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Entropy

Communicating through Probabilities: Does Quantum Theory Optimize the Transfer of Information?

William K. Wootters

Published Aug 2, 2013• 11 citations
Publisher

Abstract

A quantum measurement can be regarded as a communication channel, in which the parameters of the state are expressed only in the probabilities of the outcomes of the measurement. We begin this paper by considering, in a non-quantum-mechanical setting, the problem of communicating through probabilities. For example, a sender, Alice, wants to convey to a receiver, Bob, the value of a continuous variable, θ, but her only means of conveying this value is by sending Bob a coin in which the value of θ is encoded in the probability of heads. We ask what the optimal encoding is when Bob will be allowed to flip the coin only a finite number of times. As the number of tosses goes to infinity, we find that the optimal encoding is the same as what nature would do if we lived in a world governed by real-vector-space quantum theory. We then ask whether the problem might be modified, so that the optimal communication strategy would be consistent with standard, complex-vector-space quantum theory.

DOI
10.3390/e15083220

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A quantum measurement can be regarded as a communication channel, in which the parameters of the state are expressed only in the probabilities of the outcomes of the measurement. We begin this paper by considering, in a non-quantum-mechanical setting, the problem of communicating through probabilities. For example, a sender, Alice, wants to convey to a receiver, Bob, the value of a continuous variable, θ, but her only means of conveying this value is by sending Bob a coin in which the value of θ is encoded in the probability of heads. We ask what the optimal encoding is when Bob will be allowed to flip the coin only a finite number of times. As the number of tosses goes to infinity, we find that the optimal encoding is the same as what nature would do if we lived in a world governed by real-vector-space quantum theory. We then ask whether the problem might be modified, so that the optimal communication strategy would be consistent with standard, complex-vector-space quantum theory.
In the simplest problems of quantum communication, Alice transmits one of two quantum states, with equal probabilities, to Bob's receiver, modeled by a positive-operator-valued measure (POVM); one seeks the POVM that is optimal according to one or another criterion. We discuss four such criteria, the first three of which lead to distinctive types of POVM. By introducing a reciprocal basis for the state vectors, we shorten the derivations of known results for the two most popular criteria. A new optimization problem defined by a third criterion, intermediate between the first two, is formulated and solved. Then we turn to a fourth criterion, that of minimizing Bob's Renyi entropy for an arbitrary order α. Depending on the value of α and the separation of Alice's states, the POVM that minimizes Bob's entropy can be any of the preceding three types.
We present a scheme for quantum communication, where a set of EPR pairs, initially shared by the sender Alice and the receiver Bob, functions as a quantum channel. After insuring the safety of the quantum channel, Alice applies local measurement on her particles of the EPR pairs and informs Bob the encoding classical information publicly. According to Alice's classical information and his measurement outcomes on the EPR pairs Bob can infer the secret messages directly. In this scheme, to transmit one bit secret message, the sender Alice only needs to send one bit classical information to the receiver Bob. We also show that this scheme is completely secure if perfect quantum channel is used.
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.
Quantum resources may provide an advantage over their classical counterparts. Theoretically, in certain tasks, this advantage can be very high. In this work, we construct such a task based on a game, mediated by the Referee and played between Alice and Bob. The Referee sends Alice a value of a random variable. At the same time, the Referee also sends Bob some partial information regarding that value. Here partial information can be defined in the following way. Bob gets the information of a random set that must contain the value of the variable, which is sent to Alice by the Referee, along with other value(s). Alice is not allowed to know what information is sent to Bob by the Referee. Again, Bob does not know which value of the random variable is sent to Alice. Now, the game can be won if and only if Bob can unambiguously identify the value of the variable that is sent to Alice, with some nonzero probability, no matter what information Bob receives or which value is sent to Alice. However, to help Bob, Alice sends some limited amount of information to him, based on any strategy that is fixed by Alice and Bob before the game begins. We show that if Alice sends a limited amount of classical information, then the game cannot be won, while the quantum analog of the ``limited amount of classical information'' is sufficient for winning the game. Thus, it establishes a quantum advantage. We further analyze several variants of the game and provide certain bounds on the success probabilities. Moreover, we establish connections between the trine ensemble, mutually unbiased bases, and the encoding-decoding strategies of those variants. We also discuss the role of quantum coherence in the present context.