Quantum formalism for classical statistics

Abstract

In static classical statistical systems the problem of information transport from a boundary to the bulk finds a simple description in terms of wave functions or density matrices. While the transfer matrix formalism is a type of Heisenberg picture for this problem, we develop here the associated Schrödinger picture that keeps track of the local probabilistic information. The transport of the probabilistic information between neighboring hypersurfaces obeys a linear evolution equation, and therefore the superposition principle for the possible solutions. Operators are associated to local observables, with rules for the computation of expectation values similar to quantum mechanics. We discuss how non-commutativity naturally arises in this setting. Also other features characteristic of quantum mechanics, such as complex structure, change of basis or symmetry transformations, can be found in classical statistics once formulated in terms of wave functions or density matrices. We construct for every quantum system an equivalent classical statistical system, such that time in quantum mechanics corresponds to the location of hypersurfaces in the classical probabilistic ensemble. For suitable choices of local observables in the classical statistical system one can, in principle, compute all expectation values and correlations of observables in the quantum system from the local probabilistic information of the associated classical statistical system. Realizing a static memory material as a quantum simulator for a given quantum system is not a matter of principle, but rather of practical simplicity. 45 11.3. Strong local equivalence classes . . . . . 45 12.Connections between classical and quantum statistics 46 12.

Information transport in classical statistical systems

“…The physical interpretation of observables represented by off-diagonal operators is discussed in ref. [25].…”

Quantum computing with classical bits

“…In this section we discuss the three-spin chain for Ising spins [29]. The three classical bits s k (t) = ±1 at every layer t realize a quantum density matrix ρ(t) for a single quantum spin.…”

“…We are interested in models where the transformation is unitary, ρ(t + ) = U (t)ρ(t)U † (t). Such models can realize "static memory materials" [28,29]. If the density matrix ρ(t in ) at the initial boundary is non-trivial, the information in ρ(t in ) cannot be lost by a unitary evolution.…”

“…The appropriate theoretical framework for the transport of probabilistic information is the quantum formalism for classical statistics [28,29]. It can be based on the "classical density matrix" ρ (t), whose diagonal elements are the "local" probabilities p(t) at t. In contrast to the local probabilities, the change from ρ (t) to a neighboring layer ρ (t + ) obeys a simple linear evolution law ρ (t + ) = S(t) ρ (t) S −1 (t) .…”

“…Here S(t) is the step evolution operator at t, which corresponds to a particular normalization of the transfer matrix [31][32][33]. In the occupation number basis the step evolution operator for generalized Ising models is a real non-negative matrix [28,29]. This restriction may be removed for more general forms of probabilistic computing.…”