Measuring distances in Hilbert space by many-particle interference

The measures of distances between points in a Hilbert space are one of the basic theoretical concepts used to characterize properties of a quantum system with respect to some etalon state. These are not only used in studying fidelity of signal transmission and basic quantum phenomena but also applied in measuring quantum correlations, and also in quantum machine learning. The values of quantum distance measures are very difficult to determine without completely reconstructing the state. Here we demonstrate an interferometric approach to measuring distances between quantum states that in some cases can outperform quantum state tomography. We propose a direct experimental method to estimate such distance measures between two unknown two-qubit mixed states such as Uhlmann-Jozsa fidelity (or the Bures distance), the Hilbert-Schmidt distance, and the trace distance. The fidelity is estimated via the measurement of the upper and lower bounds of the fidelity, which are referred to as the superfidelity and subfidelity, respectively. Our method is based on the multiparticle interactions (i.e., interference) between copies of the unknown pairs of qubits.

Figure 1

The complete set of 63 graphs representing the combinations of tensor products of singlet projections needed for measuring all relevant four-particle interactions. The connected pairs (light gray lines) of dark gray or white vertices denote the modes forming a copy of ${\rho }_1$ or ${\rho }_2$ state, respectively. The red (dark) lines mark the singlet projections (Hong-Ou-Mandel antibunching events). Each graph is associated with a rate of simultaneous detection of singlet state of particles connected with red lines. This is explained in greater detail on several examples in Fig. 2.

Figure 2

From left: The conceptual scheme for measuring Hilbert-Schmidt distance in an interferometer, all possible detection events for the depicted interferometer. The interferometer can be divided into three smaller independent interferometers. The photons sent to the device belong to subsystems $a$ or $b$ of state ${\rho }_1$ or ${\rho }_2$ represented by white and dark gray dots, respectively. The relevant photons interfere on balanced beam splitters and the resulting two-photon state is detected to be (event $A$) antibunching in $C{C}_n$ ($n=1,2...,6$) blocks. If the photons do not interfere on a beam splitter (it is out of the Hong-Ou-Mandel dip) the detected rate of coincidences in $C{C}_n$ corresponds to half of the detection rate of event $B.$ The $C{C}_n$ blocks consist only of two single-photon detectors.

Figure 3

Conceptual scheme for quantum remote sensing protocol through a noisy environment. Here, two-qubit states are initially prepared as pure or mixed states $\rho$. Next, one of the states is transmitted via a reference part of the environment (Environment 1) and the other via the monitored part of the environment (Environment 2). If there are any differences between the two environments, the distance measured between the two two-qubit states will increase. The final distance between ${\rho }_1$ and ${\rho }_2$ does not depend on the global phase shifts. In this scheme there is no need to keep the overall phase between the two initial states $\rho$ stable; instead, relative phase shift will be observed between two qubits that interacted with the respective environment. This protocol will be useful in detecting changes in the monitored environment, where due to unavoidable noise the purity of both inputs states $\rho$ is decreased. Note, that the roles of the environment in this protocol are interchangeable.