Experimental characterization of quantum processes: A selective and efficient method in arbitrary finite dimensions

Q. Pears Stefano, I. Perito, J. J. M. Varga, L. Rebón, and C. Iemmi

Phys. Rev. A 103, 052438 – Published 28 May 2021

Abstract

The temporal evolution of a quantum system can be characterized by quantum process tomography, a complex task that consumes a number of physical resources scaling exponentially with the number of subsystems. An alternative approach to the full reconstruction of a quantum channel is the selective and efficient quantum process tomography, a method that allows estimating, individually and up to the required accuracy, each element of the matrix that describes the process, using only a polynomial amount of resources. The implementation of this protocol is closely related to the possibility of building a complete set of mutually unbiased bases (MUBs), whose existence is known only when the dimension of the Hilbert space is the power of a prime number. However, an extension of the method that uses tensor products of maximal sets of MUBs has been recently introduced. Here we explicitly describe how to implement the algorithm for a selective and efficient estimation of a quantum process in a nonprime power dimension and conduct, an experimental verification of the method in a Hilbert space of dimension $d = 6$ . This is the smallest space for which a complete set of MUBs is not known to exist, but it can be decomposed as a tensor product of two Hilbert spaces of dimensions $D_{1} = 2$ and $D_{2} = 3$ , in which a complete set of MUBs is already known. The six-dimensional states were codified in the discretized transverse linear momentum of photons. The state preparation and detection stages are dynamically programed with the use of only-phase spatial light modulators, in a versatile experimental setup that allows one to implement the algorithm in any finite dimension.

Received 18 November 2020
Revised 15 April 2021
Accepted 7 May 2021

DOI:https://doi.org/10.1103/PhysRevA.103.052438

Physics Subject Headings (PhySH)

Quantum channels Quantum states of light Quantum tomography

Quantum Information, Science & TechnologyAtomic, Molecular & Optical

Authors & Affiliations

Q. Pears Stefano^1,2,*, I. Perito^1,3, J. J. M. Varga^4,5, L. Rebón ⁶, and C. Iemmi^1,2

¹Departamento de Física, FCEyN, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina
²Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), 1425 Buenos Aires, Argentina
³Instituto de Física de Buenos Aires, CONICET, Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales, 1428 Buenos Aires, Argentina
⁴Centro de Física de Materiales, Paseo Manuel de Lardizabal 5, 20018 Donostia-San Sebastián, Spain
⁵Donostia International Physics Center, Paseo Manuel de Lardizabal 4, 20018 Donostia-San Sebastián, Spain
⁶Departamento de Física, IFLP–CONICET, Universidad Nacional de La Plata, Casilla de Correo 67, 1900 La Plata, Argentina

^*quimeyps@df.uba.ar

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Issue

Vol. 103, Iss. 5 — May 2021

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Images

Figure 1
(a) Circuit for a projective measurement of state $| ϕ^{A} 〉$ , after being affected by the process $E$ , onto the state $| ϕ^{B} 〉$ . (b) Circuit for measuring the survival probability of the state $| ψ 〉 = | ψ_{1} 〉 \otimes | ψ_{2} 〉$ through the modified channel $E_{i_{1} i_{2}}^{i_{1} i_{2}}$ . By sampling over $X_{\otimes}$ we can obtain the diagonal element $χ_{j_{1} j_{2}}^{i_{1} i_{2}}$ corresponding to the process matrix of $E$ .
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Figure 2
Effect of the process on the six-dimensional spatial qudit codified in the discretized transverse momentum of a photon. The process is physically implemented by a glass slab (GS) with a transparent coating covering part of its surface. This partial coating introduces a phase shift $Δ φ = 5.42$ rad on the paths that codify the states $| 0 〉$ and $| 1 〉$ of the six-dimensional canonical basis.
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Figure 3
Experimental setup. A 405-nm cw laser diode is attenuated to the single-photon level. O, microscope objective; Ls, convergent lenses; SLMs, pure-phase spatial light modulators; SFs, spatial filters. A glass slab (GS) implements the process on a spatial qudit as schematized in Fig. 2. The detection in the center of the interference pattern is performed with a fiber-coupled avalanche photodiode (APD).
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Figure 4
(a) Comparison of the absolutes values of the elements of $χ_{theo}$ (expected matrix of the target process $E_{t}$ ) and $χ_{\exp}$ (reconstructed by the tensor product SEQPT). (b) Detail of the absolute value of the elements in the $12 \times 12$ submatrix. This elements correspond to the nonzero block in the matrix $χ_{theo}$ [upper-left block in the theoretical plot of panel (a)]. The gray-scale map shows the real and imaginary part for the theoretical and the experimental reconstructed matrices.
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Figure 5
Histogram of the state fidelity $F [ρ_{out}, E_{\exp} (ρ_{in})]$ between the density matrix $ρ_{out}$ , obtained after performing QST of a given initial state $ρ_{in}$ affected by the implemented process, and the expected density matrix $E_{\exp} (ρ_{in})$ corresponding to the same initial state under the action of the map $E_{\exp}$ , obtained after performing QPT of the implemented process. Obtained fidelity in the case in which tensor product SEQPT (a) or standard QPT (b) was performed. For each histogram, 250 arbitrary states of dimension $d = 6$ were prepared and measured.
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Figure 6
Fidelity in the reconstruction of the implemented process for an increasing sampling on the elements of the tensor product of 2-design. The sampling is performed only to reconstruct the 21 nonzero elements characterizing the target process $E_{t}$ . The different markers represent the fidelity values between the reconstructed process and the different target processes $E_{target}$ . The mean value of the fidelity is indicated by a continuous line ( $E_{target} = E_{t}$ ), a dashed line ( $E_{target} = I$ ), or a dotted line ( $E_{target} = {\tilde{E}}_{t}$ ). In each case, the shaded areas correspond to the standard deviation.
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