Experimental Certification of Quantum Measurements with Maximally Mixed States

Abstract

So far, certifying quantum devices from their input-output statistics, under minimal assumptions, required the preparation of specific pure quantum states. Recently, Xu et al. [Phys. Rev. Lett. 132, 140201 (2024)] have demonstrated that certain sets of quantum observables can be certified using any state of full rank. However, their method is restricted to ideal conditions. Here, we address this problem and present an experimentally robust method that eliminates the need of preparing states with high fidelity with respect to specific pure states. We demonstrate the feasibility of the method by experimentally certifying photonic devices implementing Peres' set of 24 ququart observables [J. Phys. A 24, L175 (1991)] and Yu and Oh's set of 13 qutrit observables [Phys. Rev. Lett. 108, 030402 (2012)], using maximally mixed states as input. This approach offers a crucial advantage for certifying high-dimensional quantum systems, since it works with maximally mixed and thermal states.

Figures (4)

• Figure 1: (a) Algorithm for certifying uncharacterized projectors using a maximally mixed state. (b) Orthogonality graph for the Peres-24 set Peres24. Vertices represent measurements (represented by rank-one projectors onto the corresponding unnormalized vectors). Vectors in the same straight line or in a clique of the same color are mutually orthogonal. The Hilbert space is spanned by four orthogonal vectors: $|1\rangle, \ |2\rangle, \ |3\rangle, \ |4\rangle$. (c) Components of the 24 (unnormalized) vectors projected onto the $\{|2\rangle,\ |3\rangle,\ |4\rangle\}$ subspace. A blue arrow indicates the original vector has component $0$ in $|1\rangle$ (i.e., it lies entirely within the subspace). A red arrow indicates it has component $1$ in $|1\rangle$. Vector $|v_1\rangle=(1,0,0,0)$ is at the origin.
• Figure 2: Schematic of experimental setup. (a) Main optical system, (b) 808 nm photon source, (c) logical diagram of the experiment. In (c), P1 and P2 are preparations and M1 and M2 are measurements. Based on the outcome of M1, which measures the projector $\Pi_i'$, the motorized half-wave plates in P2 automatically prepare the corresponding state $|v_i\rangle$, using the method of preparing 4-dimensional state in P1. Abbreviations are BD (beam displacer), FC (fiber coupler), HWP (half-wave plate), MHWP (motorized half-wave plate), PBS (polarization beam splitter), PPKTP (periodically polled potassium titanyl phosphate), QC (quartz crystal), and SPAD (single photon avalanche diode).
• Figure 3: Experiment outcomes of $\epsilon_{ij}$ obtained from projectors in group 1 and group 2, shown by green and red bars, respectively. Experimental error bars are estimated as the standard deviation based on assumption of photons' Poisson statistics.
• Figure 4: Experimental results of $P_i$ obtained from group 1 and group 2 is shown by green and red bars in (a), respectively. In (b), we demonstrate $\mathcal{W}_{\rm worst}$ and $\mathcal{W}_{\rm SDP}$ with green circles and red squares, where the first and second column correspond to group 1 and 2, and other three columns correspond to three additional groups of projectors. Experimental error bars are estimated as the standard deviation based on assumption of photons' Poisson statistics. Horizontal error bars are on the order of $0.001$, whose gaps are not so obvious in the figure.