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An extended, quartic quantum theory

Karol Życzkowski

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

We propose an extended quantum theory, in which the number of degrees of freedom K behaves as fourth power the number N of distinguishable states. As the simplex of classical N–point probability distributions of mixed quantum states, one can further increase the dimensionality constructing the set of extended quantum states. The embedding proposed corresponds to an assumption that the physical system described in N dimensional Hilbert space is coupled with an auxiliary subsystem of the same dimensionality. The extended theory works for simple quantum systems and is shown to be a non-trivial generalisation of the standard quantum theory for which K = N 2. Imposing certain restrictions on initial conditions and dynamics allowed in the quartic theory one obtains quadratic theory as a special case. By imposing even stronger constraints one arrives at the classical theory, for which K = N. can be embedded inside a higher dimensional convex body M Q N

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We propose an extended quantum theory, in which the number of degrees of freedom K behaves as fourth power the number N of distinguishable states. As the simplex of classical N–point probability distributions of mixed quantum states, one can further increase the dimensionality constructing the set of extended quantum states. The embedding proposed corresponds to an assumption that the physical system described in N dimensional Hilbert space is coupled with an auxiliary subsystem of the same dimensionality. The extended theory works for simple quantum systems and is shown to be a non-trivial generalisation of the standard quantum theory for which K = N 2. Imposing certain restrictions on initial conditions and dynamics allowed in the quartic theory one obtains quadratic theory as a special case. By imposing even stronger constraints one arrives at the classical theory, for which K = N. can be embedded inside a higher dimensional convex body M Q N
We propose an extended quantum theory, in which the number K of parameters necessary to characterize a quantum state behaves as fourth power of the number N of distinguishable states. As the simplex of classical N -point probability distributions can be embedded inside a higher-dimensional convex body of mixed quantum states, one can further increase the dimensionality constructing the set of extended quantum states. The embedding proposed corresponds to an assumption that the physical system described in the N -dimensional Hilbert space is coupled with an auxiliary subsystem of the same dimensionality. The extended theory works for simple quantum systems and is shown to be a non-trivial generalization of the standard quantum theory for which K = N 2 . Imposing certain restrictions on initial conditions and dynamics allowed in the quartic theory one obtains quadratic theory as a special case. By imposing even stronger constraints one arrives at the classical theory, for which K = N .
Quantum bits can be isolated to perform useful information-theoretic tasks, even though physical systems are fundamentally described by very high-dimensional operator algebras. This is because qubits can be consistently embedded into higher-dimensional Hilbert spaces. A similar embedding of classical probability distributions into quantum theory enables the emergence of classical physics via decoherence. Here, we ask which other probabilistic models can similarly be embedded into finite-dimensional quantum theory. We show that the embeddable models are exactly those that correspond to the Euclidean special Jordan algebras: quantum theory over the reals, the complex numbers, or the quaternions, and "spin factors" (qubits with more than three degrees of freedom), and direct sums thereof. Among those, only classical and standard quantum theory with superselection rules can arise from a physical decoherence map. Our results have significant consequences for some experimental tests of quantum theory, by clarifying how they could (or could not) falsify it. Furthermore, they imply that all unrestricted non-classical models must be contextual.
We propose a model of a quantum N-dimensional system (quNit) based on a quadratic extension of the non-Archimedean field of p-adic numbers. As in the standard complex setting, states and observables of a p-adic quantum system are implemented by suitable linear operators in a p-adic Hilbert space. In particular, owing to the distinguishing features of p-adic probability theory, the states of an N-dimensional p-adic quantum system are implemented by p-adic statistical operators, i.e., trace-one selfadjoint operators in the carrier Hilbert space. Accordingly, we introduce the notion of selfadjoint-operator-valued measure (SOVM)—a suitable p-adic counterpart of a POVM in a complex Hilbert space—as a convenient mathematical tool describing the physical observables of a p-adic quantum system. Eventually, we focus on the special case where N=2, thus providing a description of p-adic qubit states and 2-dimensional SOVMs. The analogies—but also the non-trivial differences—with respect to the qubit states of standard quantum mechanics are then analyzed.
In the operational approach to general probabilistic theories one distinguishes two spaces, the state space of the "elementary systems" and the physical space in which "laboratory devices" are embedded. Each of those spaces has its own dimension- the minimal number of real parameters (coordinates) needed to specify the state of system or a point within the physical space. Within an operational framework to a physical theory, the two dimensions coincide in a natural way under the following "closeness" requirement: the dynamics of a single elementary system can be generated by the invariant interaction between the system and the "macroscopic transformation device" that itself is described from within the theory in the macroscopic (classical) limit. Quantum mechanics fulfils this requirement since an arbitrary unitary transformation of an elementary system (spin-1/2 or qubit) can be generated by the pairwise invariant interaction between the spin and the constituents of a large coherent state ("classical magnetic field"). Both the spin state space and the "classical field" are then embedded in the Euclidean three-dimensional space. Can we have a general probabilistic theory, other than quantum theory, in which the elementary system ("generalized spin") and the "classical fields" generating its dynamics are embedded in a higher-dimensional physical space? We show that as long as the interaction is pairwise, this is impossible, and quantum mechanics and the three-dimensional space remain the only solution. However, having multi-particle interactions and a generalized notion of "classical field" may open up such a possibility.
We explore a particular way of reformulating quantum theory in classical terms, starting with phase space rather than Hilbert space, and with actual probability distributions rather than quasiprobabilities. The classical picture we start with is epistemically restricted, in the spirit of a model introduced by Spekkens. We obtain quantum theory only by combining a collection of restricted classical pictures. Our main challenge in this paper is to find a simple way of characterizing the allowed sets of classical pictures. We present one promising approach to this problem and show how it works out for the case of a single qubit.
The framework of generalized probabilistic theories is a powerful tool for studying the foundations of quantum physics. It provides the basis for a variety of recent findings that significantly improve our understanding of the rich physical structure of quantum theory. This review paper tries to present the framework and recent results to a broader readership in an accessible manner. To achieve this, we follow a constructive approach. Starting from few basic physically motivated assumptions we show how a given set of observations can be manifested in an operational theory. Furthermore, we characterize consistency conditions limiting the range of possible extensions. In this framework classical and quantum theory appear as special cases, and the aim is to understand what distinguishes quantum mechanics as the fundamental theory realized in nature. It turns out non-classical features of single systems can equivalently result from higher dimensional classical theories that have been restricted. Entanglement and non-locality, however, are shown to be genuine non-classical features.