Noiseless Circuits for measuring concurrence monotones - Hilary Carteret

There has recently been a lot of interest in techniques for measuring the entanglement in an unknown density matrix, without first performing full state tomography. Entanglement measures are often defined in terms of unphysical maps, such as the Wootters 2-concurrence. This is defined in terms of the spectrum of $\rho\tilde{\rho},$ where the tilde operation is an anti-unitary map called the spin-flip. Previous proposed methods for measuring these quantities relied on full state tomography (which is very inefficient) or the Structural Physical Approximation (SPA) which suffers from large, state dependent errors. Since we know nothing about the density matrix by assumption, there is no way to correct for these. I will show how to construct a family of circuits that can measure the Wootters 2-concurrence for any 2-qubit state. These circuits do not require the deliberate addition of noise (unlike the SPA) and so will be exact up to experimental errors. If we get time, I can show how to generalise these sets of circuits to measure a multi-partite monotone defined on states on even numbers of qubits called the n-concurrence. These generalized circuits are also exact, and depend only on the dimension of the density matrix. If we are additionally given that the state is very nearly pure, its n-concurrence can be determined by a single circuit that scales linearly in the number of qubits, and requires only one-qubit gates and destructive c-SWAP gates. This talk is based on quant-ph/0309212 by the speaker, and some further work in progress in collaboration with Stephen S. Bullock. It is a sequel to the talk I gave last October on circuits for the partial-transpose spectrum, based on quant-ph/0309216, PRL vol.94, 040502 (2005).