound for key cost—into this general formalism. Then we show that the question of whether intrinsic information is equal to key cost is equivalent to the question of whether Alice and Bob can create a distribution product with Eve using I_M bits of secret key. We also propose a natural analogue of relative entropy of entanglement in privacy theory and show that it is equal to the intrinsic information. We also provide a formula analogous to the entanglement of formation for classical distributions. It is our pleasure to dedicate this paper to Asher Peres on the occasion of his seventieth birthday.     

Dynamic Entanglement and Separability Criteria for Quantum Computing Bit States

A theoretical framework is demonstrated to evaluate the degree of entanglement of bit states in quantum computing. Separability of general superposition of Hilbert space unit vectors is discussed, and criteria in amplitude as well as in phase are derived. By these criteria the possibility of different quantum gates such as controlled not (CN), controlled controlled not (CCN), controlled rotation (CR), and controlled phase shift (CPS), to create the entanglement is examined. Furthermore, the selection of measurement mode external to the quantum system is incorporated in the formula using Kronecker delta (δ _kx ), introducing the concept of dynamic entanglement. With this the process of wavefunction collapse upon measurement is understood as the result of the activation of the dynamic entanglement. A firefly in a box model is used to show a pure state of ontological uncertainty, which is in a dynamically entangled state in Hilbert space.     

The Noether Conservation Laws of Some Vaidiya Metrics

In this paper, we show that a large amount information can be extracted from a knowledge of the vector fields that leave the action integral invariant, viz., Noether symmetries. In addition to a larger class of conservation laws than those given by the isometries or Killing vectors, we may conclude what the isometries are and that these form a Lie subalgebra of the Noether symmetry algebra. We perform our analysis on versions of the Vaidiya metric yielding some previously unknown information regarding the corresponding manifold. Lastly, with particular reference to this metric, we show that the only variations on m(u) that occur are m=0, m=constant, m=u and m=m(u).     

Lectures on attractors and black holes

A basic survey on some aspects of four‐dimensional black holes (BHs) is given in these lectures. It covers thermodynamical properties as well as the Attractor Mechanism for extremal BHs in an environment of scalar field background. Some relevant formulæ for the critical points of the BH “effective potential” are discussed, and the simplest example uncovering the attractor behavior, the Maxwell‐Einstein‐dilaton supergravity, is analyzed in detail. Observations on similarities between BH entropy (as given by the Bekenstein‐Hawking entropy‐area formula) and multipartite entanglement of qubits in quantum information theory are reported, as well. Finally, among the latest developments, the moduli space of attractor points for 𝒩 ≥ 2 supergravities is also considered. Based on lectures given by S. Ferrara at the International School of Subnuclear Physics, 45th Course: Search for the “Totally Unexpected” in the LHC era, Erice, Italy, 29 August – 7 September 2007 (Directors: G. 't Hooft – A. Zichichi), and at the III Avogadro Meeting on Theoretical Physics, Alessandria, Italy, 19 – 21 December 2007.     

Entanglement Entropy Evolution under Double‐trace Deformation

In this paper, we study the bulk entanglement entropy evolution in conical BTZ black bole background using the heat kernel method. This is motivated by exploring the new examples where the quantum correction of the entanglement entropy gives the leading contribution. We find that in the large black hole limit the bulk entanglement entropy decreases under the double‐trace deformation which is consistent with the holographic c theorem and in the small black hole limit the bulk entanglement entropy increases under the deformation. We also discuss the minimal area correction.     

On Schwarzschild black hole in large N matrix theory

D0-brane gas picture of Schwarzschild black hole (SBH) is considered in the large N regime of Matrix theory. An entropy formula, which reproduces the thermodynamics of SBH in the large 0N limit for all dimensions (D≥6), is proposed. The equations of states for supersymmetric Yang-Mills theory at low temperature are obtained. We also give a proof of the Newton gravitation law between two SBHs, whose masses are not equal.     

Mutual Information of Pauli Channels with Correlated Noise

A general formula for the mutual information of the Paufi channels with memory modelled by correlated noise is derived. It is shown that the mutual information depends on the channel shrinking factor, the input state parameter and the channel memory coefficient. The analyses based on the general formula reveal that the entanglement is always a useful resource to enhance the mutual information of some Pauli channels, such as the bit flip channel and the bit-phase flip channel. Our analyses also show that the entanglement is not advantageous to the reliable transmission of classical information for some Pauli channels at any time, such as the phase flip channel and the phase damping channel.     

