Universal quantum measurements

We introduce a family of operations in quantum mechanics that one can regard as "universal quantum measurements" (UQMs). These measurements are applicable to all finite dimensional quantum systems and entail the specification of only a minimal amount of structure. The first class of UQM that we consider involves the specification of the initial state of the system—no further structure is brought into play. We call operations of this type "tomographic measurements", since given the statistics of the outcomes one can deduce the original state of the system. Next, we construct a disentangling operation, the outcome of which, when the procedure is applied to a general mixed state of an entangled composite system, is a disentangled product of pure constituent states. This operation exists whenever the dimension of the Hilbert space is not a prime, and can be used to model the decay of a composite system. As another example, we show how one can make a measurement of the direction along which the spin of a particle of spin s is oriented (s = 1/2, 1,...). The required additional structure in this case involves the embedding of CP^1 as a rational curve of degree 2s in CP^2s

Pricing Fixed-Income Securities in an Information-Based Framework

In this paper we introduce a class of information-based models for the pricing of fixed-income securities. We consider a set of continuous- time information processes that describe the flow of information about market factors in a monetary economy. The nominal pricing kernel is at any given time assumed to be given by a function of the values of information processes at that time. By use of a change-of-measure technique we derive explicit expressions for the price processes of nominal discount bonds, and deduce the associated dynamics of the short rate of interest and the market price of risk. The interest rate positivity condition is expressed as a differential inequality. We proceed to the modelling of the price-level, which at any given time is also taken to be a function of the values of the information processes at that time. A simple model for a stochastic monetary economy is introduced in which the prices of nominal discount bonds and inflation-linked notes can be expressed in terms of aggregate consumption and the liquidity benefit generated by the money supply

On the Representation of General Interest Rate Models as Square Integrable Wiener Functionals

In the setting proposed by Hughston & Rafailidis (2005) we consider general interest rate models in the case of a Brownian market information filtration $(\mathcal{F}_t)_{t\geq0}$. Let $X$ be a square-integrable $\mathcal{F}_\infty$-measurable random variable, and assume the non-degeneracy condition that for all $t<\infty$ the random variable $X$ is not $\mathcal{F}_t$-measurable. Let ${\sigma_t}$ denote the integrand appearing in the representation of $X$ as a stochastic integral, write $\pi_t$ for the conditional variance of $X$ at time $t$, and set $r_t = \sigma^2_t / \pi_t$. Then $\pi_t$ is a potential, and as such can act as a model for a pricing kernel (or state price density), where $r_t$ is the associated interest rate. Under the stated assumptions, we prove the following: (a) that the money market account process defined by $B_t = \exp (\int_0^t r_s \,ds)$ is finite almost surely at all finite times; and (b) that the product of the money-market account and the pricing kernel is a local martingale, and is a martingale provided a certain integrability condition is satisfied. The fact that a martingale is thus obtained shows that from any non-degenerate element of Wiener space satisfying the integrability condition we can construct an associated interest-rate model. The model thereby constructed is valid over an infinite time horizon, with strictly positive interest, and satisfies the relevant intertemporal relations associated with the absence of arbitrage. The results thus stated pave the way for the use of Wiener chaos methods in interest rate modelling, since any such square-integrable Wiener functional admits a chaos expansion, the individual terms of which can be regarded as parametric degrees of freedom in the associated interest rate model to be fixed by calibration to appropriately liquid sectors of the interest rate derivatives markets.Comment: 17 page

Discrete-Time Interest Rate Modelling

This paper presents an axiomatic scheme for interest rate models in discrete time. We take a pricing kernel approach, which builds in the arbitrage-free property and provides a link to equilibrium economics. We require that the pricing kernel be consistent with a pair of axioms, one giving the inter-temporal relations for dividend-paying assets, and the other ensuring the existence of a money-market asset. We show that the existence of a positive-return asset implies the existence of a previsible money-market account. A general expression for the price process of a limited-liability asset is derived. This expression includes two terms, one being the discounted risk-adjusted value of the dividend stream, the other characterising retained earnings. The vanishing of the latter is given by a transversality condition. We show (under the assumed axioms) that, in the case of a limited-liability asset with no permanently-retained earnings, the price process is given by the ratio of a pair of potentials. Explicit examples of discrete-time models are provided

Information, Inflation, and Interest

We propose a class of discrete-time stochastic models for the pricing of inflation-linked assets. The paper begins with an axiomatic scheme for asset pricing and interest rate theory in a discrete-time setting. The first axiom introduces a "risk-free" asset, and the second axiom determines the intertemporal pricing relations that hold for dividend-paying assets. The nominal and real pricing kernels, in terms of which the price index can be expressed, are then modelled by introducing a Sidrauski-type utility function depending on (a) the aggregate rate of consumption, and (b) the aggregate rate of real liquidity benefit conferred by the money supply. Consumption and money supply policies are chosen such that the expected joint utility obtained over a specified time horizon is maximised subject to a budget constraint that takes into account the "value" of the liquidity benefit associated with the money supply. For any choice of the bivariate utility function, the resulting model determines a relation between the rate of consumption, the price level, and the money supply. The model also produces explicit expressions for the real and nominal pricing kernels, and hence establishes a basis for the valuation of inflation-linked securities

Coherent States and Duality

We formulate a relation between quantum-mechanical coherent states and complex-differentiable structures on the classical phase space ${\cal C}$ of a finite number of degrees of freedom. Locally-defined coherent states parametrised by the points of ${\cal C}$ exist when there is an almost complex structure on ${\cal C}$. When ${\cal C}$ admits a complex structure, such coherent states are globally defined on ${\cal C}$. The picture of quantum mechanics that emerges allows to implement duality transformations.Comment: 7 pages, LaTe

Quantum states and space-time causality

Space-time symmetries and internal quantum symmetries can be placed on equal footing in a hyperspin geometry. Four-dimensional classical space-time emerges as a result of a decoherence that disentangles the quantum and the space-time degrees of freedom. A map from the quantum space-time to classical space-time that preserves the causality relations of space-time events is necessarily a density matrix.Comment: 9 pages, to appear in the Proceedings of the 2nd International Symposium on Information Geometry and its Application

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