We study a problem from zero-error information theory—a topic well-known for its rich connections to combinatorics [1,8,10–12,14]—in a setting where a sender and receiver may use quantum entanglement, one of the most striking features of quantum mechanics. The problem that we consider is the classical source-channel coding problem , where Alice and Bob are each given an input from a random source and get access to a noisy channel through which Alice can send messages to Bob. Their goal is to minimize the average number of channel uses per source input while allowing Bob to learn Alice’s inputs. Here we show that entanglement can allow for an unbounded decrease in the asymptotic rate of classical source-channel codes. We also consider the source problem , the case where Alice can send messages to Bob without noise. We prove a lower bound on the rate of source codes with entanglement in terms of a variant of the Lovász theta number [10,13], a graph parameter given by a semidefinite program.
We study a problem from zero-error information theory—a topic well-known for its rich connections to combinatorics [1,8,10–12,14]—in a setting where a sender and receiver may use quantum entanglement, one of the most striking features of quantum mechanics. The problem that we consider is the classical source-channel coding problem , where Alice and Bob are each given an input from a random source and get access to a noisy channel through which Alice can send messages to Bob. Their goal is to minimize the average number of channel uses per source input while allowing Bob to learn Alice’s inputs. Here we show that entanglement can allow for an unbounded decrease in the asymptotic rate of classical source-channel codes. We also consider the source problem , the case where Alice can send messages to Bob without noise. We prove a lower bound on the rate of source codes with entanglement in terms of a variant of the Lovász theta number [10,13], a graph parameter given by a semidefinite program.
We study the use of quantum entanglement in the zero-error source-channel coding problem. Here, Alice and Bob are connected by a noisy classical one-way channel, and are given correlated inputs from a random source. Their goal is for Bob to learn Alice's input while using the channel as little as possible. In the zero-error regime, the optimal rates of source codes and channel codes are given by graph parameters known as the Witsenhausen rate and Shannon capacity, respectively. The Lovász theta number, a graph parameter defined by a semidefinite program, gives the best efficiently computable upper bound on the Shannon capacity and it also upper bounds its entanglement-assisted counterpart. At the same time, it was recently shown that the Shannon capacity can be increased if Alice and Bob may use entanglement. Here, we partially extend these results to the source-coding problem and to the more general source-channel coding problem. We prove a lower bound on the rate of entanglement-assisted source-codes in terms of Szegedy's number (a strengthening of the theta number). This result implies that the theta number lower bounds the entangled variant of the Witsenhausen rate. We also show that entanglement can allow for an unbounded improvement of the asymptotic rate of both classical source codes and classical source-channel codes. Our separation results use low-degree polynomials due to Barrington, Beigel and Rudich, Hadamard matrices due to Xia and Liu, and a new application of remote state preparation.
Alice and Bob receive a bipartite state (possibly entangled) from some finite collection or from some subspace. Alice sends a message to Bob through a noisy quantum channel such that Bob may determine the initial state, with zero chance of error. This framework encompasses, for example, teleportation, dense coding, entanglement assisted quantum channel capacity, and one-way communication complexity of function evaluation. With classical sources and channels, this problem can be analyzed using graph homomorphisms. We show this quantum version can be analyzed using homomorphisms on non-commutative graphs (an operator space generalization of graphs). Previously the Lovász ϑ number has been generalized to non-commutative graphs; we show this to be a homomorphism monotone, thus providing bounds on quantum source-channel coding. We generalize the Schrijver and Szegedy numbers, and show these to be monotones as well. As an application, we construct a quantum channel whose entanglement assisted zero-error one-shot capacity can only be unlocked using a non-maximally entangled state. These homomorphisms allow definition of a chromatic number for non-commutative graphs. Many open questions are presented regarding the possibility of a more fully developed theory.
