Separable State Discrimination Using Local Quantum Operations and Classical Communication

In this thesis we study the subset of quantum operations that can be implemented using only local quantum operations and classical communication (LOCC). This restricted paradigm serves as a tool to study not only quantum correlations and other nonlocal quantum effects, but also resource transformations such as channel capacities. The mathematical structure of LOCC is complex and difficult to characterize. In the first part of this thesis we provide a precise description of LOCC and related operational classes in terms of quantum instruments. Our formalism captures both finite round protocols as well as those that utilize an unbounded number of communication rounds. This perspective allows us to measure the distance between two LOCC instruments and hence discuss the closure of LOCC in a rigorous way. While the set of LOCC is not topologically closed, we show that the operations that can be implemented using some fixed number rounds of communication constitute a compact subset of all quantum operations. We also exhibit a subset of LOCC measurements that is closed. Additionally we establish the existence of an open ball around the completely depolarizing map consisting entirely of LOCC implementable maps. In the second part of this thesis we focus on the task of discriminating states from some known set S by LOCC. Building on the work in the paper "Quantum nonlocality without entanglement", we provide a framework for lower bounding the error probability of any LOCC protocol aiming at discriminating the states from S. We apply our framework to an orthonormal product basis known as the domino states. This gives an alternative and simplified bound quantifying how well these states can be discriminated using LOCC. We generalize this result for similar bases in larger dimensions, as well as the "rotated" domino states, resolving a long-standing open question. These results give new examples of quantitative gaps between the classes of separable and LOCC operations. In the last part of this thesis, we ask what differentiates separable from LOCC operations. Both of these classes play a key role in the study of entanglement. Separable operations are known to be strictly more powerful than LOCC ones, but no simple explanation of this phenomenon is known. We show that, in the case of bipartite von Neumann measurements, the ability to interpolate is an operational principle that separates LOCC and separable operations

Introducing elements of active learning in a small course

Despite the modern consensus that traditional-style lectures do not promote deep and longlasting learning outcomes, many mathematics courses are delivered precisely in this manner.1 How does one incorporate elements of active learning in a pre-established course with traditional-style lectures? Are these additional activities introduced at the expense of covering lessmaterial and do more advanced students suffer from this style of teaching? If so, do the advantages outweigh the negative effects in the context of a master’s level mathematics course

A Generalization of Kochen-Specker Sets Relates Quantum Coloring to Entanglement-Assisted Channel Capacity

We introduce two generalizations of Kochen-Specker (KS) sets: projective KS sets and generalized KS sets. We then use projective KS sets to characterize all graphs for which the chromatic number is strictly larger than the quantum chromatic number. Here, the quantum chromatic number is defined via a nonlocal game based on graph coloring. We further show that from any graph with separation between these two quantities, one can construct a classical channel for which entanglement assistance increases the one-shot zero-error capacity. As an example, we exhibit a new family of classical channels with an exponential increase.Comment: 16 page

Graph Homomorphisms for Quantum Players

A homomorphism from a graph X to a graph Y is an adjacency preserving mapping f:V(X) -> V(Y). We consider a nonlocal game in which Alice and Bob are trying to convince a verifier with certainty that a graph X admits a homomorphism to Y. This is a generalization of the well-studied graph coloring game. Via systematic study of quantum homomorphisms we prove new results for graph coloring. Most importantly, we show that the Lovász theta number of the complement lower bounds the quantum chromatic number, which itself is not known to be computable. We also show that other quantum graph parameters, such as quantum independence number, can differ from their classical counterparts. Finally, we show that quantum homomorphisms closely relate to zero-error channel capacity. In particular, we use quantum homomorphisms to construct graphs for which entanglement-assistance increases their one-shot zero-error capacity.Published versio

