POVM - An Example: Unambiguous Quantum State Discrimination

An Example: Unambiguous Quantum State Discrimination

The task of unambiguous quantum state discrimination (UQSD) is to discern conclusively which state, of given set of pure states, a quantum system (which we call the input) is in. The impossibility of perfectly discriminating between a set of non-orthogonal states is the basis for quantum information protocols such as quantum cryptography, quantum coin-flipping, and quantum money. This example will show that a POVM has a higher success probability for performing UQSD than any possible projective measurement.

First let us consider a trivial case. Take a set that consists of two orthogonal states and . A projective measurement of the form,

will result in eigenvalue a only when the system is in and eigenvalue b only when the system is in . In addition, the measurement always discriminates between the two states (i.e. with 100% probability). This latter ability is unnecessary for UQSD and, in fact, is impossible for anything but orthogonal states. Now consider a set that consists of two states and in two-dimensional Hilbert space that are not orthogonal. i.e.,

for . These could be states of a system such as the spin of spin-1/2 particle (e.g. an electron), or the polarization of a photon. Assuming that the system has an equal likelihood of being in each of these two states, the best strategy for UQSD using only projective measurement is to perform each of the following measurements,

50% of the time. If is measured and results in an eigenvalue of 1, then it is certain that the state must have been in . However, an eigenvalue of zero is now an inconclusive result since this can come about from the system could being in either of the two states in the set. Similarly, a result of 1 for indicates conclusively that the system is in and 0 is inconclusive. The probability that this strategy returns a conclusive result is,

In contrast, a strategy based on POVMs has a greater probability of success given by,

This is the minimum allowed by the rules of quantum indeterminacy and the uncertainty principle. This strategy is based on a POVM consisting of,

where the result associated with indicates the system is in state i with certainty.

These POVMs can be created by extending the two-dimensional Hilbert space. This can be visualized as follows: The two states fall in the x-y plane with an angle of θ between them and the space is extended in the z-direction. (The total space is the direct sum of spaces defined by the z-direction and the x-y plane.) The measurement first unitarily rotates the states towards the z-axis so that has no component along the y-direction and has no component along the x-direction. At this point, the three elements of the POVM correspond to projective measurements along x-direction, y-direction and z-direction, respectively.

For a specific example, take a stream of photons, each of which are polarized along either the horizontal direction or at 45 degrees. On average there are equal numbers of horizontal and 45 degree photons. The projective strategy corresponds to passing the photons through a polarizer in either the vertical direction or -45 degree direction. If the photon passes through the vertical polarizer it must have been at 45 degrees and vice versa. The success probability is . The POVM strategy for this example is more complicated and requires another optical mode (known as an ancilla). It has a success probability of .