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Semi-Algebraic Entanglement Consistency Relations: Fundamental and Dynamical

Grant W. Allen

Published Jan 1, 2017• 0 citations
OpenAlex

Abstract

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.

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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.
The limitation on the shareability of quantum entanglement over several parties, the so-called monogamy of entanglement, is an issue that has received considerable attention from the quantum information community over the last decade. A natural question of interest in this connection is whether monogamy of correlations is true for correlations other than entanglement. This issue is examined here by choosing quantum deficit, proposed by A. K. Rajagopal and R. W. Rendell [Phys. Rev. A 66, 022104 (2002)], an operational measure of correlations. In addition to establishing the polygamous nature of the class of three-qubit symmetric pure states characterized by two distinct Majorana spinors (to which the WWW states belong), those with three distinct Majorana spinors [to which Greenberger-Horne-Zeilinger (GHZ) states belong] are shown to either obey or violate monogamy relations. While the generalized WWW states can be monogamous or polygamous, the generalized GHZ states exhibit monogamy with respect to quantum deficit. The issue of using monogamy conditions based on quantum deficit to witness the states belonging to stochastic local operations and classical communication (SLOCC) inequivalent classes is discussed in light of these results.
Multipartite entanglement includes not only the genuine entanglement but also the kkk-entanglement $(k\ensuremath{\geqslant}2)$. It is known that the most bipartite entanglement measures have been shown to be monogamous, but the monogamy relation is involved in the bipartite entanglement measures rather than the multipartite ones. So how we can explore the monogamy relation for kkk-entanglement becomes a basic open problem. In this paper we establish an axiomatic definition of the monogamy relation for the kkk-entanglement measure based on the coarser relation of the system partition. We also present the axiomatic definition of the complete kkk-entanglement measure and the associated complete monogamy relation according to the framework of the complete multipartite entanglement measure we established in [Phys. Rev. A 101, 032301 (2020)], which is shown to be an efficient tool for characterizing the multipartite quantum correlation as complementary to the monogamy relation associated with the bipartite measure. Consequently, the relation and the difference between monogamy, complete monogamy, and tightly complete monogamy are clearly depicted in light of the three types of coarser relation of the system partition. We then illustrate our approach with two classes of kkk-entanglement measures in detail. We find that all these kkk-entanglement measures are monogamous if the reduced function is strictly concave, they are not completely monogamous, and they are not complete generally.
Given a multipartite quantum system that consists of two-level particles (qubits), one may or may not have access to all the subsystems. What can we know about the entanglement of the multiqubit system and residual correlations beyond two-tangle if we have access only to two-qubits at a time? Algebraic analysis of two-qubit states yields monogamy constraints on distribution of entanglement between sub-systems of an N-qubit state and criterion to determine if the state has multipartite entanglement. Monogamy constraints, reported in this letter, are relations between well known entanglement measures such as one-tangle, two-tangles and three-tangles of an N-qubit pure state.
Unlike classical correlation, quantum entanglement cannot be freely shared among many parties. This restricted shareability of entanglement among multi-party systems is known as monogamy of entanglement, which is one of the most fundamental properties of entanglement. Here, we summarize recent theoretical progress in the field of monogamy of entanglement. We firstly review the standard CKW-type monogamy inequalities in terms of various entanglement measures. In particular, the squashed entanglement and one-way distillable entanglement are monogamous for arbitrary dimensional systems. We then introduce some generalized version of monogamy inequalities which extend and sharpen the traditional ones. We also consider the dual polygamy inequalities for multi-party systems. Moreover, we present two new definitions to define monogamy of entanglement. Finally, some challenges and future directions for monogamy of entanglement are highlighted.
The notion of nonclassical correlations is a powerful contrivance for explaining phenomena exhibited in quantum systems. It is well known, however, that quantum systems are not free to explore arbitrary correlations-the church of the smaller Hilbert space only accepts monogamous congregants. We demonstrate how to characterize the limits of what is quantum mechanically possible with a computable measure, entanglement negativity. We show that negativity only saturates the standard linear monogamy inequality in trivial cases implied by its monotonicity under local operations and classical communication, and derive a necessary and sufficient inequality which, for the first time, is a nonlinear higher degree polynomial. For very large quantum systems, we prove that the negativity can be distributed at least linearly for the tightest constraint and conjecture that it is at most linear.
We examine the various properties of the three four-qubit monogamy relations, all of which introduce the power factors in the three-way entanglement to reduce the tripartite contributions. On the analytic ground as much as possible we try to find the minimal power factors, which make the monogamy relations hold if the power factors are larger than the minimal powers. Motivated to the three-qubit monogamy inequality we also examine whether those four-qubit monogamy relations provide the SLOCC-invariant four-way entanglement measures or not. Our analysis indicate that this is impossible provided that the monogamy inequalities are derived merely by introducing weighting power factors.