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arXiv:1706.00634

A new degree bound for local unitary and $n$ -qubit SLOCC Invariants

Jacob Turner

Published Jun 2, 2017• 0 citations

Abstract

Deep connections between invariant theory and entanglement have been known for some time and been the object of intense study. This includes the study of local unitary equivalence of density operators as well as entanglement that can be observed in stochastic local operations assisted by classical communication (SLOCC). An important aspect of both of these areas is the computation of complete sets of invariants polynomials. For local unitary equivalence as well as $n$-qubit SLOCC invariants, complete descriptions of these invariants exist. However, these descriptions give infinite sets; of great interest is finding generating sets of invariants. In this regard, degree bounds are highly sought after to limit the possible sizes of such generating sets. In this paper we give new upper bounds on the degrees of the invariants, both for a certain complete set of local unitary invariants as well as the $n$-qubit SLOCC invariants. We show that there exists a complete set of local unitary invariants of density operators in a Hilbert space $\mathcal{H}$, of dimension $d$, which are generated by invariants of degree at most $d^4$. This in turn allows us to show that the $n$-qubit SLOCC invariants are generated by invariants of degree at most $2^{4n}$.

DOI: 10.48550/arxiv.1706.00634

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Deep connections between invariant theory and entanglement have been known for some time and been the object of intense study. This includes the study of local unitary equivalence of density operators as well as entanglement that can be observed in stochastic local operations assisted by classical communication (SLOCC). An important aspect of both of these areas is the computation of complete sets of invariants polynomials. For local unitary equivalence as well as

n

-qubit SLOCC invariants, complete descriptions of these invariants exist. However, these descriptions give infinite sets; of great interest is finding generating sets of invariants. In this regard, degree bounds are highly sought after to limit the possible sizes of such generating sets. In this paper we give new upper bounds on the degrees of the invariants, both for a certain complete set of local unitary invariants as well as the

n

-qubit SLOCC invariants. We show that there exists a complete set of local unitary invariants of density operators in a Hilbert space

\mathcal{H}

, of dimension

d

, which are generated by invariants of degree at most

d^4

. This in turn allows us to show that the

n

-qubit SLOCC invariants are generated by invariants of degree at most

2^{4n}

.

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.

Understanding the complexity of entangled states within the context of SLOCC (stochastic local operations and classical communications) involving several number qubits is essential for advancing our knowledge of quantum systems. This complexity is often analyzed by classifying the states via local symmetry groups. Practically, tthe resulting classes can be distinguished using invariant polynomials, but the size of these polynomials grows rapidly. Hence, it is crucial to obtain the smallest possible invariants. In this short note, we compute the basis of invariant polynomials of 7 qubits of degree 4, which are the smallest degree invariants. We obtain these polynomials using the representation theory and algebraic combinatorics.

We propose a new approach to the geometry of the four-qubit entanglement classes depending on parameters. More precisely, we use invariant theory and algebraic geometry to describe various stratifications of the Hilbert space by Stochastic Local Operations with Classical Communication (SLOCC) invariant algebraic varieties. The normal forms of the four-qubit classification of Verstraete et al. are interpreted as dense subsets of components of the dual variety of the set of separable states and an algorithm based on the invariants/covariants of the four-qubit quantum states is proposed to identify a state with a SLOCC equivalent normal form (up to qubits permutation).

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SL(3,{\mathbb {C}})

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