Generalizing Stratonovich–Weyl Axioms for Composite Systems

The statistical model of quantum mechanics is based on the mapping between operators on the Hilbert space and functions on the phase space. This map can be implemented by an operator that satisfies physically motivated Stratonovich–Weyl axioms. Arguments are given in favour of a certain extension of the axioms, provided that there is a priori knowledge about the composite nature of the quantum system.


I. INTRODUCTION
Everyday experience tells us that a "composite" is something admitting partition into an "elementary" which is considered to be primary and, therefore, not dividable any further.Moreover, it is assumed that, when a "composite" object is divided into its "elementary" parts, all properties of the "composite" object are determined by the characteristics of its constituents via certain rules of composition.Several fundamental notions of classical physics originated from these intuitive assertions and were afterwards wrapped into rigorous mathematical concepts of measure space.In particular, in classical statistical mechanics a vague idea of the "whole & part" is embodied in the measure-theoretic framework of the theory of probability where the "composition" rule is encoded in the principle of additivity of probability distributions, one of Kolmogorov's axioms of the probability space:1 A.I. Non-negativity of the probability measure, P(A) ≥ 0; of all states of the system) Ω is a 2n−dimensional symplectic manifold, the space of events F ⊂ Ω is represented by elements of σ−algebra, and the probability measure P : F → [0, 1] of event A ∈ F is given by the Lebesgue integral, P(A) = A µ(dz)ρ(z), with the probability distribution function ρ(z).
The last axiom has a profound impact on the description of composite objects in classical theory.Along with the assumption on the Boolean character of events, A.III. results in an intuitively expected representation of the classical phase space of a composite system in the form of the Cartesian product of the phase spaces of subsystems A and B, However, even at the beginning of the quantum theory era, scientists realized that the aforementioned concepts would fail to describe a quantum world (cf.[1][2][3]).In particular, in the phase-space formulation of the quantum theory one cannot maintain the non-negativity axiom A.I and the composition rule (1) as universal laws without encountering a contradiction [4][5][6].To avoid such a contradiction, a generalization of A.I-A.III was suggested through the replacement of the σ−algebra by the non-Boolean lattice which meant a fundamental transformation of conventional Boolean logic into non-Boolean quantum logic (see e.g., [1,7]).
As a result of the change of the classical paradigm, the notion of probability distribution functions has been converted into the concept of quasiprobability distributions.
In the present note, remaining in the framework of the phase-space formulation of quantum mechanics, the analog of (1) for composite finite-dimensional quantum systems will be discussed.The Stratonovich-Weyl (SW) axioms [8] will be complemented by a new axiom of additivity for systems a priori known to be compound ones.

II. COMPOSITE QUANTUM SYSTEMS
In quantum theory the analog of the Cartesian product (1) of the classical phase spaces of systems A and B is a tensor product of the corresponding Hilbert spaces The partial trace operation, allows us to extract information on each subsystem from the density matrix ̺ AB of the whole system in the form of the density matrices of subsystems ̺ A and ̺ B : Aiming to construct the quasiprobability distributions of the subsystems for a given composite system, we will propose the idea to use the partial trace operation (2) not only for the states, but for their duals, the Stratonovich-Weyl kernels.

III. THE STRATONOVICH-WEYL PRINCIPLES
The Wigner quasiprobability distribution W ̺ (Ω N ) of an N-dimensional quantum system in a mixed state is defined by the pairing of density matrix ̺ and the Stratonovich-Weyl kernel ∆(Ω N ) , which is defined on the symplectic space Ω N : The SW kernel ∆(Ω N ) determines a proper quasiprobability distribution provided the following axioms are satisfied (see e.g., [8,9] and references therein): 1. Reconstruction of state ̺ by integrating the Wigner function over a phase space: 2. Hermicity of the SW kernel, 3. Finite Norm of a state given by the integral of the Wigner distribution:

