Noiseless Circuits for measuring entanglement - Hilary Carteret

There has recently been a lot of interest in techniques for measuring the non-local properties of a density matrix as efficiently as possible. These functions are often defined in terms of unphysical maps, such as the partial transpose. Previous proposed methods for measuring these quantities relied on full state tomography (very inefficient) or the Structural Physical Approximation, which adds large amounts of noise to shift the spectrum of the partially transposed density matrix to be positive, thus incurring a corresponding loss of sensitivity. The moments of the resulting modified density operator are measured using certain sets of Mach-Zehnder interferometers and the spectrum can then be determined using a little algebra. I will show how to construct a family of simple circuits that can determine the spectrum of the partial transpose of a density matrix, without the addition of noise. These circuits depend only on the dimension of the density matrix and do not need any components that are not already required to determine the eigenspectrum of the original density matrix by interferometry. They measure the minimum amount of information required to determine the PT-spectrum completely and they will be exact up to experimental errors. If we get time, I can show how to measure the concurrence with a set of noiseless circuits. The concurrence also requires an unphysical operation in order to evaluate it. We must find the spectrum of $\\\\rho\\\\tilde{\\\\rho},$ where the tilde operation is an anti-unitary map. I have found a set of generalised interferometer circuits that can measure the concurrence spectrum for any two-qubit state. This talk is based on the following pre-prints: Partial transpose circuits -- quant-ph/0309216, Concurrence circuits -- quant-ph/0309212