Experimental proposal for symmetric minimal two-qubit state tomography

All experiments and proposals for experiments implementing SIC POMs up to date have been limited to the two-level system (qubit), with the exception of one where a SIC POM for a three-level system was approximated.

“In contrast to the major theoretical progress, up to date, all experiments and even proposals for experiments implementing SIC POMs have been limited to the very basic quantum system, the two-level system (qubit) [10, 11], with the exception of [12] where a SIC POM for a three-level system was approximated.”

• cites [10] sourceSupported
• cites [11] W.M. Pimenta, B. Marques, M.A. Carvalho, M.R. Barros, J.G. Fonseca, J. Ferraz, M. Terra Cunha, and S. P´ adua, Opt. Express 18 , 24423 (2010). sourceNot checked — no full text of the source
• cites [12] Z.E.D. Medendorp, F.A. Torres-Ruiz, L.K. Shalm, G.N.M. Tabia, C.A. Fuchs, and A.M. Steinberg, Phys. Rev. A 83 , 051801(R) (2011). sourceSupported

The qubit pair is carried by a single photon as a polarization qubit and a path qubit.

“The qubit pair is carried by a single photon as a polarization qubit and a path qubit.”

• Also stated in: § Experimental proposal for symmetric minimal two-qubit state tomography · p. 3 · plain

  “Our proposal is based on the methods of [15] where the two qubits, a polarization qubit and a path qubit, are encoded in a single photon.”

• cites [15] B.-G. Englert, C. Kurtsiefer, and H. Weinfurter, Phys. Rev. A 63 , 032303 (2001). sourceSupported

The SIC POM in dimension 4 is composed of 16 subnormalized projectors onto 16 fiducial kets.

“This SIC POM is composed of 16 subnormalized projectors onto 16 (fiducial) kets.”

• cites [16] I. Bengtsson, J. Phys.: Conf. Ser. 254 , 012007 (2010). sourceSupported

The experimental scheme uses a two-step process that breaks the SIC POM into two successive measurements, each with d outcomes: a measurement with full-rank outcomes, followed by a projective measurement on a basis chosen in accordance with the result of the first measurement.

“The experimental scheme exploits a new approach to SIC POMs that uses a two-step process: a measurement with full-rank outcomes, followed by a projective measurement on a basis that is chosen in accordance with the result of the first measurement.”

• Also stated in: § Experimental proposal for symmetric minimal two-qubit state tomography · p. 1 · hedged

  “In this approach, we 'break' the SIC POM into two successive measurements, each with d outcomes, with the intention that each measurement would be relatively easy to implement.”

• cites [13] sourceSupported

The proposal exploits a new theoretical approach for constructing SIC POMs.

“The proposal exploits a new theoretical approach for constructing SIC POMs [13].”

• cites [13] sourceSupported

The SIC POM for the qubit pair corresponds to a POM diagonal in the computational basis, followed by projections onto bases which are mutually unbiased.

“We found that the SIC POM for the qubit pair corresponds to a POM diagonal in the computational basis, followed by projections onto bases which are mutually unbiased.”

The two successive measurements that make up the SIC POM in dimension 4 have a strikingly simple form.

“Unexpectedly, we find that the two successive measurements that make up the SIC POM in dimension 4, not only have a strikingly simple form, but also hint at a close relation between the structure of mutually unbiased bases (MUB) and that of the SIC POM”

The unitary transformations U1 and U2 can be decomposed into a tensor product of two unitary transformations, one for the path qubit and one for the polarization qubit.

“We see that the unitary transformation U 1 and U 2 can be decomposed into a tensor product of two uni-”

• Also stated in: § Experimental proposal for symmetric minimal two-qubit state tomography · p. 4 · plain

  “tary transformations, one for the path qubit and one for the polarization qubit.”

Each of the matrices of fiducial vectors can be written as a diagonal matrix times a unitary matrix.

“Each of these matrices could be written as a diagonal matrix times a unitary matrix.”

The operations of Eq. (3) transform the computational basis into the MUB.

“Actually, the operations of Eq. (3) transform the computational basis into the MUB,”

The basis of the first measurement and the four bases of the second measurements are pairwise unbiased.

“The basis of the first measurement and the four bases of the second measurements are pairwise unbiased - a hint at a possibly profound link between SIC POMs and mutually unbiased bases.”

• Also stated in: § Experimental proposal for symmetric minimal two-qubit state tomography · p. 4 · plain

  “We found that the SIC POM for the qubit pair corresponds to a POM diagonal in the computational basis, followed by projections onto bases which are mutually unbiased.”

The two successive measurements that make up the SIC POM in dimension 4 hint at a close relation between the structure of mutually unbiased bases (MUB) and that of the SIC POM.

“Unexpectedly, we find that the two successive measurements that make up the SIC POM in dimension 4, not only have a strikingly simple form, but also hint at a close relation between the structure of mutually unbiased bases (MUB) and that of the SIC POM”

There is no systematic procedure for implementing SIC POMs in higher dimensions in a simple experimental set-up.

“This is, in part, due to the fact that there is no systematic procedure for implementing SIC POMs in higher dimensions, in a simple experimental set-up.”

The structure of fiducial vectors in Eq. (1) allows implementation of the SIC POM by two successive measurements: a measurement whose Kraus operators are given in Eq. (2), and depending on the measurement outcome, a measurement in one of the MUB of Eq. (4).

“The structure of fiducial vectors in Eq. (1) (which form the SIC POM in dimension 4) allows us to implement the SIC POM by two successive measurements: A measurement whose Kraus operators are given in Eq. (2), and depending on the measurement outcome, a measurement in one of the MUB of Eq. (4).”

The proposed experiment realizes a symmetric informationally complete (SIC) probability-operator measurement (POM) in the four-dimensional Hilbert space of a qubit pair.

“We propose an experiment that realizes a symmetric informationally complete (SIC) probabilityoperator measurement (POM) in the four-dimensional Hilbert space of a qubit pair.”

• Also stated in: § Experimental proposal for symmetric minimal two-qubit state tomography · p. 4 · plain

  “To conclude, we are proposing a feasible experimental scheme that implements the SIC POM for a twoqubit system.”

The settings of the optical elements ensure that the measurement of Eq. (2) is realized.

“These settings ensure that the measurement of Eq. (2) is realized.”

The experiment is feasible with current technology.

“The experiment is clearly feasible with current technology.”

• Also stated in: § Experimental proposal for symmetric minimal two-qubit state tomography · p. 4 · boosted

  “Our scheme uses linear optical elements and photodetectors, and is, therefore, well within the reach of existing technology.”

The implementation of the SIC POM is accomplished with the means of linear optics.

“The implementation of the SIC POM is accomplished with the means of linear optics.”

• Also stated in: § Experimental proposal for symmetric minimal two-qubit state tomography · p. 4 · plain

  “Our scheme uses linear optical elements and photodetectors, and is, therefore, well within the reach of existing technology.”