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arXiv:1305.3894 · Journal of Mathematical Physics

How many invariant polynomials are needed to decide local unitary equivalence of qubit states?

Tomasz Maciążek, Michał Oszmaniec, Adam Sawicki

Published Sep 1, 2013• 26 citations
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Abstract

Given L-qubit states with the fixed spectra of reduced one-qubit density matrices, we find a formula for the minimal number of invariant polynomials needed for solving local unitary (LU) equivalence problem, that is, problem of deciding if two states can be connected by local unitary operations. Interestingly, this number is not the same for every collection of the spectra. Some spectra require less polynomials to solve LU equivalence problem than others. The result is obtained using geometric methods, i.e., by calculating the dimensions of reduced spaces, stemming from the symplectic reduction procedure.

DOI
10.1063/1.4819499

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Given L-qubit states with the fixed spectra of reduced one-qubit density matrices, we find a formula for the minimal number of invariant polynomials needed for solving local unitary (LU) equivalence problem, that is, problem of deciding if two states can be connected by local unitary operations. Interestingly, this number is not the same for every collection of the spectra. Some spectra require less polynomials to solve LU equivalence problem than others. The result is obtained using geometric methods, i.e., by calculating the dimensions of reduced spaces, stemming from the symplectic reduction procedure.
We derive the necessary and sufficient conditions for the local unitary (LU) equivalence of two general (pure or mixed) nnn-qubit states and determine the local unitary operators connecting them. Almost all relevant information is contained in the one-qubit reduced matrices of the multiqubit states under investigation. Our technique relies on identifying ab initio all local symmetries and the corresponding local cyclic unitary operators. To derive the above conditions we use the reference forms of the multiqubit states whose definition requires the diagonalization of the one-qubit reduced matrices. Based on those conditions, we propose a straightforward protocol to decide whether or not two nnn-qubit states are LU equivalent.
Local-unitary (LU) equivalence asks whether two quantum states differ only by independent changes of basis on their qubits. For graph states, whether this relation can be decided in polynomial time has remained open for over a decade. We give a deterministic algorithm that decides LU equivalence for graphs on nnn labelled vertices in O~(n6.38)\widetilde O(n^{6.38})O(n6.38) bit operations and constructs exact single-qubit unitaries whenever the states are equivalent. Building on Claudet and Perdrix's quasipolynomial algorithm, we replace the enumeration of vertex subsets by a compact system of constraints generated from pairs and triples. The remaining graph transformation is found by solving linear equations over the binary field. These new steps cost O~(n5)\widetilde O(n^5)O(n5) bit operations; the inherited graph preprocessing sets the overall bound. We also count the local-Clifford (LC) classes of graph states within any LU class: their number is a power of two, computable within the same bound. For any given graph state, this decides whether single-qubit Clifford gates reach every graph state in its LU class, and supplies a counterexample when they do not. The method also decides LU equivalence of stabilizer codes encoding one logical qubit.
We show that two density operators of mixed quantum states are in the same local unitary orbit if and only if they agree on polynomial invariants in a certain Noetherian ring for which degree bounds are known in the literature.This implicitly gives a finite complete set of invariants for local unitary equivalence.This is done by showing that local unitary equivalence of density operators is equivalent to local GL equivalence and then using techniques from algebraic geometry and geometric invariant theory.We also classify the SLOCC polynomial invariants and give a degree bound for generators of the invariant ring in the case of n-qubit pure states.Of course it is well known that polynomial invariants are not a complete set of invariants for SLOCC.
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.
We derive necessary and sufficient conditions for local unitary (LU) operators to leave invariant the set of 1-qubit reduced density matrices of a multi-qubit state. LU operators with this property are tensor products of {\it cyclic local} operators, and form a subgroup, the centralizer subgroup of the set of reduced states, of the Lie group SU(2)⊗nSU(2)^{\otimes n}SU(2)⊗n. The dimension of this subgroup depends on the type of reduced density matrices. It is maximum when all reduced states are maximally mixed and it is minimum when none of them is maximally mixed. For any given multi-qubit state, pure or mixed, we compute the LU operators that fix the corresponding reduced density matrices and determine the equivalence class of the given state.