Two-dimensional Hilbert Space Research Articles

Quantum coin-tossing in a Bayesian Jeffreys framework

The Jeffreys' reparameterization-invariant prior for the classical problem of tossing a coin (with its possible binary outcomes designated ±1) is the arc-sine distribution, 1 π (1 − z 2) 1 2 , where z ∈ [−1, 1] is the (long-run) expected value intrinsic to the coin. It is demonstrated here that the analogous prior for the problem of measuring a two-level quantum system — represented by a 2 × 2 density matrix ϱ parameterized by points ( x, y, z) in the Bloch sphere (unit ball) — along two orthogonal ( Y and Z) axes is 1 2 π(1 − y 2 − z 2) 1 2 , where y is the expected value along the Y-axis and z that along the Z-axis. This contention rests upon the identification of ϱ as the covariance matrix of a bivariate complex normal distribution over the vectors (φ) of two-dimensional complex Hilbert space. The identification, in turn, is based upon quantum probabilistic arguments of Bach, supplemented by an invocation of the principle of maximum entropy. In addition, the Jeffreys' priors are presented for several sets of pairs (entangled and not) of two-level systems.

Quantum theory of Ur-objects as a theory of information

The quantum theory of ur-objects proposed by C. F. von Weizsacker has to be interpreted as a quantum theory of information. Ur-objects, or urs, are thought to be the simplest objects in quantum theory. Thus an ur is represented by a two-dimensional Hilbert space with the universal symmetry groupSU(2), and can only be characterized asone bit of potential information. In this sense it is not a spatial but aninformation atom. The physical structure of the ur theory is reviewed, and the philosophical consequences of its interpretation as an information theory are demonstrated by means of some important concepts of physics such as time, space, entropy, energy, and matter, which in ur theory appear to be directly connected with information as “the” fundamental substance. This hopefully will help to provide a new understanding of the concept of information.

Quantum theory of dissipative processes: The Markov approximation revisited

Adopting the standard mathematical framework for describing reduced dynamics, we derive two formal identities for the density operator of an open quantum system. Each of these is equivalent to the old Nakajima-Zwanzig equation. The first identity is local in time. It contains the inverse of the dynamical map which govern the evolution of the density operator. We indicate a time interval on which this inverse exists. The second identity constitutes a suitable starting point for going beyond the Markov approximation in a controlled way. On the basis of the Bloch equations we argue once more that in studying quantum dissipation one has to pay attention to the von Neumann conditions. In the Nakajima-Zwanzig equation we make the first Born approximation. The ensuing master equation possesses the correct weak-coupling limit. While proving this rather obvious but at the same time important statement, we elucidate the mathematical methods which underlie the weak-coupling limit. Moving to a two-dimensional Hilbert space, we find that both for short and for long times our approximate master equation respects the von Neumann conditions. Assuming exponential decay for correlation functions, we propose a physical limit in which the solutions for the density operator become Markovian in character. We confirm the well-known statement that, as seen from a macroscopic standpoint, the system starts from an effective initial condition. The approach to equilibrium is exponential. The accessory relaxation constants can differ from the usual Bloch parameters γ⊥ and γ∥ by more than 50%.

The completeness theorem for Rossby normal modes of a stably stratified flat ocean with an arbitrary form of side boundary

A mid-latitude flat ocean on a β \beta -plane has characteristic oscillations called Rossby normal modes, where the motion is governed by the quasigeostrophic vorticity equation. Although the relevant eigenvalue problem differs from the usual one of Hilbert-Schmidt type, a variational proof is obtained that the Rossby normal modes constitute a complete orthonormal set for a basin with an arbitrary profile of stable density stratification and an arbitrary form of side boundary. In particular, for each fixed vertical mode, the set of the horizontal modes is complete and orthonormal in a two-dimensional Hilbert space. General solutions are expressed in terms of Rossby normal modes, not only to the initial-value problem, but also to the response problem of the closed basin.

Additive isometries on a quaternionic Hilbert space

A systematic study of additive isometries on a quaternionic Hilbert space is presented. A number of new results describing the properties of such operators are proved. The work culminates in the first mathematical proof of Wigner’s theorem for quaternionic Hilbert spaces of dimension other than 2 which asserts that any operator which preserves the absolute value of the inner product on a quaternionic Hilbert space is equivalent, in the sense of differing pointwise by a mere phase factor, to a linear isometry. A complete and concise description of the exceptional situation in a two-dimensional quaternionic Hilbert space is given.

A quaternion quantum system

The (generalized) quantum mechanics of the two-dimensional quaternionic Hilbert space is constructed explicitly. In this simple example the unconventional properties of such a system as compared to complex quantum mechanics are discussed.

On the Modulii of Analytic Functions

The problem to be considered in this note, in its most concrete form, is the determination of all quartets f1, f2, g1, g2 of functions analytic on some domain and satisfying*where p > 0. When p = 2 the question can be reformulated in terms of finding a necessary and sufficient condition for (two-dimensional) Hilbert space valued analytic functions to have equal pointwise norms, and the answer (Theorem 1) justifies this point of view. If p ≠ 2, the problem is solved by reducing to the case p = 2, and the reformulation in terms of the norm equality of lp valued analytic functions gives no clue to the answer.