Permutation-invariant quantum tomography

The permutation invariant quantum tomography (PI quantum tomography) is a measurement method of quantum mechanics for the partial determination of the state of a quantum system consisting of many subsystems . For each possible measured value, the probability is given that this will occur, because quanta can only occur with certain values ​​of their physical size.

In general, the quantum mechanical state of a system consisting of subsystems is described by an exponentially large number of independent parameters. In the case of a system consisting of qubits , these are the independent complex components of the state vector or, for mixed states, the real parameters of the density matrix . The quantum tomography is a method for determining all these parameters from a series of measurements on many independent and identically prepared systems. ${\displaystyle N}$ ${\displaystyle N}$ ${\displaystyle N}$ ${\displaystyle 2^{N-1}}$ ${\displaystyle 2^{2N}-1}$

The determination of all these quantities is no longer practicable for large systems, and there is interest in methods which allow a subset of them to be determined with limited effort that still contains important information about the state. Permutation-invariant quantum tomography is such a simplified method. It is motivated by the fact that one is often interested in systems that consist of many similar subsystems (e.g. the atoms in an optical lattice or in an optical trap ). Then the state of the -atom system is invariant to a good approximation with interchanges ( permutations ) of the atoms. Accordingly, it is then sufficient to determine only that of the permutation-invariant system instead of the full density matrix (with an exponential number of independent entries). Its density matrix only has a scalable number of independent entries, which is also seen as manageable for large ones (and PI quantum tomography is therefore regarded as "scalable"). ${\displaystyle N}$ ${\displaystyle N^{3}}$ ${\displaystyle N}$

If the state is not permutation-invariant, PI quantum tomography only measures the “permutation-invariant part” of the density matrix. For the procedure it is sufficient to carry out "local measurements" on subsystems. The method is e.g. B. used to reconstruct the density matrices of systems with more than 10 particles, for example for photonic systems or systems of cold atoms .

The permutation-invariant part of a density matrix

PI state tomography reconstructs the permutation-invariant part of the density matrix, which is defined by the proportional mixture of all permutations in the density matrix

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

where the k th denotes permutation. In this respect, the density matrix is ​​obtained when the order of the particles is not to be taken into account. This corresponds to an experiment in which a subset of the particles is selected from a larger ensemble. The state of this smaller group is of course permutation invariant. ${\displaystyle \Pi _{k}}$ ${\displaystyle \varrho _{\rm {PI}}}$

The number of degrees of freedom of scales polynomially with the number of particles, where for a system of spin particles ${\displaystyle \varrho _{\rm {PI}}}$ ${\displaystyle N}$ ${\displaystyle 1/2}$

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

Degrees of freedom are to be found.

The measurement

To determine these degrees of freedom are

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

local measurements needed. Local measurement in this context means that the operator has to be measured on each particle. By repeating the measurement and collecting enough data, all two-point functions , three-point functions and higher correlations as well as the density matrix itself can be determined. ${\displaystyle A_{j}}$

Efficient determination of a physical state

While the number of measurements scales polynomially with the number of qubits - provided the state of the system is described by one - another part of the tomography scheme does not scale well with the problem size. ${\displaystyle 2^{N}\times 2^{N}}$

An important step in the determination of the state consists of the adaptation of a positive semidefinite density matrix, which only allows a physical interpretation of the data disturbed by statistical fluctuations and systematic errors. This step often represents a bottleneck in the overall process.

However, PI tomography allows the density matrix to be saved much more efficiently, which means that fitting using convex optimization is also possible efficiently. This makes the entire process scalable. In addition, the convex optimization guarantees that the solution is a global optimum.

Characteristics of the method

PI tomography is commonly used in experiments with permutation invariant states. If the density matrix obtained by PI tomography is an entangled state , the underlying system also exhibits entanglement. For this reason, the usual evidence of entanglement can be applied to the tomography result. Remarkably, the entanglement verification carried out in this way does not assume that the quantum system itself is permutation-invariant.

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• Géza Tóth: Permutationally Invariant Quantum Tomography. Retrieved January 9, 2015 . 

Individual evidence

1. ↑ ^{a b} Tobias Moroder, Philipp Hyllus, Géza Tóth, Christian Schwemmer, Alexander Niggebaum, Stefanie Gaile, Otfried Gühne, Harald Weinfurter: Permutationally invariant state reconstruction. In: New Journal of Physics. 14, No. 10, 2012, p. 105001, doi: 10.1088 / 1367-2630 / 14/10/105001 .
2. ↑ G. Tóth, W. Wieczorek, D. Gross, R. Krischek, C. Schwemmer, H. Weinfurter: Permutationally Invariant Quantum Tomography . In: Phys. Rev. Lett. tape 105 , no. 25 , 2010, p. 250403 , doi : 10.1103 / PhysRevLett.105.250403 , arxiv : 1005.3313 .