Picturing qubits in phase space

IBM Journal of Research and Development, Jan 2004 by Wootters, W K

Focusing particularly on one-qubit and two-qubit systems, I explain how the quantum state of a system of n qubits can be expressed as a real function-a generalized Wigner function-on a discrete 2^sup n^ × 2^sup n^ phase space. The phase space is based on the finite field having 2^sup n^ elements, and its geometric structure leads naturally to the construction of a complete set of 2^sup n^ 1 mutually conjugate bases.

1. Introduction

On Charlie Bennett's main webpage [1], one finds two photographs: one of Charlie himself and the other of a vortex created by a beaver dam. A vortex is a wonderful example of a structure that maintains its form by not holding on to its substance; it thrives because it continually gives its material away.

Some summers ago I was supervising four undergraduates in research projects in quantum information theory, and together we drove down to the IBM Watson Research Center for a day to talk with Charlie. He took us to the Croton Dam, one of his favorite places. As we sat there on the dam with the sound of water in the background, we discussed quantum information and wrote down quantum states on a large pad of newsprint that Charlie had brought along. The breeze was blowing a bit, and as always around Charlie, ideas were swirling. We talked for hours. In the years that have passed since that afternoon, traces of that experience and traces of those ideas have surely been carried far out into the world-who knows how far-in the lives of those four students and the people they have encountered. I offer this little story as one example of hundreds of similar acts of sharing, through which Charlie Bennett has had an influence on the world of science that could never be captured by any reckoning based on cited publications. It is a pleasure to dedicate this paper to him on the occasion of his sixtieth birthday.

A spin- ½ particle is probably not a vortex, though there may be some virtue in thinking of it more as a process than a static object. But here I will not be adventurous in that way. This paper is about qubits as normally conceived, and I will use the spin of a spin- ½ particle as my standard physical example of a qubit. We usually express the quantum state of a system of qubits as a state vector or density matrix. The main point of this paper is to show how one can represent such a quantum state as a real function on a phase space, not a continuous phase space whose axes stand for position and momentum, but a discrete phase space whose axes are associated with a pair of conjugate bases for the finite-dimensional state space. Much of the work I report here was done jointly with Kathleen Gibbons, and many of the mathematical details will be given in a paper which is currently in preparation.1 Here I want to lay out the overall contours of this phase-space construction.

Discrete phase-space representations have been proposed in a number of earlier papers [2-13]. The particular representation to be described here is different in ways that I discuss later. First, however, I would like to motivate the work by posing what might seem to be an unrelated problem, the problem of determining an unknown quantum state.

2. State determination

Imagine a device whose output is a beam of spin- ½ particles. We do not know enough about the device to predict the spin state of the particles that it produces. They might all be in the spin-up state, for example, or they might be completely unpolarized. We do, however, assume that the device does not change its operation significantly over time, so that we ought to be able to describe the whole ensemble of particles by a single state (possibly a mixed state) of a single spin- ½ particle. A general spin state of a spin- ½ particle can be pictured as a point either on the surface or in the interior of a unit sphere. Points on the surface of the sphere are pure states, points in the interior are mixed states, and the center of the sphere is the completely mixed or completely unpolarized state. Our task is to perform a set of measurements on the particles so as to determine which point represents the spin state actually produced by this mysterious device.

How do we proceed? Suppose we perform the measurement "up vs. down" on the first hundred particles that come our way. What do we get from these measurements? We get a rough estimate of the vertical height of the point that represents the state of the ensemble, because the height is what determines the probabilities of "up" and "down." However, we do not get a perfectly precise value of the height, because we have performed only a finite number of trials and therefore still have some statistical error. We know that we will have to live with some statistical error, so we now turn our attention to pinning the state down better along the horizontal dimensions. In order to do this, we perform measurements of spin along two other axes. (Of course, we have to use new particles for these measurements. The ones we have already measured hold no further information for us.) If we call the vertical direction z, we might let our two new measurements be measurements of spin along the x and y directions. In this way we can narrow the range of likely states to a small region, typically in the interior of the sphere.