Permutationally invariant quantum state tomography

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]

${\displaystyle \varrho _{\rm {PI}}={\frac {1}{N!}}\sum _{k}\Pi _{k}\varrho \Pi _{k}^{\dagger },}$

where ${\displaystyle \Pi _{k}}$ denotes the kth permutation. For instance, for ${\displaystyle N=2}$ we have two permutations. ${\displaystyle \Pi _{1}}$ leaves the order of the two particles unchanged. ${\displaystyle \Pi _{2}}$ exchanges the two particles. In general, for ${\displaystyle N}$ particles, we have ${\displaystyle N!}$ permutations.

It is easy to see that ${\displaystyle \varrho _{\rm {PI}}}$ is the density matrix that is obtained if the order of the particles is not taken into account. This corresponds to an experiment in which a subset of ${\displaystyle N}$ particles is randomly selected from a larger ensemble. The state of this smaller group is of course permutationally invariant.

The number of degrees of freedom of ${\displaystyle \varrho _{\rm {PI}}}$ scales polynomially with the number of particles. For a system of ${\displaystyle N}$ qubits (spin-${\displaystyle 1/2}$ particles) [2] the number of real degrees of freedom is

${\displaystyle {\binom {N+3}{N}}-1.}$

To determine these degrees of freedom,[1]

${\displaystyle {\binom {N+2}{N}}}$

local measurement settings are needed. Here, a local measurement settings means that the operator ${\displaystyle A_{j}}$ is to be measured on each particle. By repeating the measurement and collecting enough data, all two-point, three-point and higher order correlations can be determined.

So far we have discussed that the number of measurements scales polynomially with the number of qubits.

In order the entire tomographic procedure to be scalable, we need that we store the state in the computer in a scalable way. Clearly, the straightforward way of storing the ${\displaystyle N}$-qubit state in a ${\displaystyle 2^{N}\times 2^{N}}$ density matrix is not scalable. However, ${\displaystyle \varrho _{\rm {PI}}}$ 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.[3] This makes the whole process scalable. Furthermore, the convex optimization guarantees that the solution is a global optimum.

PI tomography is commonly used in experiments involving permutationally invariant states. If the density matrix ${\displaystyle \varrho _{\rm {PI}}}$ obtained by PI tomography is entangled, then density matrix of the system, ${\displaystyle \varrho }$ is also entangled. For this reason, the usual methods for entanglement verification, such as entanglement witnesses or the Peres-Horodecki criterion, can be applied to ${\displaystyle \varrho _{\rm {PI}}}$. Remarkably, the entanglement detection carried out in this way does not assume that the quantum system itself is permutationally invariant.

Moreover, the expectation value of any permutaionally invariant operator is the same for ${\displaystyle \varrho }$ and for ${\displaystyle \varrho _{\rm {PI}}.}$ Very relevant examples of such operators are projectors to symmetric states, such as the Greenberger–Horne–Zeilinger state, the W state and symmetric Dicke states. Thus, we can obtain the fidelity with respect to the above mentioned quantum states as the expectation value of the corresponding projectors in the state ${\displaystyle \varrho _{\rm {PI}}.}$

There are other approaches for tomography that need fewer measurements than full quantum state tomography. 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.[4] Matrix product state tomography is most suitable for, e.g., cluster states and ground states of spin models.[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,[2] and has been compared to full state tomography and compressed sensing. 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. [2]