Roger Colbeck

A Fully Quantum Asymptotic Equipartition Property

The classical asymptotic equipartition property is the statement that, in the limit of a large number of identical repetitions of a random experiment, the output sequence is virtually certain to come from the typical set, each member of which is almost equally likely. In this paper, a fully quantum generalization of this property is shown, where both the output of the experiment and side information are quantum. An explicit bound on the convergence is given, which is independent of the dimensionality of the side information. This naturally leads to a family of REacutenyi-like quantum conditional entropies, for which the von Neumann entropy emerges as a special case.

Duality Between Smooth Min- and Max-Entropies

In classical and quantum information theory, operational quantities such as the amount of randomness that can be extracted from a given source or the amount of space needed to store given data are normally characterized by one of two entropy measures, called smooth min-entropy and smooth max-entropy, respectively. While both entropies are equal to the von Neumann entropy in certain special cases (e.g., asymptotically, for many independent repetitions of the given data), their values can differ arbitrarily in the general case. In this paper, a recently discovered duality relation between (nonsmooth) min- and max-entropies is extended to the smooth case. More precisely, it is shown that the smooth min-entropy of a system A conditioned on a system B equals the negative of the smooth max-entropy of A conditioned on a purifying system C. This result immediately implies that certain operational quantities (such as the amount of compression and the amount of randomness that can be extracted from given data) are related. We explain how such relations have applications in cryptographic security proofs.

Private randomness expansion with untrusted devices

Randomness is an important resource for many applications, from gambling to secure communication. However, guaranteeing that the output from a candidate random source could not have been predicted by an outside party is a challenging task, and many supposedly random sources used today provide no such guarantee. Quantum solutions to this problem exist, for example a device which internally sends a photon through a beamsplitter and observes on which side it emerges, but, presently, such solutions require the user to trust the internal workings of the device. Here, we seek to go beyond this limitation by asking whether randomness can be generated using untrusted devices—even ones created by an adversarial agent—while providing a guarantee that no outside party (including the agent) can predict it. Since this is easily seen to be impossible unless the user has an initially private random string, the task we investigate here is private randomness expansion. We introduce a protocol for private randomness expansion with untrusted devices which is designed to take as input an initially private random string and produce as output a longer private random string. We point out that private randomness expansion protocols are generally vulnerable to attacks that can render the initial string partially insecure, even though that string is used only inside a secure laboratory; our protocol is designed to remove this previously unconsidered vulnerability by privacy amplification. We also discuss extensions of our protocol designed to generate an arbitrarily long random string from a finite initially private random string. The security of these protocols against the most general attacks is left as an open question.

Is a System's Wave Function in One-to-One Correspondence with Its Elements of Reality?

Although quantum mechanics is one of our most successful physical theories, there has been a long-standing debate about the interpretation of the wave function--the central object of the theory. Two prominent views are that (i) it corresponds to an element of reality, i.e., an objective attribute that exists before measurement, and (ii) it is a subjective state of knowledge about some underlying reality. A recent result [M. F. Pusey, J. Barrett, and T. Rudolph, arXiv:1111.3328] has placed the subjective interpretation into doubt, showing that it would contradict certain physically plausible assumptions, in particular, that multiple systems can be prepared such that their elements of reality are uncorrelated. Here we show, based only on the assumption that measurement settings can be chosen freely, that a systems wave function is in one-to-one correspondence with its elements of reality. This also eliminates the possibility that it can be interpreted subjectively.

Free randomness can be amplified

Bell’s equations enable scientists to test the fundamental implications of quantum physics. A central tenet of this idea is that the choice of measurement is truly random. Researchers now show that some Bell experiments can even increase randomness in cases where choice is not entirely free. The concept could increase the usefulness of weakly random sources for more thorough tests of quantum mechanics.

no extension of quantum theory can have improved predictive power

According to quantum theory, measurements generate random outcomes, in stark contrast with classical mechanics. This raises the question of whether there could exist an extension of the theory that removes this indeterminism, as suspected by Einstein, Podolsky and Rosen. Although this has been shown to be impossible, existing results do not imply that the current theory is maximally informative. Here we ask the more general question of whether any improved predictions can be achieved by any extension of quantum theory. Under the assumption that measurements can be chosen freely, we answer this question in the negative: no extension of quantum theory can give more information about the outcomes of future measurements than quantum theory itself. Our result has significance for the foundations of quantum mechanics, as well as applications to tasks that exploit the inherent randomness in quantum theory, such as quantum cryptography.

Experimental investigation of the uncertainty principle in the presence of quantum memory and its application to witnessing entanglement

The uncertainty principle tells us that two associated properties of a particle cannot be simultaneously known with infinite precision. However, if the particle is entangled with a quantum memory, the uncertainty of a measurement is reduced. This concept is now observed experimentally.

Uncertainty Relations from Simple Entropic Properties

Uncertainty relations provide constraints on how well the outcomes of incompatible measurements can be predicted, and as well as being fundamental to our understanding of quantum theory, they have practical applications such as for cryptography and witnessing entanglement. Here we shed new light on the entropic form of these relations, showing that they follow from a few simple properties, including the data-processing inequality. We prove these relations without relying on the exact expression for the entropy, and hence show that a single technique applies to several entropic quantities, including the von Neumann entropy, min- and max-entropies, and the Rényi entropies.

Memory attacks on device-independent quantum cryptography.

Device-independent quantum cryptographic schemes aim to guarantee security to users based only on the output statistics of any components used, and without the need to verify their internal functionality. Since this would protect users against untrustworthy or incompetent manufacturers, sabotage, or device degradation, this idea has excited much interest, and many device-independent schemes have been proposed. Here we identify a critical weakness of device-independent protocols that rely on public communication between secure laboratories. Untrusted devices may record their inputs and outputs and reveal information about them via publicly discussed outputs during later runs. Reusing devices thus compromises the security of a protocol and risks leaking secret data. Possible defenses include securely destroying or isolating used devices. However, these are costly and often impractical. We propose other more practical partial defenses as well as a new protocol structure for device-independent quantum key distribution that aims to achieve composable security in the case of two parties using a small number of devices to repeatedly share keys with each other (and no other party).

Simple channel coding bounds

New channel coding converse and achievability bounds are derived for a single use of an arbitrary channel. Both bounds are expressed using a quantity called the “smooth 0-divergence”, which is a generalization of Rényis divergence of order 0. The bounds are also studied in the limit of large block-lengths. In particular, they combine to give a general capacity formula which is equivalent to the one derived by Verdú and Han.