Quantum tomography
Whereas, randomized benchmarking scalably obtains a figure of merit of the overlap between the error prone physical quantum process and its ideal counterpart.
The general principle behind quantum state tomography is that by repeatedly performing many different measurements on quantum systems described by identical density matrices, frequency counts can be used to infer probabilities, and these probabilities are combined with Born's rule to determine a density matrix which fits the best with the observations.
The position and momentum of the oscillator at any given point can be measured and therefore the motion can be completely described by the phase space.
By performing this measurement for a large number of identical oscillators we get a probability distribution in the phase space (figure 2).
The only difference is that the Heisenberg's uncertainty principle mustn't be violated, meaning that we cannot measure the particle's momentum and position at the same time.
The particle's momentum and its position are called quadratures (see Optical phase space for more information) in quantum related states.
[4][5] One can imagine a situation in which a person Bob prepares many identical objects (particles or fields) in the same quantum states and then gives them to Alice to measure.
To fully reconstruct the density matrix for a mixed state in a finite-dimensional Hilbert space, the following technique may be used.
yields the probabilities: Linear inversion corresponds to inverting this system using the observed relative frequencies
In infinite dimensional Hilbert spaces, e.g. in measurements of continuous variables such as position, the methodology is somewhat more complex.
Using balanced homodyne measurements, one can derive the Wigner function and a density matrix for the state of the light.
[9][10] Electromagnetic field amplitudes (quadratures) can be measured with high efficiency using photodetectors together with temporal mode selectivity.
Balanced homodyne tomography is a reliable technique of reconstructing quantum states in the optical domain.
The other photon is directed into the homodyne tomography detector, in order to reconstruct its quantum state.
The signal photon (this is the quantum state we want to reconstruct) interferes with the local oscillator, when they are directed onto a 50-50% beamsplitter.
The system can be seen as an interferometer with such a high intensity reference beam (the local oscillator) that unbalancing the interference by a single photon in the signal is measurable.
The measurement is repeated again a large number of times and a marginal distribution is retrieved from the current difference.
Making assumptions about the structure and using a finite measurement basis leads to artifacts in the phase space density.
[5] Maximum likelihood estimation (also known as MLE or MaxLik) is a popular technique for dealing with the problems of linear inversion.
By restricting the domain of density matrices to the proper space, and searching for the density matrix which maximizes the likelihood of giving the experimental results, it guarantees the state to be theoretically valid while giving a close fit to the data.
This can be seen as related to linear inversion giving states which lie outside the valid space (the Bloch sphere).
This is where the problem arises, in that it is not logical to conclude with absolute certainty after a finite number of measurements that any eigenvalue (that is, the probability of a particular outcome) is 0.
Given a reasonable prior knowledge function, BME will yield a state strictly within the n-dimensional Bloch sphere.
In the case of a coin flipped N times to get N heads described above, with a constant prior knowledge function, BME would assign
[4] BME provides a high degree of accuracy in that it minimizes the operational divergences of the estimate from the actual state.
[17] Permutationally Invariant Quantum Tomography has been combined with compressed sensing in a six-qubit photonic experiment.
Not surprisingly, this suffers from the same pitfalls as in quantum state tomography: namely, non-physical results, in particular negative probabilities.
Bayesian methods as well as Maximum likelihood estimation of the density matrix can be used to restrict the operators to valid physical results.
However, under weak decoherence assumption, a quantum dynamical map can find a sparse representation.
The method of compressed quantum process tomography (CQPT) uses the compressed sensing technique and applies the sparsity assumption to reconstruct a quantum dynamical map from an incomplete set of measurements or test state preparations.
