von Neumann Entropy

Notre objectif dans ce papier (partiel) est d’étudier le lien qui existe entre l’approximation linéaire des équations d’Einstein (ou approximation des champs faibles) à l’équation de Klein-Gordon. La partie 1 présente ce qu’est l’équation de Klein-Gordon ainsi que l’intégration de la théorie de l’information quantique en son sein. La partie 2 traite du tenseur énergie impulsion quantique, dans lequel je détail l’approximation des champs faibles de l’équation d’Einstein, puis dans lequel je développe le tenseur énergie impulsion quantique à partir de l’équivalence entre l’équation d’einstein à l’approximation des champs faibles et l’équation de Schrödinger relativiste décrit par l’équation de Klein-Gordon.

In this paper, we will examine the Von-Neumann's argument related to a quantum gas undergoing a reversible process, and we will show how there is a flaw when information entropy of the observer is neglected, this process can be modified by considering the observer as a separate system and the process in which quantum gas in undergoing reversible operation as another system , both of these systems are subsystem of a system, We will consider here the physical significance of 'entanglement entropy' as information entropy. We show how inclusion of observer as system affects the information entropy and thus the overall process which leads to irreversibility in the system which was undergoing reversible operation before.

We study entanglement properties of mixed density matrices obtained from combinatorial Laplacians. This is done by introducing the notion of the density matrix of a graph. We characterize the graphs with pure density matrices and show that the density matrix of a graph can be always written as a uniform mixture of pure density matrices of graphs. We consider the von Neumann entropy of these matrices and we characterize the graphs for which the minimum and maximum values are attained. We then discuss the problem of separability by pointing out that separability of density matrices of graphs does not always depend on the labelling of the vertices. We consider graphs with a tensor product structure and simple cases for which combinatorial properties are linked to the entanglement of the state. We calculate the concurrence of all graph on four vertices representing entangled states. It turns out that for some of these graphs the value of the concurrence is exactly fractional.

Abstract: We develop the argument that the Gibbs-von Neumann entropy is the appropriate statistical mechanical generalisation of the thermodynamic entropy, for macroscopic and microscopic systems, whether in thermal equilibrium or not, as a consequence of Hamiltonian dynamics. The mathematical treatment utilises well known results [Gib02, Tol38, Weh78, Par89], but most importantly, incorporates a variety of arguments on the phenomenological properties of thermal states [Szi25, TQ63, HK65, GB91] and of ...

Kishore T. Kapale Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, MS 169-315, Pasadena, CA 91109 Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803 Kishor.T.Kapale@jpl.nasa.gov

Our goal in this paper (partial) is to study the relationship between the linear approximation of Einstein's equations (or linearized gravity) to the Klein-Gordon equation. Part 1 presents what the Klein-Gordon equation and the integration of the theory of quantum information in it. Part 2 deals with the Stress Energy tensor quantum, wherein the detail I linearized gravity of Einstein equation, and wherein I develop the tensor quantum energy pulse from the equivalence of equation einstein the linearized gravity and the Schrödinger equation relativistic described by Klein-Gordon equation.

We investigate bipartite entanglement in spin-1∕2 systems on a generic lattice. For states that are an equal superposition of elements of a group G of spin flips acting on the fully polarized state ∣0⟩ ⊗n , we find that the von Neumann entropy depends only on the boundary ...

We investigate decoherence induced by a quantum channel in terms of minimal output entropy and map entropy. The latter is the von Neumann entropy of the Jamiołkowski state of the channel. Both quantities admit q-Renyi versions. We prove additivity of the map entropy for all q. For the case q = 2, we show that the depolarizing channel has the smallest map entropy among all channels with a given minimal output Renyi entropy of order two. This allows us to characterize pairs of channels such that the output entropy of their tensor product acting on a maximally entangled input state is larger than the sum of the minimal output entropies of the individual channels. We conjecture that for any channel Φ1 acting on a finite dimensional system, there exists a class of channels Φ2 sufficiently close to a unitary map such that additivity of minimal output entropy for Ψ1 ⊗ Ψ2 holds.

We compute, for massive particles, the explicit Wigner rotations of one-particle states for arbitrary Lorentz transformations; and the explicit Hermitian generators of the infinite-dimensional unitary representation. For a pair of spin 1/2 particles, Einstein-Podolsky-Rosen-Bell entangled states and their behaviour under the Lorentz group are analysed in the context of quantum field theory. Group theoretical considerations suggest a convenient definition of the Bell states which is slightly different from the conventional assignment. The behaviour of Bell states under arbitrary Lorentz transformations can then be described succinctly. Reduced density matrices applicable to identical particles are defined through Yang's prescription. The von Neumann entropy of each of the reduced density matrix is Lorentz invariant; and its relevance as a measure of entanglement is discussed, and illustrated with an explicit example. A regularization of the entropy in terms of generalized zeta fu...

