Quantum resources can be more powerful than classical resources-a quantum computer can solve certain problems exponentially faster than a classical computer, and computing a function of two parties' inputs can be done with exponentially less communication with quantum messages than with classical ones. Here we consider a task between two players, Alice and Bob where quantum resources are infinitely more powerful than their classical counterpart. Alice is given a string of length n, and Bob's task is to exclude certain combinations of bits that Alice might have. If Alice must send classical messages, then she must reveal nearly n bits of information to Bob, but if she is allowed to send quantum bits, the amount of information she must reveal goes to zero with increasing n. Next, we consider a version of the task where the parties may have access to entanglement. With this assistance, Alice only needs to send a constant number of bits, while without entanglement, the number of bits Alice must send grows linearly with n. The task is related to the Pusey-Barrett-Rudolph theorem which arises in the context of the foundations of quantum theory.
Quantum resources can be more powerful than classical resources-a quantum computer can solve certain problems exponentially faster than a classical computer, and computing a function of two parties' inputs can be done with exponentially less communication with quantum messages than with classical ones. Here we consider a task between two players, Alice and Bob where quantum resources are infinitely more powerful than their classical counterpart. Alice is given a string of length n, and Bob's task is to exclude certain combinations of bits that Alice might have. If Alice must send classical messages, then she must reveal nearly n bits of information to Bob, but if she is allowed to send quantum bits, the amount of information she must reveal goes to zero with increasing n. Next, we consider a version of the task where the parties may have access to entanglement. With this assistance, Alice only needs to send a constant number of bits, while without entanglement, the number of bits Alice must send grows linearly with n. The task is related to the Pusey-Barrett-Rudolph theorem which arises in the context of the foundations of quantum theory.
Communication complexity is the amount of communication needed to compute a function when the function inputs are distributed over multiple parties. In its simplest form, one-way communication complexity, Alice and Bob compute a function f(x,y) , where x is given to Alice and y is given to Bob, and only one message from Alice to Bob is allowed. A fundamental question in quantum information is the relationship between one-way quantum and classical communication complexities, i.e., how much shorter the message can be if Alice is sending a quantum state instead of bit strings? We make some progress towards this question with the following results.Let f:X×Y→Z∪{⊥} be a partial function and μ be a distribution with support contained in f−1(Z) . Denote d=|Z| . Let Rϵ1,μ(f) be the classical one-way communication complexity of f ; Qϵ1,μ(f) be the quantum one-way communication complexity of f and Qϵ1,μ,∗(f) be the entanglement-assisted quantum one-way communication complexity of f , each with distributional error (average error over μ ) at most ϵ . We show:1) If μ is a product distribution, η>0 and 0≤ϵ≤1−1/d , then, R2ϵ−dϵ2/(d−1)+η1,μ(f)≤2Qϵ1,μ,∗(f)+O(loglog(1/η)). 2)If μ is a non-product distribution and Z={0,1} , then ∀ϵ,η>0 such that ϵ/η+η<0.5 , R3η1,μ(f)=O(Qϵ1,μ(f)⋅CS(f)/η3), where CS(f)=maxyminz&am
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.
We have now seen in Chapters 20–22 how Alice can communicate classical or quantum information to Bob, perhaps even with the help of shared entanglement. One might argue that these communication tasks are the most fundamental tasks in quantum Shannon theory, given that they have furthered our understanding of the nature of information transmission over quantum channels. However, when discussing the communication of classical information, we made no stipulation as to whether this classical information should be public, so that any third party might have partial or full access to it, or private, so that no third party has access. This chapter establishes the private classical capacity theorem, which gives the maximum rate at which Alice can communicate classical information privately to Bob without anyone else in the universe knowing what she sent to him. A variation of the information-processing task corresponding to this theorem was one of the earliest studied in quantum information theory, with the Bennett– Brassard-84 quantum key distribution protocol being the first proposed protocol for exploiting quantum mechanics to establish a shared secret key between two parties. The private classical capacity theorem is important for quantum key distribution because it establishes the maximum rate at which two parties can generate a shared secret key. Another equally important, but less obvious utility of private classical communication is in establishing a protocol for quantum communication at the coherent information rate. Section 22.2 demonstrated a somewhat roundabout way of arriving at the conclusion that it is possible to communicate quantum information reliably at the coherent information rate—recall that we “coherified” the entanglement-assisted classical capacity theorem and then exploited the coherent communication identity and catalytic use of entanglement. Establishing achievability of the coherent information rate via private classical coding is another way of arriving at the same result, with the added benefit that the resulting protocol does not require the catalytic use of entanglement. The intuition for quantum communication via privacy arises from the nocloning theorem. Suppose that Alice is able to communicate private classical messages to Bob, so that the channel's environment (Eve) is not able to distinguish which message Alice is transmitting to Bob.
