Certifying the dimensionality of any quantum channel with minimal assumptions

TL;DR

The paper addresses the problem of certifying whether a quantum channel preserves entanglement dimensionality above a threshold $k$, i.e., whether it is not $k$-SNB, under a minimal-assumptions framework. It introduces semiquantum signaling games with trusted input preparation and leverages the Choi–Jamiołkowski representation to construct witnesses that separate any non-$k$-SNB channel from the convex set of $k$-SNBC channels, with extensions to non-NPT-breaking channels. The main contributions are a faithful certification method applicable to all non-$k$-SNB channels, an explicit experimental implementation plan, and illustrative examples (e.g., depolarizing/dephasing channels on 3-dimensional systems) demonstrating the method’s practicality. The approach provides robust, device-independent-feel certification of high-dimensional entanglement preservation in quantum memories and communication links under minimal assumptions, advancing the practical deployment of high-dimensional quantum resources.

Abstract

High-dimensional entanglement offers significant advantages over low-dimensional ones in various information-processing tasks. However, to harness these advantages, it is crucial that the quantum channels used to store or transmit the subsystems of an entangled system not only preserve entanglement but also maintain its dimensionality above a certain threshold. The maximum entanglement dimension that a channel can preserve is referred to as its effective dimensionality, since the channel cannot be used to transmit information of dimension greater than that in a single use. In this work, we present a method to certify whether a quantum channel can preserve entanglement dimension above a given threshold. Unlike existing approaches, our method is faithful, i.e., it can be applied to any channel, and avoids common assumptions such as reliable preparation of entangled states, auxiliary side channels, or perfect measurement devices. Moreover, the method can be extended to faithfully certify other classes of non-resource-breaking channels, such as non-NPT-breaking channels. Finally, we discuss possible experimental realizations of our certification scheme through explicit examples.

Figures (4)

• Figure 1: A schematic setup for the certification of non-$k$-SNB channels in a semiquantum signaling game with trusted quantum inputs.
• Figure 2: Circuit for experimental realization of the certification scheme for an unknown channel $\mathcal{N}$.
• Figure 3: Variation of the average payoff $\mathcal{J}_{\text{avg}}$ with respect to the noise parameter ($\lambda$) for depolarizing and dephasing channels, illustrating the transition to non-EB and non-$2$-SNB behaviors. The blue (red) line corresponds to the dephasing (depolarizing) channel. A negative payoff in the solid (dotted) blue line denotes the range of noise parameters for which the dephasing channel is non-$2$-SNB (non–entanglement breaking). As the noise parameter increases, the channel becomes $2$-SNB and eventually transitions to being entanglement-breaking (EB). An analogous behavior is observed for the depolarizing channel, where the negative payoff in the fine-dotted and dotted red lines represents the regions corresponding to the non-$2$-SNB and non-EB regimes, respectively.
• Figure 4: Variation of the average payoff $\mathcal{\tilde{J}}_{\text{avg}}$ with respect to the parameter ($\alpha$) for the channel given by Eq. \ref{['channel']}. A negative payoff denotes the range of parameters for which the channel is non-NPT-breaking.

Theorems & Definitions (15)

• Lemma 1
• Lemma 2
• proof
• Theorem 1
• proof
• Remark 1
• Example 1
• Definition 1
• Definition 2
• Lemma 3