Quantum Information

There are many results showing that the probability of entanglement is high for large dimensions. Recently, Arveson showed that the probability of entanglement is zero when the rank of a bipartite state is no larger than half the dimension of the smaller space. We show that that the probability of entanglement is zero when the rank of a bipartite state is no larger than half the maximum of the rank of its two reduced density matrices. Our approach is quite different from that of Arveson and uses a different measure. But both approaches show that the separable states lie in a lower dimensional manifold given a reasonable parameterization of the separable states. This is joint work with Elisabeth Werner, using on characterizations of the extreme points of qubit channels given by Ruskai, Szarek and Werner and the extreme points of entanglement breaking channels given by M. Horodecki, Shor and Ruskai.

I will report on efforts to implement a new method for simulating concatenated quantum error correction, where many levels of concatenation are simulated together explicitly. That is, the approach involves a Monte Carlo simulation of a noisy circuit involving many thousands of qubits, rather than tens of qubits previously. The new approach allows the threshold and resource usage of concatenated quantum error correction to be determined more accurately than before. Also, the approach makes it possible to better study the effects of circuit optimizations and message-passing algorithms [Poulin, PRA, 2006] on the performance of fault-tolerant concatenated quantum error correction. Such studies are necessary in order make a proper comparison with competing families of error-correction protocols, such as those involving surface codes. In the talk, a range of new numerical results will be presented.

One simple way to think about physics is in terms of information. We gain information about physical systems by observing them, and with luck this data allows us to predict what they will do next. Quantum mechanics doesn't just change the rules about how physical objects behave - it changes the rules about how information behaves. In this talk we explore what quantum information is, and how strangely it differs from our intuitions. In particular we see how information about quantum particles can become entangled, leading to seemingly impossibly coordinated behaviour for separate objects, and to phenomena such as quantum teleportation.

One of the quintessential features of quantum information is its exclusivity, the inability of strong quantum correlations to be shared by many physical systems. Likewise, complementarity has a similar status in quantum mechanics as the sine qua non of quantum phenomena. We show that this is no coincidence, and that the central role of exclusivity in quantum information theory stems from the phenomenon of complementarity. We adopt an information-theoretic approach to complementarity, which leads to a new and simple definition of private states and new proofs of the achievable asymptotic rates of both secret key and entanglement distillation. From the latter follows a new proof of the direct part of the quantum noisy channel coding theorem.

We discuss two methods to encode one qubit into six physical qubits. Each of our two examples corrects an arbitrary single-qubit error. Our first example is a degenerate six-qubit quantum error-correcting code. We explicitly provide the stabilizer generators, encoding circuits, codewords, logical Pauli operators, and logical CNOT operator for this code. We also show how to convert this code into a non-trivial subsystem code that saturates the subsystem Singleton bound. We then prove that a six-qubit code without entanglement assistance cannot simultaneously possess a Calderbank-Shor-Steane (CSS) stabilizer and correct an arbitrary single-qubit error. A corollary of this result is that the Steane seven-qubit code is the smallest single-error correcting CSS code. Our second example is the construction of a non-degenerate six-qubit CSS entanglement-assisted code. This code uses one bit of entanglement (an ebit) shared between the sender and the receiver and corrects an arbitrary single-qubit error. The code we obtain is globally equivalent to the Steane seven-qubit code and thus corrects an arbitrary error on the receiver\'s half of the ebit as well. We prove that this code is the smallest code with a CSS structure that uses only one ebit and corrects an arbitrary single-qubit error on the sender\'s side. We discuss the advantages and disadvantages for each of the two codes.

The world as experienced by single atoms is radically different from the everyday world we, as gigantic humans, are used to: the laws of quantum mechanics replace the laws of classical physics. Because the quantum realm is alien from our daily experience, many common quantum effects are surprising and unintuitive. To understand the quantum world, it may be better to just start from scratch, as a child might, and develop intuition about the behavior of quantum objects by simply looking at a series of quantum examples rather than trying to analogize with classical physics. I will present a short introduction to quantum mechanics from this point of view, and make ties to current research in the field of quantum information.