Marco Piani course @ UFF – Lecture 2

Lecture 2 – Tue 27 Aug

Today’s lecture started with the introduction of a number of technical tools. First, Marco reviewed the general formalism of Positive Operator-Valued Maps (POVMs), the most general description for the evolution of a quantum system interacting with systems it has not previously been correlated with.

He also noted that these evolutions are in fact Completely Positive (CP) maps, meaning that they are still positive even when acting only on a part of a larger system. This apparently innocuous observation has in fact important consequences for the detection of entanglement, as explained below.

He then demonstrated a very handy identity showing that, if a bipartite system AB is in a certain special state, the action of any operator C on subsystem A is equivalent to that of its transpose C^T acting on subsystem B. Using this identity and the Singular Value decomposition of a matrix, Marco demonstrated that any pure bipartite state of AB may be written in the so-called Schmidt form. In particular, this form makes it easy to see that the reduced density operators of either subsystem have identical spectra.

Based on these considerations, Marco introduced the notion of a measure of (pure-state) entanglement as a quantity measuring the amount of ‘spreadness’, or disorder, of this spectrum. An example is the Shannon entropy, which can in this case also be expressed as the Von Neumann entropy S(\rho)= - \text{Tr} \rho_A \ln \rho_A.

Moving to the mixed-state case, Marco noted first that the question of measuring or even detecting entanglement is much more complicated, since local entropy can also be due in part to the global mixedness of the state. Nevertheless, it is still true that a state that is more mixed locally than globally must be entangled.

Looking again at the set of separable (non-entangled) from a geometric point of view, Marco noted that it is convex. Incidentally, by using this fact along with a well-known theorem from convex analysis due to Carathéodory, he showed that it is possible to upper-bound the number of product states that are necessary to express any separable state in an explicitly separable form.

Convexity has another important consequence in this case: Marco used it to show that for any entangled mixed state \rho_{ent} there is always an Entanglement Witness: a Hermitian operator W whose expectation value is \geq 0 for any separable state, but < 0 for \rho_{ent}. Measuring such observables is thus a powerful technique for detecting mixed-state entanglement, even though no witness can detect the entanglement of an arbitrary state.

Finally, Marco showed another way to detect entanglement, that comes from noticing that a separable density operator is always mapped onto another positive operator if a Positive map \Lambda acts onto one of its subsystems. This is true even if \Lambda is Positive but not CP (PnCP). Thus, any density operator that, when acted on by a PnCP map, is mapped into another operator that is not positive, must have been entangled. An example of such a PnCP map is the Transposition map.

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