Certifying high-dimensional quantum channels

Abstract

The use of high-dimensional systems for quantum communication opens interesting perspectives, such as increased information capacity and noise resilience. In this context, it is crucial to certify that a given quantum channel can reliably transmit high-dimensional quantum information. Here we develop efficient methods for the characterization of high-dimensional quantum channels. We first present a notion of dimensionality of quantum channels, and develop efficient certification methods for this quantity. We consider a simple prepare-and-measure setup, and provide witnesses for both a fully and a partially trusted scenario. In turn we apply these methods to a photonic experiment and certify dimensionalities up to 59 for a commercial graded-index multi-mode optical fiber. Moreover, we present extensive numerical simulations of the experiment, providing an accurate noise model for the fiber and exploring the potential of more sophisticated witnesses. Our work demonstrates the efficient characterization of high-dimensional quantum channels, a key ingredient for future quantum communication technologies.

Figures (6)

• Figure 1: We discuss the task of bounding the dimensionality of an unknown high-dimensional quantum channel $\Lambda$. A prepare-and-measure setup is considered, where a sender prepares various input states $\rho_y$. We develop witnesses for two scenarios with varying level of trust on the receiver's device: a) A fully trusted model, where the measurement operators $M_{a|x}$ are characterized, and b) A partially trusted model, where the measurement device is uncharacterized.
• Figure 2: Experimental setup. The experiment consists of three parts: state generation, state propagation through the channel and state measurement. To generate the input state, a spatially-coherent light source (810 nm) is incident on a programmable phase-only spatial light modulator (SLM1). This is then coupled into a graded-index (GRIN) multi-mode fiber (MMF), representing a noisy channel. The state at the output of the MMF is incident on SLM2 followed by a CCD camera, the combination of which allows one to perform projective measurements on the output state. The modes generated correspond to the eigenmode basis of the fiber and a MUB with respect to the eigenmode basis. (a) and (b) insets display four examples of input and output modes, arranged in descending order of eigenvalue magnitude. (c) shows an example of a phase pattern displayed on the SLM's.
• Figure 3: Certified dimension $n$ of 2 m and 5 m-long graded-index multi-mode fibers, using the FT witness (Eq. \ref{['eq:Mubbywitness']}) and the PT witness (Eq. \ref{['eq:channel schmidt number bound']}) when utilising subspaces of different dimension $d$. In a) we show the results of the FT witness (purple) and in b) the PT witness (blue). For all cases, we present the certified dimension for three scenarios: simulation of an idealised fiber accounting for dispersion effects in the MMF (dashed lines), simulation of an idealised fiber with additional white noise (solid lines), and the experimental results (crosses).
• Figure 4: Certified dimension for the simulated idealised 5 m MMF with additional white noise, using the witness of Eq. \ref{['eq:FTwitness']} for the fully trusted scenario. The plot shows simulation results as scatter points, with lines added to guide the eye. Notably, considering additional MUBs provides a significant enhancement in the certified dimension.
• Figure 5: Experimentally certified dimension $n$ of 2 m and 5 m graded-index multi-mode fibers, using the FT Bavaresco witness, Eq. \ref{['eq:Mubbywitness']} (blue circles) and the FT Morelli witness, Eq. \ref{['eq:FTwitness']} (purple crosses) when utilising subspaces of different dimensions $d$.