From Classical to Quantum Information – Or: When You Have Less Than No Uncertainty

by Serge Fehr (CWI)

Over the last few years, significant progress has been made in understanding the peculiar behaviour of quantum information. An important step in this direction was taken with the discovery of the quantum Rényi entropy. This understanding will be vital in a possible future quantum information society, where quantum techniques are used to store, communicate, process and protect information.

Information theory is the area of computer science that develops and studies abstract mathematical measures of “information” – or, from a more pessimistic perspective, measures of “uncertainty”, which capture the lack of information. It is clear that when you toss a coin there will be uncertainty in the outcome: it can be either “head” or “tail”, and you have no clue what it will be. Similarly, there is uncertainty in the face that will show up when you throw a dice. It is even intuitively clear that there is more uncertainty in the latter than in the coin toss. However, such comparisons become less clear for more complicated cases. If we want to compare, say, tossing three coins on the one hand with throwing two dice and taking the sum of the two faces on the other hand, it is not immediately clear in which of the two there is more uncertainty: there are fewer possible outcomes in the former, namely eight, but, on the other hand, the eleven possible outcomes in the latter are biased.

Information theory offers quantitative measures that express precisely how much uncertainty there is. Similarly, it also offers measures of conditional uncertainty given that one holds some “side information”. For instance, how much uncertainty is there in the faces of the two dice given that I know the sum of the two? How much uncertainty is there in a message that was communicated over a noisy channel given that I hold the received noisy version? How much uncertainty is there in a digital photo given that I hold a compressed version? How much uncertainty does an eavesdropper have on secret data given that he got to see an encryption? Information theory allows us to answer such questions in a precise manner and to make rigorous predictions about the behaviour of information in all kinds of information processing tasks. As such, information theory had – and still has – a major impact on the development of today’s information and communication infrastructure.

What makes information theory very powerful is its independence of how information is physically represented: whether the information is represented by coins that show “head” or “tail”, or by the tiny indentations on a DVD, or whether the information is stored on a flash drive or communicated over WiFi, the predictions of information theory hold universally – well, until we hit the realm of quantum mechanics. If, say, we encode information into the polarisation of photos, then information starts to behave very differently. Therefore, a quantum version of information theory is necessary in order to rigorously study the behaviour of information in the quantum realm, and, for instance, to be able to quantify the amount of information an eavesdropper may have on secret data when the data is protected by means of quantum cryptography, or to quantify the amount of error correction needed in order to counter the loss of information in quantum computation caused by decoherence.

From a mathematical perspective, given that quantum mechanics is described by non-commuting mathematical objects, quantum information theory can be understood as a non-commutative extension of its classical commutative counterpart. This insight can serve as a guideline for coming up with quantum versions of classical information measures, but it also shows a typical predicament: a commutative expression can be generalised in various ways into a non-commutative one. For instance, an expression like A⁵B⁴ can be generalised to A⁵B⁴ or to B⁴A⁵ for non-commuting A and B, or to ABABABABA, or to B²A⁵B², etc. One of the challenging questions is to understand which of the possible generalisations of classical information measures are suitable measures of quantum information and have operational significance.

Building upon new insights and new results [1] that we discovered on the classical notion of Rényi entropy, and in collaboration with several partners, we succeeded in lifting the entire family of Rényi entropies to the quantum setting [2,3]. The Rényi entropies form a continuous spectrum of information measures and cover many important special cases; as such, our extension to the quantum setting offers a whole range of new quantum information measures. We showed that our newly proposed definition satisfies various mathematical properties that one would expect from a good notion of information and which make it convenient to work with the definition. It is due to these that our quantum Rényi entropies have quickly turned into an indispensable tool for studying the behaviour of quantum information in various contexts.

One very odd aspect of quantum information is that uncertainty may become negative: it may be that you have less than no uncertainty in your target of interest. This peculiarity is an artifact of entanglement, which is one of the most bizarre features of quantum mechanics. Entanglement is a form of correlation between quantum information that has no classical counterpart. A sensible explanation of negative uncertainty can be given as follows. By Heisenberg’s uncertainty principle, even when given full information in the form of a perfect description of the quantum state of interest, there is still uncertainty in how the state behaves under different measurements. However, if you are given another quantum state that is entangled with the state of interest, then you can actually predict the behaviour of the state of interest under any measurement by means of performing the same measurement on your state. Indeed, by what Einstein referred to as “spooky action at a distance”, the measurement on your entangled state will instantaneously affect the other state as to produce the same measurement outcome.

The above aspect nicely illustrates that quantum information theory is much more than a means for understanding the behaviour of information within possible future quantum communication and computation devices: it sheds light on the very foundations of quantum mechanics itself.

References:
[1] S. Fehr, S. Berens: “On the Conditional Rényi Entropy”, in IEEE Trans. Inf. Theory 60(11):6801 (2014).
[2] M. Müller-Lennert, F. Dupuis, O. Szehr, S. Fehr, M. Tomamichel: “On Quantum Rényi Entropies: A New Generalization and Some Properties”, in J. Math. Phys 54:122203 (2013).
[3] M. Wilde, A. Winter, D. Yang: “Strong Converse for the Classical Capacity of Entanglement-Breaking and Hadamard Channels via a Sandwiched Rényi Relative Entropy”, in Commun. Math. Phys 331:593 (2014).

Serge Fehr, CWI, The Netherlands
+31 20 592 42 57, This email address is being protected from spambots. You need JavaScript enabled to view it.

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