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Multipartite Entanglement in Physically Motivated Settings

Gunn, David Kenworthy

Published Jan 1, 2024• 0 citations
OpenAlex

Abstract

Entanglement is a central phenomenon in modern physics, ubiquitous in quantum systems. From a quantum information theory perspective, entanglement is a resource when spatially separated labs are only able to act locally. The set of quantum operations in this setting is known as local operations and classical communication (LOCC). Studying LOCC transformations allows one to quantify entanglement. With this approach, bipartite entanglement has been well understood. However, multipartite entanglement presents a greater challenge. In particular, for large enough systems, states are generically incomparable via LOCC. Motivated by these results, we study multipartite entanglement via two approaches: firstly, by considering physically motivated modifications to LOCC; secondly, by focusing on physically relevant states. This thesis is divided into two parts. The first part takes the first of the two approaches and studies physically motivated modifications of LOCC. In particular, we study multi-state LOCC and approximate LOCC. In multi-state LOCC, we consider transformations of multiple states. This setting is physically motivated; multi-state transformations are natural if one can store states and act on them simultaneously. We show this setting is considerably more powerful than single-state LOCC, with even local unitaries (LUs) able to change the stochastic LOCC (SLOCC) class of the individual states. We also demonstrate novel multipartite features, including multipartite catalysis, and characterise LU transformations of multiple bipartite states. In approximate LOCC, we study transformations which nearly reach the desired state, rather than exactly reaching it. This setting is also physically motivated as, in any real lab, one never transforms states exactly to a desired state, but instead transforms states to an ensemble of states that are near the desired state. Here, we show certain features of LOCC simplify in approximate transformations. We then define a hierarchy of physically relevant approximate transformations and show that, unlike in the bipartite setting, this hierarchy does not collapse for multipartite approximate transformations. The second part of this thesis takes the second approach and considers a physically relevant class of states: matrix product states (MPS). Two MPS are said to belong to the same phase with respect to a symmetry if they can be converted into one another via short-depth, symmetric circuits. With this definition, new exotic phases of matter emerge, including symmetry-protected topological (SPT) order. We study how symmetric measurements and feedforward alter the phase classification of MPS. We demonstrate that for all on-site finite abelian symmetries, the phase classification trivialises. Moreover, for certain non-abelian symmetry groups, we provide protocols that implement asymptotically deterministic transformations between any two phases. We also study the connection between SPT phases and SLOCC classes of MPS. In particular, we characterise the normal MPS SLOCC classes, with bond and physical dimension equal to two, that are capable of corresponding to SPT phases.