Bits,time, carriers,and matter

The formal structure of quantum information theory is based on the well-founded concepts and postulates of quantum mechanics. In the present contribution, I am inverting the usual approach presented in textbooks by beginning with the use of bit states as basic and fundamental units of information and establish a dynamical map for them. The condition of reversibility, imposed on an ordered sequence of actions operating on a bit state, introduces, by necessity, the unitarity property of actions. I also verify that the uniformity of time, as a parameter for ordering events, is due to the admission of a composition law for the actions. In the limit of infinitesimal intervals between actions, a reversible and linear equation arises for the dynamical changes in time of a qubit (superposition of bit states). The admission that a bit of information is stored or carried by a massive particle necessarily leads to the Schrödinger–Pauli equation (SPE); the bit is associated to a spin 1/2. Within this approach, I verify that the particle dynamical equation becomes “enslaved” by the spin dynamics. In other words, the bit (or spin) precedes in status the particle dynamical evolution, being at the root of the quantum character of the standard Schr¨odinger equation, even when spin and spatial degrees of freedom are uncoupled.     

D-branes and fat black holes

The application of D-brane methods to large black holes whose Schwarzschild radius is larger than the compactification scale is problematic. Callan and Maldacena have suggested that despite apparent problems of strong interactions when the number of branes becomes large, the open string degrees of freedom may remain very dilute due to the growth of the horizon area which they claim grows more rapidly than the average number of open strings. Such a picture of a dilute weakly coupled string system conflicts with the picture of a dense string soup that saturates the bound of one string per Planck area. A more careful analysis shows that Callan and Maldacena were not fully consistent in their estimates. In the form that their model was studied it can not be used to extrapolate to large mass without being in conflict with the Hawking-Bekenstein entropy formula. A somewhat modified model can reproduce the correct entropy formula. In this “improved model” the number of string bits on the horizon scales like the entropy in agreement with earlier speculations of Susskind.     

Entanglement distribution and entangling power of quantum gates

Quantum gates, which play a fundamental role in quantum computation and other quantum information processes, are unitary evolution operators Û that act on a composite system, changing its entanglement. In the present contribution, we study some aspects of these entanglement changes. By recourse to a Monte Carlo procedure, we compute the so-called “entangling power” for several paradigmatic quantum gates and discuss results concerning the action of the CNOT gate. We pay special attention to the distribution of entanglement among the several parties involved.     

Entanglement can Increase Asymptotic Rates of Zero-Error Classical Communication over Classical Channels

It is known that the number of different classical messages which can be communicated with a single use of a classical channel with zero probability of decoding error can sometimes be increased by using entanglement shared between sender and receiver. It has been an open question to determine whether entanglement can ever increase the zero-error communication rates achievable in the limit of many channel uses. In this paper we show, by explicit examples, that entanglement can indeed increase asymptotic zero-error capacity, even to the extent that it is equal to the normal capacity of the channel.     

Urbaryon Model Approach to Unified Understanding of Large and Small PT Spectra

Single-particle distributions in the entire Peyrou plot at CERN-PS ∼ ISR energies are unifiedly described by the whole-region formula, which incorporates both parton model results at large p_T and all of the Mueller-Regge forms at small p_T. The quantum number dependence of the distributions is explained by the formula as due to the intimate relation between the energy-momentum flow and the quantum-number flow carried by valence urbaryons. A field theoretical basis of the formula is provided by a covariant model of the hadron-urbaryon amplitude which is parametrized in terms of valence urbaryon lines and pair effects. Two-body reactions at all angles are also described by the formula as the exclusive limit of inclusive reactions. The construction of a formula for inclusive two-particle distribution is briefly discussed.     

Null Wave Front and Ryu–Takayanagi Surface

The Ryu–Takayanagi formula provides the entanglement entropy of quantum field theory as an area of the minimal surface (Ryu–Takayanagi surface) in a corresponding gravity theory. There are some attempts to understand the formula as a flow rather than as a surface. In this paper, we consider null rays emitted from the AdS boundary and construct a flow representing the causal holographic information. We present a sufficient and necessary condition that the causal information surface coincides with Ryu–Takayanagi surface. In particular, we show that, in spherical symmetric static spacetimes with a negative cosmological constant, wave fronts of null geodesics from a point on the AdS boundary become extremal surfaces and therefore they can be regarded as the Ryu–Takayanagi surfaces. In addition, from the viewpoint of flow, we propose a wave optical formula to calculate the causal holographic information.