{"id":2694,"date":"2018-08-22T17:53:03","date_gmt":"2018-08-22T16:53:03","guid":{"rendered":"https:\/\/www.quantum-bits.org\/?p=2694"},"modified":"2022-08-12T17:14:22","modified_gmt":"2022-08-12T16:14:22","slug":"on-superdense-coding","status":"publish","type":"post","link":"https:\/\/www.quantum-bits.org\/?p=2694","title":{"rendered":"On superdense coding"},"content":{"rendered":"

Alice and Bob are fictional characters often used to describe cryptographic protocols. We introduced them in a previous article on quantum cryptography.<\/a> Picture a situation where Alice and Bob are living in different parts of the world and where Alice would like to communicate two classical bits<\/em> of information to Bob by sending him only a single quantum bit<\/em>.<\/p>\n

As usual, simply formulated problems are more complicated than they sound. Superdense coding<\/strong> is a solution to this problem. This is a quantum communication protocol used to transmit two classical bits from Alice to Bob by sending only one qubit, at the condition that Alice and Bob have pre-shared an entangled Bell pair<\/a><\/em>.<\/p>\n

It can be thought of as the opposite of quantum teleportation<\/a>, in which one transfers one qubit from Alice to Bob by communicating two classical bits, as long as Alice and Bob have a pre-shared Bell pair.<\/p>\n

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The HSW theorem and Holevo’s bound
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As we guessed earlier, though seemingly simple, the enounced problem turns out to more subtle. The HSW theorem<\/a> (named after Holevo, Schumacher and Westmoreland) is an important limitative theorem in quantum computing, published in the early 70’s. It is often called the Holevo’s bound<\/b>, since it establishes an upper bound to the amount of information that can be known about a quantum state<\/em>.<\/p>\n

As in many other fields, in quantum information theory, accessible information is best understood in terms of a 2-party Alice-Bob communication.<\/p>\n

Let’s imagine that Alice holds a classical random variable<\/a> X, which can take the values \\{1, 2, \\cdots, n\\} with corresponding probabilities \\{p_1, p_2, \\cdots , p_n\\}. Alice then prepares a quantum state, represented by the density matrix<\/a> \\rho_X chosen from a set \\{\\rho_1, \\rho_2, \\cdots, \\rho_n\\}, and gives this state to Bob.<\/p>\n

Bob’s goal is to find the value of X. In order to do that, Bob performs a measurement on the state \\rho_X, obtaining a classical outcome, which we denote with Y.<\/p>\n

In this context, the amount of accessible information, that is, the amount of information that Bob can get about the variable X, is the maximum value of the mutual information<\/a> I(X : Y) between the random variables X and Y over all the possible measurements that Bob can do:<\/p>\n

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The mutual information<\/b> of two random variables X and Y is a measure of the mutual dependence between the two variables. It quantifies the amount of information (bits) obtained about one random variable, through the other random variable. The concept of mutual information is intricately linked to that of entropy<\/a> of a random variable.<\/p>\n

There is currently no known formula to compute the accessible information. There are however several upper bounds, the best-known of which is the Holevo<\/a> bound, which is specified in the HSW theorem:<\/p>\n