Quantum Random Access Codes with Shared Randomness

We consider a communication method, where the sender encodes n classical bits into 1 qubit and sends it to the receiver who performs a certain measurement depending on which of the initial bits must be recovered. This procedure is called (n,1,p) quantum random access code (QRAC) where p > 1/2 is its success probability. It is known that (2,1,0.85) and (3,1,0.79) QRACs (with no classical counterparts) exist and that (4,1,p) QRAC with p > 1/2 is not possible. We extend this model with shared randomness (SR) that is accessible to both parties. Then (n,1,p) QRAC with SR and p > 1/2 exists for any n > 0. We give an upper bound on its success probability (the known (2,1,0.85) and (3,1,0.79) QRACs match this upper bound). We discuss some particular constructions for several small values of n. We also study the classical counterpart of this model where n bits are encoded into 1 bit instead of 1 qubit and SR is used. We give an optimal construction for such codes and find their success probability exactly--it is less than in the quantum case. Interactive 3D quantum random access codes are available on-line at http://home.lanet.lv/~sd20008/racs .Comment: 51 pages, 33 figures. New sections added: 1.2, 3.5, 3.8.2, 5.4 (paper appears to be shorter because of smaller margins). Submitted as M.Math thesis at University of Waterloo by M

Characterization of non-universal two-qubit Hamiltonians

It is known that almost all 2-qubit gates are universal for quantum computing (Lloyd 1995; Deutsch, Barenco, Eckert 1995). However, an explicit characterization of non-universal 2-qubit gates is not known. We consider a closely related problem of characterizing the set of non-universal 2-qubit Hamiltonians. We call a 2-qubit Hamiltonian n-universal if, when applied on different pairs of qubits, it can be used to approximate any unitary operation on n qubits. It follows directly from the results of Lloyd and Deutsch, Barenco, Eckert, that almost any 2-qubit Hamiltonian is 2-universal. Our main result is a complete characterization of 2-non-universal 2-qubit Hamiltonians. There are three cases when a 2-qubit Hamiltonian H is not universal: (1) H shares an eigenvector with the gate that swaps two qubits; (2) H acts on the two qubits independently (in any of a certain family of bases); (3) H has zero trace. The last condition rules out the Hamiltonians that generate SU(4)---it can be omitted if the global phase is not important. A Hamiltonian that is not 2-universal can still be 3-universal. We give a (possibly incomplete) list of 2-qubit Hamiltonians that are not 3-universal. If this list happens to be complete, it actually gives a classification of n-universal 2-qubit Hamiltonians for all n >= 3

Bounds on entanglement assisted source-channel coding via the Lovasz theta number and its variants

We study zero-error entanglement assisted source-channel coding (communication in the presence of side information). Adapting a technique of Beigi, we show that such coding requires existence of a set of vectors satisfying orthogonality conditions related to suitably defined graphs $G$ and $H$. Such vectors exist if and only if $\vartheta(\overline{G}) \le \vartheta(\overline{H})$ where $\vartheta$ represents the Lov\'asz number. We also obtain similar inequalities for the related Schrijver $\vartheta^-$ and Szegedy $\vartheta^+$ numbers. These inequalities reproduce several known bounds and also lead to new results. We provide a lower bound on the entanglement assisted cost rate. We show that the entanglement assisted independence number is bounded by the Schrijver number: $\alpha^*(G) \le \vartheta^-(G)$. Therefore, we are able to disprove the conjecture that the one-shot entanglement-assisted zero-error capacity is equal to the integer part of the Lov\'asz number. Beigi introduced a quantity $\beta$ as an upper bound on $\alpha^*$ and posed the question of whether $\beta(G) = \lfloor \vartheta(G) \rfloor$. We answer this in the affirmative and show that a related quantity is equal to $\lceil \vartheta(G) \rceil$. We show that a quantity $\chi_{\textrm{vect}}(G)$ recently introduced in the context of Tsirelson's conjecture is equal to $\lceil \vartheta^+(\overline{G}) \rceil$. In an appendix we investigate multiplicativity properties of Schrijver's and Szegedy's numbers, as well as projective rank.Comment: Fixed proof of multiplicativity; more connections to prior work in conclusion; many changes in expositio