Equilibrium Statistical Ensembles and Structure of the Entropy Functional in Generalized Quantum Dynamics

We review the microcanonical and canonicalensembles constructed on an underlying generalizedquantum dynamics and the algebraic properties of theconserved quantities. We discuss the structure imposed on the microcanonical entropy by theequilibrium conditions.     

Multiple Entropy Measures for Multi-particle Pure Quantum State

We propose to use a set of averaged entropies, the multiple entropymeasures (MEMS), to partially quantify quantum entanglement ofmultipartite quantum state. The MEMS is vector-like with m=[N/2]components: [S₁, S₂,..., S_m], and the $i$-th component S_i is the geometric mean of i-qubits partial entropy of the system. The S_i measures how strong an arbitrary i qubits from the system are correlated with the rest of the system. It satisfies the conditions for a good entanglement measure. We have analyzed the entanglement properties of the GHZ-state, the W-states, and cluster-states under MEMS.     

Quantum Swap Gate with Molecular Ensembles via Cavity Quantum Electrodynamics

We present a simple method to realize a swap gate at one step with two molecular ensembles in a stripline cavity. In this scheme, we can benefit from the enhancement of the coherent coupling and acquire a long coherent time with encoding qubits in different spin states of the rotational ground state in the molecular ensembles. As a by-product, a scheme to create an entangled state with one excitation stored in two ensembles is proposed.     

Quantum Information and Entropy

Thermodynamic entropy is not an entirely satisfactory measure of information of a quantum state. This entropy for an unknown pure state is zero, although repeated measurements on copies of such a pure state do communicate information. In view of this, we propose a new measure for the informational entropy of a quantum state that includes information in the pure states and the thermodynamic entropy. The origin of information is explained in terms of an interplay between unitary and non-unitary evolution. Such complementarity is also at the basis of the so-called interaction-free measurement.     

Controlled Coherent Quantum Tunneling of Atomic Ensembles

We investigate the coherent tunneling phenomenon of the laser-driven atomic ensembles confined in a well-separated double-well potential. By generalizing the Frohlich canonical transformation to adiabatically eliminate the light field variable, a BCS-like effective Hamiltonian is obtained to depict the residual interaction between the two atomic ensembles. The number of the tunneling collective low excitations and its relationship to the ratios gr/gl and Nr/Nl are given.     

Entropy and the Shannon-McMillan-Breiman Theorem for Beta Random Matrix Ensembles

We show that beta ensembles in Random Matrix Theory with generic real analytic potential have the asymptotic equipartition property. In addition, we prove a Central Limit Theorem for the density of the eigenvalues of these ensembles.     

On Quantum Entropy

Quantum physics, despite its intrinsically probabilistic nature, lacks a definition of entropy fully accounting for the randomness of a quantum state. For example, von Neumann entropy quantifies only the incomplete specification of a quantum state and does not quantify the probabilistic distribution of its observables; it trivially vanishes for pure quantum states. We propose a quantum entropy that quantifies the randomness of a pure quantum state via a conjugate pair of observables/operators forming the quantum phase space. The entropy is dimensionless, it is a relativistic scalar, it is invariant under canonical transformations and under CPT transformations, and its minimum has been established by the entropic uncertainty principle. We expand the entropy to also include mixed states. We show that the entropy is monotonically increasing during a time evolution of coherent states under a Dirac Hamiltonian. However, in a mathematical scenario, when two fermions come closer to each other, each evolving as a coherent state, the total system’s entropy oscillates due to the increasing spatial entanglement. We hypothesize an entropy law governing physical systems whereby the entropy of a closed system never decreases, implying a time arrow for particle physics. We then explore the possibility that as the oscillations of the entropy must by the law be barred in quantum physics, potential entropy oscillations trigger annihilation and creation of particles.     

Multipartite Entanglement Preparation and Quantum Communication with Atomic Ensembles

We propose an experimentally feasible scheme to generate Greenberger-Horne-Zeilinger type of maximal entanglement among three atomic ensembles, and show one of the applications, controlled secure direct communication. The scheme involves laser manipulation of atomic ensembles, quarter- and half-wave plates, beam splitters, polarizing beam splitters and single-photon detectors, which are within the reach of current experimental technology.     

Information and Distinguishability of Ensembles of Identical Quantum States

We consider a fixed quantum measurement performed over n identical copies of quantum states. Using a rigorous notion of distinguishability based on Shannon’s 12th theorem, we show that in the case of a single qubit, the number of distinguishable states is , where (α₁,α₂) is the angle interval from which the states are chosen. In the general case of an N-dimensional Hilbert space and an area Ω of the domain on the unit sphere from which the states are chosen, the number of distinguishable states is . The optimal distribution is uniform over the domain in Cartesian coordinates.     

Quantum Interface between a Superconducting Qubit and Spin Ensembles

An experimentally feasible strong coupling system between a spin ensemble and a superconducting qubit is studied. The coupling strength can be exponentially enhanced by applying the squeezing transformations to the system. By means of the two spin ensembles commonly coupled to a superconducting qubit, a set of universal nonadiabatic holonomic single‐qubit quantum gates can be realized in a decoherence‐free subspace. Furthermore, this proposal is robust with respect to decay of the system parameters, and it is experimentally feasible with currently available technology.     

