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Physics of Atomic Nuclei

Constraints on SU(2) ⊗ SU(2) invariant polynomials for a pair of entangled qubits

Vladimir P. Gerdt, A. Khvedelidze, Yu. G. Palii

Published Jun 1, 2011• 6 citations
Publisher

Abstract

We discuss the entanglement properties of two qubits in terms of polynomial invariants of the adjoint action of SU(2) ⊕ SU(2) group on the space of density matrices $\mathfrak{P}_ +$ . Since elements of $\mathfrak{P}_ +$ are Hermitian, non-negative fourth-order matrices with unit trace, the space of density matrices represents a semi-algebraic subset, $\mathfrak{P}_ + \in \mathbb{R}^{15}$ . We define $\mathfrak{P}_ +$ explicitly with the aid of polynomial inequalities in the Casimir operators of the enveloping algebra of SU(4) group. Using this result the optimal integrity basis for polynomial SU(2) ⊕ SU(2) invariants is proposed and the well-known Peres-Horodecki separability criterion for 2-qubit density matrices is given in the form of polynomial inequalities in three SU(4) Casimir invariants and two SU(2) ⊕ SU(2) scalars; namely, determinants of the so-called correlation and the Schlienz-Mahler entanglement matrices.

DOI
10.1134/s1063778811060111

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We discuss the entanglement properties of two qubits in terms of polynomial invariants of the adjoint action of group on the space of density matrices . Since elements of are Hermitian, non-negative fourth-order matrices with unit trace, the space of density matrices represents a semi-algebraic subset, . We define explicitly with the aid of polynomial inequalities in the Casimir operators of the enveloping algebra of group. Using this result the optimal integrity basis for polynomial invariants is proposed and the well-known Peres–Horodecki separability criterion for 2-qubit density matrices is given in the form of polynomial inequalities in three Casimir invariants and two scalars; namely, determinants of the so-called correlation and the Schlienz–Mahler entanglement matrices.
A method allowing to increase a computational efficiency of evaluation of non-local characteristics of a pair of qubits is described. The method is based on the construction of coordinates on a generic section of 2-qubit’s entanglement space E2×2{{\mathcal{E}}_{{2 \times 2}}}E2×2​ represented as the direct product of an ordered 3-dimensional simplex and the double coset SU(2)×SU(2)\SU(4)/T3.{\text{SU(2)}} \times {\text{SU(2)}}{\kern 1pt} \backslash {\kern 1pt} {\text{SU(4)}}{\kern 1pt} /{\kern 1pt} {{{\text{T}}}^{{\text{3}}}}{\kern 1pt} .SU(2)×SU(2)\SU(4)/T3. Within this framework, the subset SE2×2⊂E2×2\mathcal{S}{{\mathcal{E}}_{{2 \times 2}}} \subset {{\mathcal{E}}_{{2 \times 2}}}SE2×2​⊂E2×2​ corresponding to the rank-4 separable 2-qubit states is described as a semialgebraic variety given by a system of 3rd and 4th order polynomial inequalities in eigenvalues of the density matrix, whereas the polynomials coefficients are trigonometric functions defined over a direct product of two regular octahedra.
In the present paper few steps are undertaken towards the description of the qubit-qutrit pair - quantum bipartite system composed of two and three level subsystems. The computational difficulties with the construction of the local unitary polynomial invariants are discussed. Calculations of the Molien functions and Poincare series for the qubit-qubit and qubit-qutrit local unitary invariants are outlined and compared with the known results. The requirement of positive semi-definiteness of the density operator is formulated explicitly as a set of inequalities in five Casimir invariants of the algebra su(6).
Randomized measurement protocols give experimental access to low-degree polynomial invariants of a quantum state rather than to the state itself, which raises the question of what the best entanglement criterion is that can be built from finitely many such invariants. In this work we answer this question through convex duality. For each degree we introduce the cone of local unitary trace inequalities that are valid for separable bipartite states in all local dimensions, together with its one-matrix counterparts for Hermitian and positive semidefinite matrices. Each cone is the polar dual of a moment cone; we also determine the minimal combinatorial parametrization of the separable cone. The framework organizes the known moment criteria: the PPT criterion using the first three moments of the partial transposition is the image under an embedding of a Hankel determinant from the positive semidefinite cone to the bipartite cone, which we show to be an extremal ray, and we determine the first nontrivial separable cone completely, finding exactly four extremal rays. The framework allows us to analyze the realignment criterion and its centered (or enhanced) variant. We prove that the even singular value moments of the (centered) realignment matrix are trace invariants, associated with explicit fixed-point-free involutions. Extracting the optimal trace-norm bound from mmm such moments thus becomes a dimension-free truncated moment problem, which we solve in closed form for m=2m = 2m=2 and relax to a semidefinite program of size O(m)O(m)O(m) for arbitrary mmm. The dual solutions are polynomial minorants of x\sqrt{x}x​ on [0,1][0,1][0,1], and two explicit families, binomial and L1L^1L1-weighted, yield hierarchies of separability inequalities. Finally, we compare the criteria against one another and on PPT entangled states.
A new geometric representation of qubit and qutrit states based on probability simplexes is used to describe the separability and entanglement properties of density matrices of two qubits. The Peres-Horodecki positive partial transpose (ppt) -criterion and the concurrence inequalities are formulated as the conditions that the introduced probability distributions must satisfy to present entanglement. A four-level system, where one or two states are inaccessible, is considered as an example of applying the elaborated probability approach in an explicit form. The areas of three Triadas of Malevich's squares for entangled states of two qubits are defined through the qutrit state, and the critical values of the sum of their areas are calculated. We always find an interval for the sum of the square areas, which provides the possibility for an experimental checkup of the entanglement of the system in terms of the probabilities.