Measuring non-local parameters of a two-qubit unitary

Daniel started his talk by quoting a paper by Nielsen in which he rallied the quantum information community to better understand and quantify different useful properties of quantum dynamics.

In that vein, one can ask: what can you do with a given unitary evolution? Some answers include: create and destroy entanglement or quantum discord, communication, simulate other operations. Quantifying how a given unitary help in each of these tasks involves the definition of nonlinear functions of the operators, which are typically invariant under local basis rotations.

Zanardi (2000) proposed an interesting quantity with these properties, which provided the initial inspiration for Daniel’s work. That is the entangling power, measuring the average entanglement created from initially separable qubits. The average, here, is done uniformly over all two-qubit separable states. Interestingly, Zanardi also provides a simple circuit that evaluates the entangling power using two black boxes that implement U. The average can then be done by picking input states from a 2-design.

Daniel described how two-qubit unitaries only have three non-local parameters, i.e. which are independent of which local single-qubit unitaries are performed before or after it. Zanardi’s entangling power is one particular combination of those parameters.

Then he described a new, well-motivated function of the non-local parameters, which he called the purity-swapping power P.
Roughly speaking, the two-qubit unitary U is applied to the first qubit in a initial state, and a second qubit belonging to an EPR pair, and P is a measure of the average final purity of the second qubit, which was initially in the EPR state. As with the entangling power, this new function also has a simple circuit that calculates it, using two U black boxes.

Daniel showed some nice picture of the trade-offs between these two non-local functions of two-qubit unitaries, and different gates which extremize different combinations of those. The third such non-local parameter, a phase, is apparently more complicated to deal with, and requires move involved circuits for its evaluation.

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