This dissertation contains results on three quite different topics. First, I investigate the entanglement resources required for two parties to jointly implement a unitary operation using local operations and classical communication (LOCC). If this resource is of smallest feasible Schmidt rank then it must be maximally entangled. If the Schmidt rank is higher, less entanglement may suffice. Second, I investigate the source of the “quantum speedup”. I quantify quantum interference and show that in order for a quantum computer to be significantly faster than a classical computer, it must make use of operations capable of producing large amounts of interference (or a large number of operations that can produce small amounts of interference). A quantum computer not making use of such a resource can be efficiently simulated by a classical computer. Third, I investigate zero-error source-channel coding. In this scenario, Alice wishes to convey a message to Bob through a noisy channel, with zero chance of error, in the case where Bob already has some prior knowledge regarding the message that is to be sent. For classical messages and classical channels, it was known that three graph invariants of Lovász, Szegedy, and Schrijver provide necessary conditions for this task. We show that this applies also when Alice and Bob make use of an entangled state, unifying and extending a series of previous results. Finally, I introduce a fully quantum version of source-channel coding where the message to be sent, the channel, and the side information are all quantum. Whereas the classical case makes use of graphs, the quantum case makes use of non-commutative graphs. I generalize the concept of graph homomorphism, as well as the Szegedy and Schrijver numbers, to non-commutative graphs and show that the necessary conditions for the classical case generalize to the quantum case. Using this theory, I construct a quantum channel whose one-shot zero-error entanglement assisted capacity can only be unlocked using a non-maximally entangled state, showing that in this case less is more when it comes to entanglement resources.
Bounds on Entanglement-Assisted Source-Channel Coding via the Lovász ϑ 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 ϑ(G̅) ≤ ϑ(H̅), where ϑ represents the Lovász number. We also obtain similar inequalities for the related Schrijver ϑ-and Szegedy ϑ+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: α*(G) ≤ ϑ-(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 Lovász number. Beigi introduced a quantity β as an upper bound on α* and posed the question of whether β(G) = ⌊ϑ(G)⌋. We answer this in the affirmative and show that a related quantity is equal to ⌊ϑ(G)⌋. We show that a quantity χvect(G) recently introduced in the context of Tsirelson's problem is equal to ⌊ϑ+(G)⌋. In an appendix, we investigate multiplicativity properties of Schrijver's and Szegedy's numbers, as well as projective rank.
We study the effects of quantum entanglement on the performance of two classical zero-error communication tasks among multiple parties. Both tasks are generalizations of the two-party zero-error channel-coding problem, where a sender and a receiver want to perfectly communicate messages through a one-way classical noisy channel. If the two parties are allowed to share entanglement, there are several positive results that show the existence of channels for which they can communicate strictly more than what they could do with classical resources. In the first task, one sender wants to communicate a common message to multiple receivers. We show that if the number of receivers is greater than a certain threshold then entanglement does not allow for an improvement in the communication for any finite number of uses of the channel. On the other hand, when the number of receivers is fixed, we exhibit a class of channels for which entanglement gives an advantage. The second problem we consider features multiple collaborating senders and one receiver. Classically, cooperation among the senders might allow them to communicate on average more messages than the sum of their individual possibilities. We show that whenever a channel allows single-sender entanglement-assisted advantage, then the gain extends also to the multisender case. Furthermore, we show that entanglement allows for a peculiar amplification of information which cannot happen classically, for a fixed number of uses of the channels.
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 theta(G) <= theta(H) where theta represents the Lovász number. We also obtain similar inequalities for the related Schrijver theta^- and Szegedy theta^+ 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) <= theta^-(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 Lovász number. Beigi introduced a quantity beta as an upper bound on alpha^* and posed the question of whether beta(G) = \lfloor theta(G) \rfloor. We answer this in the affirmative and show that a related quantity is equal to \lceil theta(G) \rceil. We show that a quantity chi_{vect}(G) recently introduced in the context of Tsirelson's conjecture is equal to \lceil theta^+(G) \rceil.