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arXiv:2305.16178 · Journal of Physics A Mathematical and Theoretical

Effective rationality for local unitary invariants of mixed states of two qubits

Luca Candelori, Vladimir Chernyak, John R. Klein, Nick Rekuski

Published Mar 7, 2024• 1 citation
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Abstract

Abstract We calculate the field of rational local unitary invariants for mixed states of two qubits, by employing methods from algebraic geometry. We prove that this field is rational (i.e. purely transcendental), and that it is generated by nine algebraically independent polynomial invariants. We do so by constructing a relative section, in the sense of invariant theory, whose Weyl group is a finite abelian group. From this construction, we are able to give explicit expressions for the generating invariants in terms of the Bloch matrix representation of mixed states of two qubits. We also prove similar rationality results for the local unitary invariants of symmetrically mixed states of two qubits. We also provide a sketch of how to generalize our results to the case of an arbitrary number of qubits. Our results apply to both complex-valued and real-valued invariants.

DOI
10.1088/1751-8121/ad318a

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Abstract We calculate the field of rational local unitary invariants for mixed states of two qubits, by employing methods from algebraic geometry. We prove that this field is rational (i.e. purely transcendental), and that it is generated by nine algebraically independent polynomial invariants. We do so by constructing a relative section, in the sense of invariant theory, whose Weyl group is a finite abelian group. From this construction, we are able to give explicit expressions for the generating invariants in terms of the Bloch matrix representation of mixed states of two qubits. We also prove similar rationality results for the local unitary invariants of symmetrically mixed states of two qubits. We also provide a sketch of how to generalize our results to the case of an arbitrary number of qubits. Our results apply to both complex-valued and real-valued invariants.
For an nnn-qubit system, a rational function on the space of mixed states which is invariant with respect to the action of the group of local symmetries may be viewed as a detailed measure of entanglement. We show that the field of all such invariant rational functions is purely transcendental over the complex numbers and has transcendence degree 4n−2n−14^n - 2n-14n−2n−1. An explicit transcendence basis is also exhibited.
Local unitary equivalence serves as the cornerstone for classifying entanglement in bipartite quantum systems. Mathematically, it reduces to the study of polynomial invariants of the density matrix under the action of local unitary groups. The collection of all such polynomial invariants forms a ring, known as the invariant ring. However, identifying the complete generators of the invariant ring is the central issue. In 2007, for the two-qubit system, King et al fully characterized the structure of the invariant ring and determined its Cohen--Macaulay decomposition. In this paper, we revisit their work, with a focus on the computation of the Molien series and the construction of invariants. On one hand, we rigorously derive the Molien series via explicit contour integration over the maximal torus, filling in all previously omitted computational steps. On the other hand, we systematically construct all invariants using a graphical method, and then reduce the candidate set by applying various identities and algebraic relations, obtaining a generating set consisting of 21 invariants. This paper aims to make this important result more widely accessible to researchers in quantum information and invariant theory through the above discussions.
We present a complete set of local unitary invariants for generic multiqubit systems which gives necessary and sufficient conditions for two states being local unitary equivalent. These invariants are canonical polynomial functions in terms of the generalized Bloch representation of the quantum states. In particular, we prove that there are at most 12 polynomial local unitary invariants for two-qubit states and at most 90 polynomials for three-qubit states. Comparison with Makhlin's 18 local unitary invariants is given for two-qubit systems.
We study the local unitary equivalence for two and three-qubit mixed states by investigating the invariants under local unitary transformations. For two-qubit system, we prove that the determination of the local unitary equivalence of 2-qubits states only needs 14 or less invariants for arbitrary two-qubit states. Using the same method, we construct invariants for three-qubit mixed states. We prove that these invariants are sufficient to guarantee the LU equivalence of certain kind of three-qubit states. Also, we make a comparison with earlier works.
Local unitary equivalence is an important ingredient for quantifying and classifying entanglement. Verifying whether or not two quantum states are local unitary equivalent is a crucial problem, where only the case of multipartite pure states is solved. For mixed states, however, the verification of local unitary equivalence is still a challenging problem. In this paper, based on the coefficient matrices of generalized Bloch representations of quantum states, we find a variety of local unitary invariants for arbitrary-dimensional bipartite quantum states. These invariants are operational and can be used as necessary conditions for verifying the local unitary equivalence of two quantum states. Furthermore, we extend the construction to the arbitrary-dimensional multipartite case. We finally apply these invariants to estimate concurrence, a vital entanglement measure, showing the practicability of local unitary invariants in characterizing entanglement.
The mixed two qutrit system: local unitary invariants, entanglement monotones, and the SLOCC group SL(3,C)SL(3,{\mathbb {C}})SL(3,C)
We consider local unitary invariants and entanglement monotones for the mixed two qutrit system. Character methods for the local SU(3)xSU(3) transformation group are used to establish the count of algebraically independent polynomial invariants up to degree 5 in the components of the density operator. These are identified up to quartic degree in the standard basis of Gell-Mann matrices, with the help of the calculus of f and d coefficients. Next, investigating local measurement operations, we study a SLOCC qutrit group, which plays the role of a `relativistic' transformation group analogous to that of the Lorentz group SL(2,C)_R ~ SO(3,1) for the qubit case. This is the group SL(3,C)_R, presented as a group of real 9x9 matrices acting linearly on the 9-dimensional space of projective coordinates for the qutrit density matrix. The counterpart, for qutrits, of the invariant 4x4 Minkowski metric of the qubit case, proves to be a certain 9x9x9 totally symmetric three-fold tensor generalizing the Gell-Mann d coefficient. Using this structure, we provide a count of the corresponding LSL polynomial invariants using group character methods. Finally, we give an explicit construction of the lowest degree quantity (the cubic invariant) and its expansion in terms of SU(3)xSU(3) invariants, and we indicate how to construct higher degree analogues. These quantities are proven to yield entanglement monotones. This work generalizes and partly extends the paper of King, Welsh and Jarvis (Journal of Physics A, 40(33):10083, 2007) on the mixed two qubit system, which is reviewed in an appendix.