Extended Quantum Conditional Entropy and Quantum Uncertainty Inequalities

Quantum states can be subjected to classical measurements, whose incompatibility, or uncertainty, can be quantified by a comparison of certain entropies. There is a long history of such entropy inequalities between position and momentum. Recently these inequalities have been generalized to the tensor product of several Hilbert spaces and we show here how their derivations can be shortened to a few lines and how they can be generalized. Our proofs utilize the technique of the original derivation of strong subadditivity of the von Neumann entropy.     

Quantum Chaos and Dynamical Entropy

We review the notion of dynamical entropy by Connes, Narnhofer and Thirring and relate it to Quantum Chaos. A particle in a periodic potential is used as an example. This is worked out in the classical and the quantum mechanical framework, for the single particle as well as for the corresponding gas. The comparison does not only support the general assertion that quantum mechanics is qualitatively less chaotic than classical mechanics. More specifically, the same dynamical mechanism by which a periodic potential leads to a positive dynamical entropy of the classical particle may reduce the dynamical entropy of the quantum gas in comparison to free motion. Received: 26 June 1997 / Accepted: 13 April 1998     

Pre- and Post-Selected Ensembles and Time-Symmetry in Quantum Mechanics

It has recently been argued (Shimony, Erkenntnis 45:337, 1997) that time-symmetry does not hold for pre- and post-selected ensembles in quantum mechanics. That conclusion depends on what is meant by “time-symmetry” in relation to those types of ensembles. It is shown that on the conventional view of time-symmetry, pre- and post-selected ensembles are time-symmetric as was originally proposed.     

Entropy of Quantum Dynamical Systems and Sufficient Families in Orthomodular Lattices with Bayessian State

The purpose of the present paper is to study the entropy hs（Ф） of a quantum dynamical systems Ф = （ L, s, Ф）, where s is a bayessian state on an orthomodular lattice L. Having introduced the notion of entropy hs（ Ф, A） of partition A of a Boolean algebra B with respect to a state s and a state preserving homomorphism Ф, we prove a few results on that, define the entropy of a dynamical system hs（Ф）, and show its invariance. The concept of sufficient families is also given and we establish that hs （Ф） comes out to be equal to the supremum of hs （Ф,A）, where A varies over any sufficient family. The present theory has then been extended to the quantum dynamical system （ L, s, Ф）, which as an effect of the theory of commutators and Bell inequalities can equivalently be replaced by the dynamical system （B, s0, Ф）, where B is a Boolean algebra and so is a state on B.     

Entropy of Quantum Dynamical Systems and Sufficient Families in Orthomodular Lattices with Bayessian State

The purpose of the present paper is to study the entropy h_s(Φ) of a quantum dynamical systems Φ=(L,s,φ), where s is a bayessian state on an orthomodular lattice L. Having introduced the notion of entropy h_s(φ,A) of partition A of a Boolean algebra B with respect to a state s and a state preserving homomorphism φ, we prove a few results on that, define the entropy of a dynamical system h_s(Φ), and show its invariance. The concept of sufficient families is also given and we establish that h_s(Φ) comes out to be equal to the supremum of h_s(φ,A), where A varies over any sufficient family. The present theory has then been extended to the quantum dynamical system (L,s,φ), which as an effect of the theory of commutators and Bell inequalities can equivalently be replaced by the dynamical system (B, s₀,φ), where B is a Boolean algebra and s₀ is a state on B.     

Entropy of Quantum States

Given the algebra of observables of a quantum system subject to selection rules, a state can be represented by different density matrices. As a result, different von Neumann entropies can be associated with the same state. Motivated by a minimality property of the von Neumann entropy of a density matrix with respect to its possible decompositions into pure states, we give a purely algebraic definition of entropy for states of an algebra of observables, thus solving the above ambiguity. The entropy so-defined satisfies all the desirable thermodynamic properties and reduces to the von Neumann entropy in the quantum mechanical case. Moreover, it can be shown to be equal to the von Neumann entropy of the unique representative density matrix belonging to the operator algebra of a multiplicity-free Hilbert-space representation.     

Entropy of Quantum Limits

 In this paper we show that any measure arising as a weak^* limit of microlocal lifts of eigenfunctions of the Laplacian on certain arithmetic manifolds have dimension at least 11/9, and in particular all ergodic components of this measure with respect to the geodesic flow have positive entropy. Received: 14 March 2002 / Accepted: 24 June 2002 Published online: 13 January 2003     

Quantum Transport of Particles and Entropy

A unified view on macroscopic thermodynamics and quantum transport is presented. Thermodynamic processes with an exchange of energy between two systems necessarily involve the flow of other balancable quantities. These flows are first analyzed using a simple drift-diffusion model, which includes the thermoelectric effects, and connects the various transport coefficients to certain thermodynamic susceptibilities and a diffusion coefficient. In the second part of the paper, the connection between macroscopic thermodynamics and quantum statistics is discussed. It is proposed to employ not particles, but elementary Fermi- or Bose-systems as the elementary building blocks of ideal quantum gases. In this way, the transport not only of particles but also of entropy can be derived in a concise way, and is illustrated both for ballistic quantum wires, and for diffusive conductors. In particular, the quantum interference of entropy flow is in close correspondence to that of electric current.     

A Note on Quantum Entropy

Incremental information, as measured by the quantum entropy, is increasing when two ensembles are united. This result was proved by Lieb and Ruskai, and it is the foundation for the proof of strong subadditivity of quantum entropy. We present a truly elementary proof of this fact in the context of the broader family of matrix entropies introduced by Chen and Tropp.