Narrow Results

Monotonicity under coarse-graining is a crucial property of the quantum relative entropy. The aim of this paper is to investigate the condition of equality in the monotonicity theorem and in its consequences as the strong sub-additivity of von Neumann entropy, the Golden–Thompson trace inequality and the monotonicity of the Holevo quantitity. The relation to quantum Markov states is briefly indicated.

This is a review of the geometry of quantum states using elementary methods and pictures. Quantum states are represented by a convex body, often in high dimensions. In the case of $n$ qubits, the dimension is exponentially large in $n$. The space of states can be visualized, to some extent, by its simple cross sections: Regular simplexes, balls and hyper-octahedra.^a When the dimension gets large, there is a precise sense in which the space of states resembles, almost in every direction, a ball. The ball turns out to be a ball of rather low purity states. We also address some of the corresponding, but harder, geometric properties of separable and entangled states and entanglement witnesses.

“All convex bodies behave a bit like Euclidean balls.”

Keith Ball

We study Sorkin's proposal of a generalization of quantum mechanics and find that the theories proposed derive their probabilities from kth order polynomials in additive measures, in the same way that quantum mechanics uses a probability bilinear in the quantum amplitude and its complex conjugate. Two complementary approaches are presented, a C* and a Hopf-algebraic one, illuminating both algebraic and geometric aspects of the problem.

We discuss the Lewis and Riesenfeld invariant method for cases where the invariant has continuous eigenvalues and use it to find the Schrödinger wave functions of an inverted pendulum under time-dependent gravitation. As a particular case, we consider an inverted pendulum with exponentially increasing mass and constant gravitation. We also obtain the exact solutions for a generalized time-dependent inverted pendulum.

Edge states have been a backbone of our understanding of the experimental basis of the quantum Hall effect for quite some time. Interestingly, this comprises a quantum system with well defined currents and particle trajectories. The role of trajectories in quantum mechanics has been a problematic question of interpretation for quite some time, and the open quantum dot is a natural system in which to probe this question. Contrary to early speculation, a set of well defined quantum states survives in the open quantum dot. These states are the pointer states and provide a transition into the classical states that can be found in these structures. These states provide resonances, which are observable as oscillatory behavior in the magnetoconductance of the dots. But, they have well defined current directions within the dots. Consequently, one expects trajectories to be a property of these states as well. As one crosses from the low to the high field regime, quite steady trajectories and consequent wave functions can easily be identified and examined. In this talk, we review the current understanding and the support for the decoherence theory.

The amplitude squeezing properties of the electromagnetic field is investigated in the fundamental, Stokes (probe) and signal mode in six-wave mixing process under short-time approximation based on a fully quantum mechanical approach. The interaction is looked upon as a process involving absorption of two pump photons and emission of three probe photons and one signal photon at different frequencies. The coupled Heisenberg quantum mechanical equations of motion involving real and imaginary parts of the quadrature operators are established and solved under short-time scale. The occurrence of amplitude squeezing effects in both the quadrature of the radiation field in the fundamental mode as well as in the Stokes (probe) mode is investigated using the required conditions of squeezing in each of the cases. It is also shown that the squeezing in the signal mode is found to be dependent on amplitude-squared squeezing of the initial pump field. This gives a method of converting higher-order (amplitude-squared) squeezing into normal squeezing.

