In general, the number of parameters needed to describe the quantum mechanical state of a system consisting of
real parameters are needed to describe the density matrix of a mixed state.
Quantum state tomography is a method to determine all these parameters from a series of measurements on many independent and identically prepared systems.
Thus, in the case of full quantum state tomography, the number of measurements needed scales exponentially with the number of particles or qubits.
For the procedure, it is sufficient to carry out local measurements on the subsystems.
If the state is close to being permutationally invariant, which is the case in many practical situations, then
can still be used for entanglement detection and computing relevant operator expectations values.
Thus, the procedure does not assume the permutationally invariance of the quantum state.
Thus, permutationally invariant quantum tomography is considered manageable even for large
In other words, permutationally invariant quantum tomography is considered scalable.
PI state tomography reconstructs the permutationally invariant part of the density matrix, which is defined as the equal mixture of the quantum states obtained after permuting the particles in all the possible ways [1] where
is the density matrix that is obtained if the order of the particles is not taken into account.
By repeating the measurement and collecting enough data, all two-point, three-point and higher order correlations can be determined.
However, for using the method in practice, the entire tomographic procedure must be scalable.
Thus, we need to store the state in the computer in a scalable way.
is a blockdiagonal matrix due to its permutational invariance and thus it can be stored much more efficiently.
[3] Moreover, it is well known that due to statistical fluctuations and systematic errors the density matrix obtained from the measured state by linear inversion is not positive semidefinite and it has some negative eigenvalues.
An important step in a typical tomography is fitting a physical, i. e., positive semidefinite density matrix on the tomographic data.
This step often represents a bottleneck in the overall process in full state tomography.
However, PI tomography, as we have just discussed, allows the density matrix to be stored much more efficiently, which also allows an efficient fitting using convex optimization, which also guarantees that the solution is a global optimum.
[3] PI tomography is commonly used in experiments involving permutationally invariant states.
obtained by PI tomography is entangled, then density matrix of the system,
Remarkably, the entanglement detection carried out in this way does not assume that the quantum system itself is permutationally invariant.
As we have discussed, PI tomography is typically most useful for quantum states that are close to being permutionally invariant.
Compressed sensing is especially suited for low rank states.
[5] Permutationally invariant tomography can be combined with compressed sensing.
In this case, the number of local measurement settings needed can even be smaller than for permutationally invariant tomography.
[2] Permutationally invariant tomography has been tested experimentally for a four-qubit symmetric Dicke state,[1] and also for a six-qubit symmetric Dicke in photons, and has been compared to full state tomography and compressed sensing.
[2] A simulation of permutationally invariant tomography shows that reconstruction of a positive semidefinite density matrix of 20 qubits from measured data is possible in a few minutes on a standard computer.
[3] The hybrid method combining permutationally invariant tomography and compressed sensing has also been tested.