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Entanglement is a central phenomenon in modern physics, ubiquitous in quantum systems. From a quantum information theory perspective, entanglement is a resource when spatially separated labs are only able to act locally. The set of quantum operations in this setting is known as local operations and classical communication (LOCC). Studying LOCC transformations allows one to quantify entanglement. With this approach, bipartite entanglement has been well understood. However, multipartite entanglement presents a greater challenge. In particular, for large enough systems, states are generically incomparable via LOCC. Motivated by these results, we study multipartite entanglement via two approaches: firstly, by considering physically motivated modifications to LOCC; secondly, by focusing on physically relevant states. This thesis is divided into two parts. The first part takes the first of the two approaches and studies physically motivated modifications of LOCC. In particular, we study multi-state LOCC and approximate LOCC. In multi-state LOCC, we consider transformations of multiple states. This setting is physically motivated; multi-state transformations are natural if one can store states and act on them simultaneously. We show this setting is considerably more powerful than single-state LOCC, with even local unitaries (LUs) able to change the stochastic LOCC (SLOCC) class of the individual states. We also demonstrate novel multipartite features, including multipartite catalysis, and characterise LU transformations of multiple bipartite states. In approximate LOCC, we study transformations which nearly reach the desired state, rather than exactly reaching it. This setting is also physically motivated as, in any real lab, one never transforms states exactly to a desired state, but instead transforms states to an ensemble of states that are near the desired state. Here, we show certain features of LOCC simplify in approximate transformations. We then define a hierarchy of physically relevant approximate transformations and show that, unlike in the bipartite setting, this hierarchy does not collapse for multipartite approximate transformations. The second part of this thesis takes the second approach and considers a physically relevant class of states: matrix product states (MPS). Two MPS are said to belong to the same phase with respect to a symmetry if they can be converted into one another via short-depth, symmetric circuits. With this definition, new exotic phases of matter emerge, including symmetry-protected topological (SPT) order. We study how symmetric measurements and feedforward alter the phase classification of MPS. We demonstrate that for all on-site finite abelian symmetries, the phase classification trivialises. Moreover, for certain non-abelian symmetry groups, we provide protocols that implement asymptotically deterministic transformations between any two phases. We also study the connection between SPT phases and SLOCC classes of MPS. In particular, we characterise the normal MPS SLOCC classes, with bond and physical dimension equal to two, that are capable of corresponding to SPT phases.
Understanding multipartite entanglement is a key goal in quantum information. Entanglement in pure states can be characterized by considering transformations under local operations assisted by classical communication (LOCC). However, it has been shown that, for $n\ensuremath{\ge}5$ parties, multipartite pure states are generically isolated, i.e., they can be neither reached nor transformed under LOCC. Nonetheless, in any real laboratory, one never deterministically transforms a pure initial state exactly to a pure target state. Instead, one transforms a mixed state near the initial state to an ensemble that is on average close to the target state. This motivates studying approximate LOCC transformations. After reviewing in detail the known results in the bipartite case, we present the gaps that remain open in the multipartite case. While the analysis of the multipartite setting is much more technically involved due to the existence of different stochastic LOCC (SLOCC) classes, certain features simplify in the approximate setting. In particular, we show that it is sufficient to consider pure initial states, that it is sufficient to consider LOCC protocols with finitely many rounds of communication, and that approximate transformations can be approximated by ensemble transformations within an SLOCC class. Then we formally define a hierarchy of different forms of approximate transformations that are relevant from a physical point of view. Whereas this hierarchy collapses in the bipartite case, we show that this is not the case for the multipartite setting, which is fundamentally richer. To wit, we show that optimal multipartite approximate transformations are not generally deterministic, that ensemble transformations within an SLOCC class can achieve a higher fidelity than deterministic transformations within an SLOCC class, and that there are approximate transformations with no deterministic transformations nearby.
Understanding multipartite entanglement is vital, as it underpins a wide range of phenomena across physics. The study of transformations of states via Local Operations assisted by Classical Communication (LOCC) allows one to quantitatively analyse entanglement, as it induces a partial order in the Hilbert space. However, it has been shown that, for systems with fixed local dimensions, this order is generically trivial, which prevents relating multipartite states to each other with respect to any entanglement measure. In order to obtain a non-trivial partial ordering, we study a physically motivated extension of LOCC: multi-state LOCC. Here, one considers simultaneous LOCC transformations acting on a finite number of entangled pure states. We study both multipartite and bipartite multi-state transformations. In the multipartite case, we demonstrate that one can change the stochastic LOCC (SLOCC) class of the individual initial states by only applying Local Unitaries (LUs). We show that, by transferring entanglement from one state to the other, one can perform state conversions not possible in the single copy case; provide examples of multipartite entanglement catalysis; and demonstrate improved probabilistic protocols. In the bipartite case, we identify numerous non-trivial LU transformations and show that the source entanglement is not additive. These results demonstrate that multi-state LOCC has a much richer landscape than single-state LOCC.