In previous papers and letters on quantum amplitudes in black-hole evaporation, a boundary-value approach was developed for calculating (for example) the quantum amplitude to have a prescribed slightly non-spherical configuration of a massless scalar field ϕ on a final hypersurface Σ_F at a very late time T, given initial almost-stationary spherically symmetric gravitational and scalar data on a space-like hypersurface Σ_I at time t = 0. For definiteness, we assumed that the gravitational data are also spherically symmetric on Σ_F. Such boundary data can correspond to a classical solution for the Einstein/scalar system, describing gravitational collapse from an early low-density configuration to a nearly Schwarzschild black hole. This approach provides the quantum amplitude (not just the probability) for a transition from an initial to a final state. For a real Lorentzian time-interval T, the classical boundary-value problem refers to a set of hyperbolic equations (modulo gauge), and is badly posed. Instead, the boundary-value approach of the previous letters and papers requires (following Feynman) a rotation into the complex: T → |T| exp(-iθ), for 0 < θ ≤ π/2, of the time-separation-at-infinity T. The classical boundary-value problem, for a complex solution of the coupled nonlinear classical field equations, is expected to be well-posed for 0 < θ ≤ π/2. For a locally supersymmetric Lagrangian, containing supergravity coupled to supermatter, the classical Lorentzian action S_class, a functional of the boundary data (which include the complexified T), yields a quantum amplitude proportional to exp(iS_class), apart from possible loop corrections which are negligible for boundary data with frequencies below the Planck scale. Finally (still following Feynman), one computes the Lorentzian quantum amplitude by taking the limit of exp(iS_class) as θ → 0₊. In the present paper, a connection is made between the above boundary-value approach and the original approach to quantum evaporation in gravitational collapse to a black hole, via Bogoliubov coefficients. This connection is developed through consideration of the radial equation obeyed by the (adiabatic) non-spherical classical perturbations. When one studies the probability distribution for configurations of the final scalar field, based on our quantum amplitudes above, one finds that this distribution can also be interpreted in terms of the Wigner quasi-probability distribution for a harmonic oscillator.

This paper presents a non-string-theoretic calculation of the microcanonical entropy of relic integer-spin Hawking radiation, at fixed total energy E, from an evanescent, neutral, non-rotating four-dimensional black hole. The only conserved macroscopic quantity is the total energy E which, for a black hole that evaporates completely, is the total energy of the relic radiation. Through a boundary-value approach, in which data for massless, integer-spin perturbations are set on initial and final space-like hypersurfaces, the statistical-mechanics problem becomes, in effect, a one-dimensional problem, with the "volume" of the system determined by the real part of the time separation at spatial infinity — the variable conjugate to the total energy. We count the number of field configurations on the final space-like hypersurface that have total energy E, assuming that initial perturbations are weak. We find that the density of states resembles the well-known Cardy formula. The Bekenstein–Hawking entropy is recovered if the real part of the asymptotic time separation is of the order of the semi-classical black-hole lifetime. We thereby obtain a statistical interpretation of black-hole entropy. Corrections to the microcanonical entropy are computed, and we find agreement with other approaches in terms of a logarithmic correction to the black-hole area law, which is universal (independent of black-hole parameters). This result depends crucially upon the discreteness of the energy levels. We discuss the similarities of our approach with the transition from the black-hole to the fundamental-string regime in the final stages of black-hole evaporation. In addition, we find that the squared coupling, g², which regulates the transition from a black hole to a highly-excited string state, and vice versa, can be related to the angle, δ, in the complex-time plane, through which we continue analytically the time separation at spatial infinity. Thus, in this scenario, the strong-coupling regime corresponds to a Euclidean black hole, while the physical limit of a Lorentzian space–time (the limit as δ → 0₊) corresponds to the weak-coupling regime. This resembles the transition of a black hole to a highly-excited string-like state, which subsequently decays into massless particles, thereby avoiding the naked singularity.

There are discernible and fundamental differences between clocks, waves and physical states in classical physics. These fundamental concepts find a common expression in the context of quantum physics in gravitational fields; matter and light waves, quantum states and oscillator clocks become quantum synonymous through the Planck–Einstein–de Broglie relations and the equivalence principle. With this insight, gravitational effects on quantum systems can be simply and accurately analyzed. Apart from providing a transparent framework for conceptual and quantitative thinking on matter waves and quantum states in a gravitational field, we address and resolve with clarity the recent controversial discussions on the important issue of the relation and the crucial difference between gravimetery using atom interferometers and the measurement of gravitational time dilation.

