PI tomography - pitomography

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Permutationally Invariant Quantum Tomography

Permutionally Invariant Quantum Tomography or PI Quantum Tomography is the process of reconstructing the permutationally invariant part of thequantum state (density matrix) of a multi-particle system by local measurements on the system [1]. It is a scaleable alternative of full quantum state tomography, which make density matrix reconstruction possible for systems with more than 10 particles for photons, cold trapped ions, and any other realization where the particles are locally accessible.

The permutationally invariant part of the density matrix

PI state tomography reconstructs the permutationally invariant part of the density matrix, which is defined as the equal mixture of all permutations of the density matrix,

$\varrho_{\rm PI}=\frac{1}{N!} \sum_k \Pi_k \varrho \Pi_k^\dagger,$

where $\Pi_k$ is the kth permutation. In fact, $\varrho_{\rm PI}$ is the density matrix that we obtain after the we dispose the information on the order of particles. This is a typical situation when in an experiment a small group of particles is selected from a large ensemble. The state of this small group will naturally be permutationally invariant.

The number of degrees of freedom of $\varrho_{\rm PI}$ scales polynomially with the number of particles. For a system of $N$ spin- $1/2$ particles it is

$\binom{N+3}{N}.$

The measurements

In order to determine these degrees of freedom, we need to measure

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

local measurement settings. A single local measurement settings in this case means that at each particle operator $A_j$ is measured. After repeating such a measurement and collecting enough statistics, all two-point, three-point etc. correlations can be calculated, and even the density matrix can be obtained.

Efficient fitting of a physical state

While the number of measurements scales polynomially with the number of qubits, as long as the state of the system is represented by a $2^N\times2^N$ density matrix, there is still a part of the tomographic procedure that does not scale well with the problem size. An important procedure in obtaining a density matrix from state tomography is fitting a physical, i.e., positive semidefinite, density matrix on the measured data. This is often the bottleneck of the entire tomographic procedure. With PI tomography, it is possible to store the density matrix efficiently. Moreover, fitting a physical density matrix on the measured data with convex optimization can also be done in an efficient way [2]. Thus, the entire procedure including collecting the data and processing it is scalable. Moreover, convex optimization is guaranteed to find the global optimum.

Characteristics of the method

PI tomography can be used typically in experiments in which permutationally invariant states are produced. If the density matrix obtained from PI tomography is entangled, the density matrix of the system was also entangled. Thus the PI density matrix reconstructed can be used as the input of any usual entanglement detection procedure. In order to detect entanglement this way, we do not need to assume that the quantum state was permutationally invariant. The expectation values of any permutationally invariant operator, including projectors to symmetric states, can be obtained exactly from the permutationally invariant density matrix.

References

Tóth, G.; Wieczorek, W.; Gross, D.; Krischek, R.; Schwemmer, C.; Weinfurter, H. (2010). "Permutationally Invariant Quantum Tomography". Physical Review Letters105 (25). doi:10.1103/PhysRevLett.105.250403.
Moroder, T.; Hyllus, P.; Tóth, G. Z.; Schwemmer, C.; Niggebaum, A.; Gaile, S.; Gühne, O.; Weinfurter, H. (2012). "Permutationally invariant state reconstruction".New Journal of Physics 14 (10): 105001. doi:10.1088/1367-2630/14/10/105001.
Schwemmer, C.; Tóth, G.; Niggebaum A.; Moroder T.; Gross D.; Gühne O.; Weinfurter H. (2014). "Efficient Tomographic Analysis of a Six Photon State". Physical Review Letters113, 040503. doi:10.1103/PhysRevLett.113.040503