The zero-entropy-density conjecture states that the entropy density, defined as the limit of S(N)/N at infinity, vanishes for all translation-invariant pure states on the spin chain. Or equivalently, S(N), the von Neumann entropy of such a state restricted to N consecutive spins, is sublinear. In this paper it is proved that this conjecture cannot be sharpened, i.e., translation-invariant states give

Based on the concept of many-letter theory, an observable is defined measuring the raw quantum information content of single messages. A general characterization of quantum codes using the Kraus representation is given. Compression codes are defined by their property of decreasing the expected raw information content of a given message ensemble. Lossless quantum codes, in contrast to lossy codes, provide the retrieval of the original data with perfect fidelity. A general lossless coding scheme is given that translates between two quantum alphabets. It is shown that this scheme is never compressive. Furthermore, a lossless quantum coding scheme, analog to the classical Huffman scheme but different from the Braunstein scheme, is implemented, which provides optimal compression. Motivated by the concept of lossless quantum compression, an observable is defined that measures the core quantum information content of a particular message with respect to a given a priori message ensemble. Th...

Relations between Shannon entropy and Rényi entropies of integer order are discussed. For any N-point discrete probability distribution for which the Rényi entropies of order two and three are known, we provide a lower and an upper bound for the Shannon entropy. The average of both bounds provide an explicit extrapolation for this quantity. These results imply relations between the von Neumann entropy of a mixed quantum state, its linear entropy and traces.

The discrete-time quantum walk is a quantum counterpart of the random walk. It is expected that the model plays important roles in the quantum field. In the quantum information theory, entanglement is a key resource. We use the von Neumann entropy to measure the entanglement between the coin and the particle's position of the quantum walks. Also we deal with the Shannon entropy which is an important quantity in the information theory. In this paper, we show limits of the von Neumann entropy and the Shannon entropy of the quantum walks on the one dimensional lattice starting from the origin defined by arbitrary coin and initial state. In order to derive these limits, we use the path counting method which is a combinatorial method for computing probability amplitude.

It is well known that projective measurement will not decrease the von Neumann entropy of a quantum state. In this paper, it is shown that projective measurement will not decrease the quantum Tsallis entropy of a quantum state, either. Using a similar analysis, it can be shown that projective measurement will not decrease the quantum unified (r, s)-entropy in general.

We consider a quantum linear oscillator coupled at an arbitrary strength to a bath at an arbitrary temperature. We find an exact closed expression for the oscillator density operator. This state is non-canonical but can be shown to be equivalent to that of an uncoupled linear oscillator at an effective temperature T_eff with an effective mass and an effective spring constant. We derive an effective Clausius inequality delta Q_eff =< T_eff dS, where delta Q_eff is the heat exchanged between the effective (weakly coupled) oscillator and the bath, and S represents a thermal entropy of the effective oscillator, being identical to the von-Neumann entropy of the coupled oscillator. Using this inequality (for a cyclic process in terms of a variation of the coupling strength) we confirm the validity of the second law. For a fixed coupling strength this inequality can also be tested for a process in terms of a variation of either the oscillator mass or its spring constant. Then it is neve...

In this paper, we study the entanglement properties of a spin-1 model the exact ground state of which is given by a Matrix Product state. The model exhibits a critical point transition at a parameter value a=0. The longitudinal and transverse correlation lengths are known to diverge as a tends to zero. We use three different entanglement measures S(i) (the one-site von Neumann entropy), S(i,j) (the two-body entanglement) and G(2,n) (the generalized global entanglement) to determine the entanglement content of the MP ground state as the parameter a is varied. The entanglement length, associated with S(i,j), is found to diverge in the vicinity of the quantum critical point a=0. The first derivative of the entanglement measure E (=S(i), S(i,j)) w.r.t. the parameter a also diverges. The first derivative of G(2,n) w.r.t. a does not diverge as a tends to zero but attains a maximum value at a=0. At the QCP itself all the three entanglement measures become zero. We further show that multipa...

A definition of quantum chaos is given in terms of entropy production rates for a quantum system coupled weakly to a reservoir. This allows the treatment of classical and quantum chaos on the same footing. In the quantum theory the entropy considered is the von Neumann entropy and in classical systems it is the Gibbs entropy. The rate of change of the coarsegrained Gibbs entropy of the classical system with time is given by the Kolmogorov–Sinai (KS) entropy. The relation between KS entropy and the rate of change of von Neumann entropy is investigated for the kicked rotator. For a system which is classically chaotic there is a linear relationship between these two entropies. Moreover it is possible to construct contour plots for the local KS entropy and compare it with the corresponding plots for the rate of change of von Neumann entropy. The quantative and qualitative similarities of these plots are discussed for the standard map (kicked rotor) and the generalised cat maps.