Quantum Tomography - Methods of Quantum State Tomography - Linear Inversion

Linear Inversion

Using Born's rule, one can derive the simplest form of quantum tomography. If it is known in advance that the state is represented by a pure state, a single measurement can be performed repeatedly to build up a histogram which can then be used to express the pure state in the basis of the measurement. Generally, being in a pure state is not known, and a state may be mixed. In this case, many different measurements will have to be performed, many times each. To fully reconstruct the density matrix for a mixed state in a finite-dimensional Hilbert space, the following technique may be used.

Born's rule states, where is a particular measurement outcome projector and is the density matrix of the system. Given a histogram of observations for each measurement, one has an approximation to for each .

Given linear operators and, define the inner product

where is representation of the operator as a column vector and a row vector such that is the inner product in of the two.

Define the matrix as

.

Then applying this to yields the probabilities:

.

Linear inversion corresponds to inverting this system using the observed relative frequencies to derive (which is isomorphic to ).

This system is not going to be square in general, as for each measurement being made there will generally be multiple measurement outcome projectors . For example, in a 2-D Hilbert space with 3 measurements, each measurement has 2 outcomes, leaving to be 6 x 4. To solve the system, multiply on the left by :

.

Now solving for yields the pseudoinverse:

.

This works in general only if the measurements were tomographically complete. Otherwise, the matrix will not be invertible.