Quantum Tomography - Quantum Tomography of Pre-measurement States

Quantum Tomography of Pre-measurement States

The main tool of the retrodictive approach of quantum physics is the pre-measurement state which allows predictions about state preparations of the measured system leading to a given measurement result. As it was shown in a recent work, such a state reveals interesting quantum properties of the corresponding measurement such as its non-classicality or its projectivity. However, we cannot realize the tomography of this state with the usual methods based on measurements, since it needs non-destructive measurements which are some particularly measurements. The experimental procedure, proposed in, is based on the retrodictive approach of quantum physics, in which we have an expression of retrodictive probabilities similar to Born's rule:

\mathrm{Pr}\left(m \vert n\right) = \mathrm{Tr}\lbrace\hat{\rho}_{retr}^{}\hat{\Theta}_{m}\rbrace,

where and are respectively the pre-measurement state, corresponding to the measurement characterized by some a POVM element, and a hermitian and positive operator corresponding to the preparation of the measured system in a state . In the frame of the mathematical foundations of quantum physics, such a operator is a proposition about the state of the system, as a POVM element, and for having an exhaustive set of propositions, these operators must be a resolution of the Hilbert space:

\sum_{m}\,\hat{\Theta}_{m}=\hat{1}.

From Born's, we can derive with Bayes' theorem, the expressions of the pre-measurement state and proposition operators . The pre-measurement state simply corresponds to the normalized POVM element:

\hat{\rho}_{retr}^{}=\frac{\hat{\Pi}_{n}}{\mathrm{Tr}\lbrace\Pi_{n}\rbrace},

and the proposition operators are linked to the possible preparations of the system by:

\hat{\Theta}_{m}=D\mathcal{P}_{m}\hat{\rho}_{m},

where is the dimension of the Hilbert space and is the probability of preparing the state .

Thus, we can probe the measurement apparatus with a statistical mixture:

\hat{\rho}^{}=\sum_{m}\,\mathcal{P}_{m}\hat{\rho}_{m}=\hat{1}/D,

in order to measure the retrodictive probability . This mixture could be obtained by preparations based on random choices 'm' with the probabilities . Then, we replace the POVM elements describing the measurements in a usual method for the tomography of states by the operators . The method will give the state giving the probabilities which are the most closest to those measured. This is the pre-measurement state with which we can have some interesting properties of the measurement giving the result 'n', as explained in.

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