Local operations assisted by classical communication (LOCC) play a fundamental role in the resource-theoretic formulation of entanglement theory inducing an operationally meaningful ordering in the set of entangled states. The particular class within LOCC of local unitary (LU) transformations is obviously closed under inversion. Thus, LU-equivalent states are interconvertible by LOCC and have therefore the same entanglement. For this reason LU-equivalence has been thoroughly studied in the context of entanglement theory. However, as I note here with a simple example, the converse is not true: there exist entangled states that are interconvertible by LOCC but which are not LU-equivalent. This motivates studying under which conditions LOCC interconvertibility is characterized by LU-equivalence. I provide a sufficient condition for this to hold. When one considers multipartite states with full entanglement dimensionality (as quantified by the Schmidt number vector) LOCC interconvertibility can only hold under LU-equivalence.
The study of state transformations by spatially separated parties with local operations assisted by classical communication (LOCC) plays a crucial role in entanglement theory and its applications in quantum information processing. Transformations of this type among pure bipartite states were characterized long ago and have a revealing theoretical structure. However, it turns out that generic fully entangled pure multipartite states cannot be obtained from nor transformed to any inequivalent fully entangled state under LOCC. States with this property are referred to as isolated. Nevertheless, multipartite states are classified into families, the so-called SLOCC classes, which possess very different properties. Thus, the above result does not forbid the existence of particular SLOCC classes that are free of isolation, and therefore, display a rich structure regarding LOCC convertibility. In fact, it is known that the celebrated n -qubit GHZ and W states give particular examples of such classes and in this work, we investigate this question in general. One of our main results is to show that the SLOCC class of the 3-qutrit totally antisymmetric state is isolation-free as well. Actually, all states in this class can be converted to inequivalent states by LOCC protocols with just one round of classical communication (as in the GHZ and W cases). Thus, we consider next whether there are other classes with this property and we find a large set of negative answers. Indeed, we prove weak isolation (i.e., states that cannot be obtained with finite-round LOCC nor transformed by one-round LOCC) for very general classes, including all SLOCC families with compact stabilizers and many with non-compact stabilizers, such as the classes corresponding to the n -qunit totally antisymmetric states for n≥4 . Finally, given the pleasant feature found in the family corresponding to the 3-qutrit totally antisymmetric state, we explore in more detail the structure induced by LOCC and the entanglement properties within this class.
Organising the space of entanglement structures of a multipartite quantum system is a much more challenging task than its bipartite version: while the local unitary (LU) orbit of a bipartite pure state can be conveniently characterized by its entanglement spectrum, invariants of multipartite entanglement structures are comparatively difficult to define and work with. The root cause of this difference is that the bipartite problem can be reduced to the analysis of matrix invariants, while its multipartite version is governed by a much richer space of tensor invariants. The present work explores the latter through the lens of so-called trace-invariants, which are in one-to-one correspondence with combinatorial objects known as colored graphs. We first explain why trace-invariant evaluations can serve as labels of LU-orbits of multipartite pure states, how this strategy extends to random states, and how the effect of local operations (LO) can be analyzed through such data. We then focus on entanglement classification within an (infinite-dimensional) subspace of reference states, whose basic building blocks are GHZ states of various dimensions. We show that relatively simple subclasses of trace-invariants are sufficient to separate the LU-orbits of reference states, and enable a complete (resp. an incomplete) characterization of their relations in the LO (resp. LOCC) resource theory of entanglement. Finally, we investigate how a (still infinite) subclass of reference states of local dimension N can be efficiently distinguished at leading and subleading orders in an asymptotic large-N expansion (among themselves, or from Haar-random states). This analysis relies crucially on combinatorial quantities associated to colored graphs, some of which have already played instrumental roles in the recent literature on random tensors. Results of broader relevance are reported along the way.
We introduce deterministic state-transformation protocols between many-body quantum states which can be implemented by low-depth Quantum Circuits (QC) followed by Local Operations and Classical Communication (LOCC). We show that this gives rise to a classification of phases in which topologically-ordered states or other paradigmatic entangled states become trivial. We also investigate how the set of unitary operations is enhanced by LOCC in this scenario, allowing one to perform certain large-depth QC in terms of low-depth ones.