The problem of liquid-state NMR entanglement

To address the problem of entanglement at room temperature NMR liquid-state experiments, let us start from our generic density matrix  

Remember that this form is motivated from a high temperature approximation for the NMR equilibrium density matrix, for which e = hcoL/2 kBT. But, whatever the situation, one must have Tr(l) = 2 and Tr(/0i) = 1. Consequently, Tr(/Oe) = 1, as it must be for density matrices. The matrix p can represent an equilibrium mixed state, or a pseudopure state. In particular, it can represent an entangled state. For instance, for two-qubits it could be the cat-state  

In this case, we will refer to pe as a pseudo-cat state. Generally, if p represents an entangled state, we say that Pe is pseudo-entangled. Now, if pi is a cat-state, the question is whether P( is entangled or not. We have to keep in mind that the density matrix of the whole spin system is Pe, and not p, but remember that NMR signals are proportional to pi, and not Pe. 

Suppose that in a NMR experiment we produce an initial pseudopure state, and apply the quantum circuit that generates a cat state (see Chapter 3). Suppose also that we perform quantum state tomography on this state. We will find a matrix which will be similar to Equation (6.1.2), upon which one has to add the background to build the complete matrix  

This way of writing p suggests a straightforward interpretation a maximally mixed state added to a fraction e of entangled state. However, a fraction x of entanglement can be extracted from the maximally mixed state itself  

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