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arXiv:2106.14891 · Physical Review A

Algebraic-geometric characterization of tripartite entanglement

Masoud Gharahi, Stefano Mancini

Published Oct 4, 2021• 12 citations
arXivHTMLPDF

Abstract

On the basis of an algebraic-geometric approach, the authors provide a rigorous characterization of pure states of three-qudit systems in terms of a finite number of families and subfamilies. The characterization has operational relevance for quantifying entanglement as a resource.
DOI
10.1103/physreva.104.042402

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We provide an in-depth study of tripartite entanglement of qudits. We start with a short review of tripartite entanglement invariants, prove a theorem about the complete list of all allowed values of three (out of the total of four) such invariants, and give several bounds on the allowed values of the fourth invariant. After introducing several operations on entangled states (that allow us to build new states from old states) and deriving general properties pertaining to their invariants, we arrive at the decomposition theorem as one of our main results. The theorem relates the algebraic invariants of any entanglement class with the invariants of its corresponding components in each of its direct sum decompositions. This naturally leads to the definition of reducible and irreducible entanglement classes. We explicitly compute algebraic invariants for several families of irreducible classes and show how the decomposition theorem allows computations of invariants for compounded classes to be carried out efficiently. This theorem also allows us to compute the invariants for the infinite number of entanglement classes constructed from irreducible components. We proceed with the complete list of the entanglement classes for three tribits with decompositions of each class into irreducible components, and provide a visual guide to interrelations of these decompositions. We conclude with numerous examples of building classes for higher-spin qudits.
Quantum Entanglement is one of the key manifestations of quantum mechanics that separate the quantum realm from the classical one. Characterization of entanglement as a physical resource for quantum technology became of uppermost importance. While the entanglement of bipartite systems is already well understood, the ultimate goal to cope with the properties of entanglement of multipartite systems is still far from being realized. This paper covers characterization of multipartite entanglement using algebraic-geometric tools. First, we establish an algorithm to classify multipartite entanglement by [Formula: see text]-secant varieties of the Segre variety and [Formula: see text]-multilinear ranks that are invariant under Stochastic Local Operations with Classical Communication (SLOCC). We present a fine-structure classification of multiqubit and tripartite entanglement based on this algorithm. Another fundamental problem in quantum information theory is entanglement transformation that is quite challenging regarding to multipartite systems. It is captivating that the proposed entanglement classification by algebraic geometry can be considered as a reference to study SLOCC and asymptotic SLOCC interconversions among different resources based on tensor rank and border rank, respectively. In this regard, we also introduce a new class of tensors that we call persistent tensors and construct a lower bound for their tensor rank. We further cover SLOCC convertibility of multipartite systems considering several families of persistent tensors.
We introduce a purely geometric formulation for two different measures addressed to quantify the entanglement between different parts of a tripartite qubit system. Our approach considers the entanglement-polytope defined by the smallest eigenvalues of the reduced density matrices of the qubit-components. The measures identify global and genuine entanglement, and are respectively associated with the projection and rejection of a given point of the polytope on the corresponding biseparable segments. Solving the so called 'inverse problem', we also discuss a way to force the system to behave in a particular form, which opens the possibility of controlling and manipulating entanglement for practical purposes.
Encoding the state of physical systems, the many-body wavefunction lives in a Hilbert space that conveniently offers a tensor factor to each constituent of the system. Consequentially, the wavefunction inherits structural properties such as entanglement, a new type of quantum correlation across the tensor factors. Entanglement can be quantified by forming polynomials in the wavefunction that remain invariant under local operations, which in turn naturally imply the properties we intuit about entanglement. It follows from the Tarski-Seidenberg theorem that these polynomial entanglement measures must obey polynomial constraints. It is among these constraints that the core features separating quantum correlations from classical correlations are illuminated, such as their monogamous nature. The smallest non-trivial system which exhibits entanglement monogamy tradeoffs governed by polynomial constraints is three qubits, for which we find novel constraints that are stronger than the previously known constraints, and which may as well be a complete description of three-qubit entanglement since the number of degrees of freedom matches the number of entanglement measures. A remarkable redundancy known in three qubits is that any party's total entanglement decomposes exactly into bipartite and tripartite type entanglement; we find that this behavior is merely part of the GHZ family tradition, and that any party within the n-qubit GHZ SLOCC equivalence class also has total entanglement which decomposes exactly into all k-partite type entanglements, thus providing insight into strong versions of monogamy relations. Refocusing back to tripartite systems, we then suppose arbitrarily many degrees of freedom into our subsystems, which appear to refuse all temptations of polygamy. Up to this point we have only been considering idealized isolated systems as it is known that monogamy relations aren't sensitive to the total purity, however we then find entanglement constraints which are violated by globally mixed states, ultimately suggesting that environments can enable entanglement to grow beyond what is otherwise allowed. Finally, we address the dynamical behavior of entanglement under several key interactions, and characterize the couplings according to their ability to generate entanglement.