Information Content for Quantum States

A method of representing probabilistic aspects of quantum systems is introduced by means of a density function on the space of pure quantum states. In particular, a maximum entropy argument allows us to obtain a natural density function that only reflects the information provided by the density matrix. This result is applied to derive the Shannon entropy of a quantum state. The information theoretic quantum entropy thereby obtained is shown to have the desired concavity property, and to differ from the the conventional von Neumann entropy. This is illustrated explicitly for a two-state system.Comment: RevTex file, 4 pages, 1 fi

The Quantum Canonical Ensemble

The phase space of quantum mechanics can be viewed as the complex projective space endowed with a Kaehlerian structure given by the Fubini-Study metric and an associated symplectic form. We can then interpret the Schrodinger equation as generating a Hamiltonian dynamics. Based upon the geometric structure of the quantum phase space we introduce the corresponding natural microcanonical and canonical ensembles. The resulting density matrix for the canonical ensemble differs from density matrix of the conventional approach. As an illustration, the results are applied to the case of a spin one-half particle in a heat bath with an applied magnetic field.Comment: 8 pages, minor corrections. to appear in JMP vol. 3

Thermalisation of Quantum States

An exact stochastic model for the thermalisation of quantum states is proposed. The model has various physically appealing properties. The dynamics are characterised by an underlying Schrodinger evolution, together with a nonlinear term driving the system towards an asymptotic equilibrium state and a stochastic term reflecting fluctuations. There are two free parameters, one of which can be identified with the heat bath temperature, while the other determines the characteristic time scale for thermalisation. Exact expressions are derived for the evolutionary dynamics of the system energy, the system entropy, and the associated density operator.Comment: 8 pages, minor corrections. To appear in JM

Determination of the Lévy Exponent in Asset Pricing Models

We consider the problem of determining the Lévy exponent in a Lévy model for asset prices given the price data of derivatives. The model, formulated under the real-world measure P, consists of a pricing kernel {π_t} together with one or more non-dividend-paying risky assets driven by the same Lévy process. If {S_t} denotes the price process of such an asset then {π_t S_t} is a P-martingale. The Lévy process {ξ_t} is assumed to have exponential moments, implying the existence of a Lévy exponent ψ(α) = 1/t log E(e^{α ξ_t}) for α in an interval A ⊂ R containing the origin as a proper subset. We show that if the prices of power-payoff derivatives, for which the payoff is H_T = (ζ_T )^q for some time T > 0, are given at time 0 for a range of values of q, where {ζ_t} is the so-called benchmark portfolio defined by ζ_t = 1/π_t, then the Lévy exponent is determined up to an irrelevant linear term. In such a setting, derivative prices embody complete information about price jumps: in particular, the spectrum of the price jumps can be worked out from current market prices of derivatives. More generally, if H_T = (S_T )^q for a general non-dividend-paying risky asset driven by a Lévy process, and if we know that the pricing kernel is driven by the same Lévy process, up to a factor of proportionality, then from the current prices of power-payoff derivatives we can infer the structure of the Lévy exponent up to a transformation ψ(α) → ψ(α + μ) − ψ(μ) + cα, where c and μ are constants

Pricing with Variance Gamma Information

In the information-based pricing framework of Brody, Hughston & Macrina, the market filtration {Ft}_t≥0 is generated by an information process {ξ_t}t≥0 defined in such a way that at some fixed time T an F_T -measurable random variable X_T is “revealed”. A cash flow H_T is taken to depend on the market factor X_T , and one considers the valuation of a financial asset that delivers H_T at T. The value of the asset S_t at any time t ∈ [0, T ) is the discounted conditional expectation of H_T with respect to F_t, where the expectation is under the risk neutral measure and the interest rate is constant. Then S_T− = H_T , and S_t = 0 for t ≥ T. In the general situation one has a countable number of cash flows, and each cash flow can depend on a vector of market factors, each associated with an information process. In the present work we introduce a new process, which we call the normalized variance-gamma bridge. We show that the normalized variance-gamma bridge and the associated gamma bridge are jointly Markovian. From these processes, together with the specification of a market factor X_T , we construct a so-called variance-gamma information process. The filtration is then taken to be generated by the information process together with the gamma bridge. We show that the resulting extended information process has the Markov property and hence can be used to develop pricing models for a variety of different financial assets, several examples of which are discussed in detail

Quantum Measurement of Space-Time Events

Abstract. The phase space of a relativistic system can be identified with the future tube of complexified Minkowski space. As well as a complex structure and a symplectic structure, the future tube, seen as an eight-dimensional real manifold, is endowed with a natural positive- definite Riemannian metric that accommodates the underlying geometry of the indefinite Minkowski space metric, together with its symmetry group. A unitary representation of the 15-parameter group of conformal transformations can then be constructed that acts upon the Hilbert space of square-integrable holomorphic functions on the future tube. These structures are enough to allow one to put forward a quantum theory of phase-space events. In particular, a theory of quantum measurement can be formulated in a relativistic setting, based on the use of positive operator valued measures, for the detection of phase-space events, hence allowing one to assign probabilities to the outcomes of joint space-time and four- momentum measurements in a manifestly covariant framework. This leads to a localization theorem for phase-space events in relativistic quantum theory, determined by the associated Compton wavelength

Metric approach to quantum constraints

A new framework for deriving equations of motion for constrained quantum systems is introduced, and a procedure for its implementation is outlined. In special cases the framework reduces to a quantum analogue of the Dirac theory of constrains in classical mechanics. Explicit examples involving spin-1/2 particles are worked out in detail: in one example our approach coincides with a quantum version of the Dirac formalism, while the other example illustrates how a situation that cannot be treated by Dirac's approach can nevertheless be dealt with in the present scheme.Comment: 13 pages, 1 figur

Lévy Models for Collapse of the Wave Function

Recently there has been much progress in the development of stochastic models for state reduction in quantum mechanics. In such models, the collapse of the wave function is a physical process, governed by a nonlinear stochastic differential equation that generalizes the Schrödinger equation. The present paper considers energy-based stochastic extensions of the Schrödinger equation. Most of the work carried out hitherto in this area has been concerned with models where the process driving the stochastic dynamics of the quantum state is Brownian motion. Here, the Brownian framework is broadened to a wider class of models where the noise process is of the Lévy type, admitting stationary and independent increments. The properties of such models are different from those of Brownian reduction models. In particular, for Lévy models the decoherence rate depends on the overall scale of the energy. Thus, in Lévy reduction models, a macroscopic quantum system will spontaneously collapse to an eigenstate even if the energy level gaps are small