We introduce deterministic state-transformation protocols between many-body quantum states that can be implemented by low-depth quantum circuits followed by local operations and classical communication. 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 local operations and classical communication in this scenario, allowing one to perform certain large-depth quantum circuits in terms of low-depth ones.
We introduce deterministic state-transformation protocols between many-body quantum states that can be implemented by low-depth quantum circuits followed by local operations and classical communication. 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 local operations and classical communication in this scenario, allowing one to perform certain large-depth quantum circuits in terms of low-depth ones.
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.
We consider generic pure n-qubit states and a general class of pure states of arbitrary dimensions and arbitrarily many subsystems. We characterize those states which can be reached from some other state via local operations assisted by finitely many rounds of classical communication (LOCC_{N}). For n qubits with n>3, we show that this set of states is of measure zero, which implies that the maximally entangled set is generically of full measure if restricted to the practical scenario of LOCC_{N}. Moreover, we identify a class of states for which any LOCC_{N} protocol can be realized via a concatenation of deterministic steps. We show, however, that in general there exist state transformations which require a probabilistic step within the protocol, which highlights the difference between bipartite and multipartite 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.
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.
We introduce a classification of quantum phases of mixed states based on the ability to preserve and fault tolerantly transfer information. We formulate phase equivalence between sets of states that encode logical quantum information and are connected by shallow local channel circuits assisted by local measurements and global classical communication of measurement outcomes, where all quantum operations, including measurements, must be robust to noise. We show that the existence of a recovery threshold for local noise is a property of an entire phase. We prove that any two sets of states with nonzero thresholds that are related by finite-depth local channel circuits belong to the same fault tolerant phase. Our approach leads to a classification of subsystem codes without favoring any state of the gauge subsystem, and we demonstrate phase equivalence between 2D and 3D color codes using dimensional jumps within the 3D gauge color code. Lastly, we study how syndrome structure affects fault tolerance in circuits with measurements. We show that qudit stabilizer codes with extended syndromes permit single-shot state preparation, whereas a broad class of shallow circuits measuring point-like syndromes, including state preparation for 2D topological codes and non-Abelian topological orders with solvable anyon theories, is not fault tolerant.
Recent advances have led towards first prototypes of quantum networks in which entanglement is distributed by sources producing bipartite entangled states. This raises the question of which states can be generated in quantum networks based on bipartite sources using local operations and classical communication. In this work, we study state transformations under finite rounds of local operations and classical communication (LOCC) in networks based on maximally entangled two-qubit states. We first derive the symmetries for arbitrary network structures, as these determine which transformations are possible. Then, we show that contrary to tree graphs, for which it has already been shown that any state within the same entanglement class can be reached, there exist states which can be reached probabilistically but not deterministically if the network contains a cycle. Furthermore, we provide a systematic way to determine states which are not reachable in networks consisting of a cycle. Moreover, we provide a complete characterization of the states which can be reached in a cycle network with a protocol where each party measures only once, and each step of the protocol results in a deterministic transformation. Finally, we present an example which cannot be reached with such a simple protocol, and constitutes, up to our knowledge, the first example of a LOCC transformation among fully entangled states requiring three rounds of classical communication.