Covariance:
The unitary transformation According to [9], the axioms (1-4) are fulfilled if 3) is an element of the dual space: The space of solutions to (6), i.e., the moduli space, is set by an isotropy group, For a regular SU(N)−stratum P N is a polyhedron on S N −2 (1).
(ii) The phase space Ω N is identified with the orbit Ω N = SU(N)/Iso SU(N) (∆) .
Now, leaving (1-4) in force, the 5-th axiom on composite quantum systems is proposed: The extended system of axioms (1-5) allows us to render the following assertions: (i).The dual state space P * A×B of a binary composite system with N A − and N B − dimensional subsystems is (N 2 A N 2 B − 4)−dimensional subspace of P * N , defined as (ii).The phase space Ω A×B of a composite system is determined by the Local Unitary (LU) group, LU := SU(N A ) × SU(N B ) ⊂ SU(N) , corresponding to the factorization H A × H B , and by the isotropy group of the SW kernel: Ω A×B := LU/Iso LU (∆) ; (iii).The moduli space is the factor space P A×B := P * A×B /LU .
Below, the statements (i)-(iii) are illustrated for a simplest binary system of a pair of qubits.
Similarly, vectors η A and η B and a real 3 × 3 matrix E define the SW kernel ∆ ∈ P 4 : If a 4-level system is elementary, the master equations ( 6) impose the following condition: However, if it is known that the system consists of two qubits, according to (7), instead of (10) the individual norms of vectors η A , η B and the matrix E are fixed: Therefore, the 2-qubit's dual space is P * 2×2 ⊂ P * 4 such that dim(P * 2×2 ) = 15 − 3 = 12.The phase space Ω 4 , corresponding to an elementary 4-level system, is one of the SU(4)−orbits Ω 4 = SU(4)/Iso SU(4) (∆) on P 4 .When a 4-level system is considered to be a joint of 2 qubits, then according to assertion (ii) the LU transformations and isotropy group Iso LU (∆) of the SW kernel (9) define the phase space of 2-qubit Ω 2×2 , as well as its moduli space P 2×2 .
In accordance with the 2-qubit Hilbert space factorization, the LU transformations form the subgroup K = SU(2) × SU(2) ⊂ SU(4) and the orbits of K on P 4 define Ω 2×2 .Hence, the issue of describing the 2 qubit phase space is reduced to the mathematical problem of classifying admissible types of K−orbits on P 4 .Being restricted by the required volume of this note I will illustrate the construction of a 2-qubit SW kernel for only one class of 6-dimensional phase space Ω 2×2 and the moduli space P 2×2 .In order to explicitly describe these spaces, it is convenient to decompose g ∈ SU(4) into three factors: In (12) a and a ′ are the Abelian subalgebras in the direct sum decomposition of the algebra: such that the following commutator relations hold: Using (12) for unitary factor U in SVD of 4∆ 4 = U I 4 + √ 30 λα∈k ′ λ α µ α U † , one can be convinced that the master equations (11) describe a bundle of a unit 2-sphere and two ellipsoids E A and E B in the space P 4 with the Cartesian coordinates µ = {µ 3 , µ 6 , µ 15 }: The 3×3 matrices A and B in (15) are constructed out of the adjoint matrix Aλ ν A † = O νµ λ µ : The moduli space P 2×2 of a 2-qubit system is determined from analyzing the following pairwise characteristic polynomials of ellipsoids E A , E B and a unit 2-sphere: According to [10], each characteristic polynomial in (17) always has at least one negative root.Moreover, the ellipsoids and the 2-sphere overlap iff characteristic polynomials have no positive roots.More on the geometric properties of P 2×2 will be given elsewhere.

V. SUMMARY
Discussing the phase-space approach to elementary vs composite quantum systems, it is worth emphasizing the common principle used: the underlying symmetry of a system dictates the construction of the basic quantities in both cases.In the first case, the global SU(N) symmetry sets the phase space Ω N := SU(N)/Iso SU(N) (∆), as well as the moduli space P N , via the master equations ( 6) on the orbit space P * N /SU(N).In the second case, dealing with another pair of spaces (P A×B , P * A×B ), the local sub-symmetry, LU := SU(N A ) × SU(N B ) ⊂ SU(N), comes into play once more.The latter defines the phase space Ω N A ×N B and the moduli space P N A ×N B via the extended master equations (7) on the orbit space P * A×B /LU.

5 .
Composite systems axiom: Let ̺ AB and ∆(Ω N ) be a density matrix and a SW kernel of composite system H A×B , respectively, then the Wigner functions of states ̺ A and ̺ B are constructed by pairing (3) with partially reduced matrices ∆ A := Tr B ∆ and ∆ B := Tr A ∆, i.e., W ̺ A = Tr (̺ A ∆ N A ) and W ̺ B = Tr (̺ B ∆ N B ) .