Zurek's existential interpretation of quantum mechanics suffers from three classical prejudices, including the belief that space and time are intrinsically and infinitely differentiated. These compel him to relativize the concept of objective existence in two ways. The elimination of these prejudices makes it possible to recognize the quantum formalism's ontological implications — the relative and contingent reality of spatiotemporal distinctions and the extrinsic and finite spatiotemporal differentiation of the physical world — which in turn makes it possible to arrive at an unqualified objective existence. Contrary to a widespread misconception, viewing the quantum formalism as being fundamentally a probability algorithm does not imply that quantum mechanics is concerned with states of knowledge rather than states of Nature. On the contrary, it makes possible a complete and strongly objective description of the physical world that requires no reference to observers. What objectively exists, in a sense that requires no qualification, is (i) the trajectories of macroscopic objects, whose fuzziness is empirically irrelevant, (ii) the properties and values of whose possession these trajectories provide indelible records, and (iii) the fuzzy and temporally undifferentiated states of affairs that obtain between measurements and are described by counterfactual probability assignments.

Providing quantum theory, and thus quantum information, with some meaning, can only be the result of adopting a well-defined approach to the notion of probability. From a subjectivistic Bayesian approach to the latter, it follows that not only quantum states, but also quantum operations, are structural elements of a subjective nature. Furthermore, we provide a representation of any quantum operation as a stochastic matrix.

An elementary introduction into algebraic approach to unified quantum information theory and operational approach to quantum entanglement as generalized encoding is given. After introducing compound quantum state and two types of informational divergences, namely, Araki–Umegaki (a-type) and of Belavkin–Staszewski (b-type) quantum relative entropic information, this paper treats two types of quantum mutual information via entanglement and defines two types of corresponding quantum channel capacities as the supremum via the generalized encodings. It proves the additivity property of quantum channel capacities via entanglement, which extends the earlier results of Belavkin to products of arbitrary quantum channels for quantum relative entropy of any type.

Sub- and super-fidelity describe respectively the lower and super bound of fidelity of quantum states. In this paper, we obtain several properties of sub- and super-fidelity for both finite- and infinite-dimensional quantum systems. Furthermore, let H be a separable complex Hilbert space and ϕ : 𝒮(H) → 𝒮(H) a map, where 𝒮(H) denotes the convex set of all states on H. We show that, if dim H < ∞, or, if dim H = ∞ and ϕ is surjective, then the following statements are equivalent: (1) ϕ preserves the super-fidelity; (2) ϕ preserves the fidelity; (3) ϕ preserves the sub-fidelity; (4) there exists a unitary or an anti-unitary operator U on H such that ϕ(ρ) = UρU^† for all ρ ∈ 𝒮(H).

On the affine space containing the space ${𝒮}$ of quantum states of finite-dimensional systems, there are contravariant tensor fields by means of which it is possible to define Hamiltonian and gradient vector fields encoding the relevant geometrical properties of ${𝒮}$. Guided by Dirac’s analogy principle, we will use them as inspiration to define contravariant tensor fields, Hamiltonian and gradient vector fields on the affine space containing the space of fair probability distributions on a finite sample space and analyze their geometrical properties. Most of our considerations will be dealt with for the simple example of a three-level system.

Motivated by the geometrical structures of quantum mechanics, we introduce an almost complex structure $J$ on the product $M\times M$ of any parallelizable statistical manifold $M$. Then, we use $J$ to extract a pre-symplectic form and a metric-like tensor on $M\times M$ from a divergence function. These tensors may be pulled back to $M$, and we compute them in the case of an N-dimensional symplex with respect to the Kullback–Leibler relative entropy, and in the case of (a suitable unfolding space of) the manifold of faithful density operators with respect to the von Neumann–Umegaki relative entropy.

Entanglement plays a major role in quantum information process and is a principal resource for various quantum applications like quantum teleportation, quantum key distribution, quantum communication, etc. Strong entangled pairs are required for efficient information process but system impurities during the transportation diminishes the entanglement by reducing the fidelity of the entangled pair. In order to reduce this, purification techniques can be used. In this paper, we propose an efficient entanglement purification method to distill the entanglement using entanglement swapping. The proposed method increases the fidelity of the entanglement and can be a proficient for various applications of quantum computing.