In this work we introduce an intermediate setting between quantum nonlocality and communication complexity problems. More precisely, we study the value of XOR games G when Alice and Bob are allowed to use a limited amount of one-way classical communication ωo.w.−c(G) (resp. one-way quantum communication ωo.w.−c∗(G)), where c denotes the number of bits (resp. qubits). The key quantity here is the quotient ωo.w.−c∗(G)/ωo.w.−c(G). We provide a universal way to obtain Bell inequality violations of general Bell functionals from XOR games for which the quotient ωo.w.−c∗(G)/ωo.w.−2c(G) is larger than 1. This allows, in particular, to find (unbounded) Bell inequality violations from communication complexity problems in the same spirit as the recent work by Buhrman et al. (2016). We also provide an example of a XOR game for which the previous quotient is optimal (up to a logarithmic factor) in terms of the amount of information c. Interestingly, this game has only polynomially many inputs per player. For the related problem of separating the classical vs quantum communication complexity of a function, the known examples attaining exponential separation require exponentially many inputs per party.
Quantum resources may provide an advantage over their classical counterparts in communication tasks. We refer to this as quantum advantage. Here, we consider a single communication task to study different approaches of observing quantum advantage. We call this setting a unified scenario. Our task is described as the following. There are three parties---the Manager, Alice, and Bob. The Manager sends a value of a random variable to Alice, and at the same time Bob receives some partial information regarding that value. Initially, neither Alice nor Bob knows the input of the other, which is received from the Manager. The goal of the task is achieved if and only if the value of the random variable, sent to Alice by the Manager, is identified by Bob with a success probability greater than half all the time. Here, nonzero error probability is allowed. However, to help Bob, Alice sends a limited amount of classical or quantum information to him (cbit or qubit). We show that the goal of the task can be achieved when Alice sends a qubit. On the other hand, a cbit communication is not sufficient for achieving the goal. Thus, it establishes quantum advantage. We further show that the optimal success probability in the overall process for a qubit communication can be higher than the optimal success probability for a cbit communication. In fact, this success probability is higher for qubit communication even if along with cbit communication Alice and Bob share randomness which allows mixing of classical strategies. Clearly, it not only establishes quantum advantage, it also demonstrates a more prominent nonclassical feature. Actually, we obtain higher success probability compared to other tasks. This also suggests the experiment-friendly nature of our task. For applications, we connect our task with semi-device-independence and show how our task can be useful to detect quantumness of a communication in a secure way. Moreover, within the same setting, we provide a way to detect universal coherence present in an ensemble. For a high-dimensional random variable, to achieve the goal, it may require a high-dimensional classical communication, while it can be achieved with only a qubit communication. This establishes a large separation between the quantum and the classical communication.
We study a new type of separations between quantum and classical communication complexity, separations that are obtained using quantum protocols where all parties are efficient , in the sense that they can be implemented by small quantum circuits, with oracle access to their inputs. Our main result qualitatively matches the strongest known separation between quantum and classical communication complexity Gavinsky (2016) and is obtained using a quantum protocol where all parties are efficient. More precisely, we give an explicit partial Boolean function f over inputs of length N , such that: f can be computed by a simultaneous-message quantum protocol with communication complexity polylog( N ) (where at the beginning of the protocol Alice and Bob also have polylog( N ) entangled EPR pairs). Any classical randomized protocol for f , with any number of rounds, has communication complexity at least Ω~(N1/4) . All parties in the quantum protocol of Item (1) (Alice, Bob and the referee) can be implemented by quantum circuits of size polylog( N ) (where Alice and Bob have oracle access to their inputs). Items (1), (2) qualitatively match the strongest known separation between quantum and classical communication complexity, proved by Gavinsky (2016). Item (3) is new. (Our result is incomparable to the one of Gavinsky. While he obtained a quantitatively better lower bound of Ω(N1/2) in the classical case, the referee in his quantum protocol is inefficient). Exponential separations of quantum and classical communication complexity have been studied in numerous previous works, but to the best of our knowledge the efficiency of the parties in the quantum protocol has not been addressed, and in most previous separations the quantum parties seem to be inefficient. The only separations that we know of that have efficient quantum parties are the recent separations that are based on lifting Göös et al. (2017), Chattopadhyay et al. (2019a). However, these separations seem to require quantum protocols with at least two rounds of communication, so they imply a separation of two-way quantum and classical communication complexity, but they do not give the stronger separations of simultaneous-message quantum communication complexity vs. two-way classical communication complexity (or even one-way quantum communication complexity vs. two-way classical communication complexity). Our proof technique is completely new, in the context of communication complexity, and is based on techniques from Raz & Tal (2019). Our function f is based on a lift of the forrelation problem, using xor as a gadget.