Characterization-free classification and identification of the environment between two quantum players

TL;DR

A characterization-free protocol enabling two isolated players to classify and identify the definite-order strategy adopted by an unknown environment mediating their channels is introduced, providing a strong and robust tool for causal inference in quantum networks.

Abstract

Classifying the causal structure of quantum channels is essential for verifying quantum networks and certifying quantum resources. We introduce a characterization-free protocol enabling two isolated players, Alice and Bob, to classify and identify the definite-order strategy adopted by an unknown environment mediating their channels. Without assuming knowledge of their devices or the environment, the players infer the causal order solely from input-output statistics by testing Markovian conditions that we prove are necessary and sufficient for each strategy class. Remarkably, we prove that even with a minimal random channel consisting of two-outcome POVMs and two-state preparations, the protocol retains full performance with probability one. We experimentally demonstrate the protocol on an optical platform, reliably distinguishing between several strategies. Our results provide a strong and robust tool for causal inference in quantum networks.

Figures (4)

• Figure 1: Fig. (\ref{['parallel']}) and Fig. (\ref{['sequential']}) specify the structure of Charlie's strategy classes within the general class $S_{G}$, which can be represented as a quantum comb (the gray part). chiribella2008quantum. Fig. (\ref{['parallel']}) specifies the three parallel strategies. When the type of memory is trivial (T)/ classical (C)/ quantum (Q), the strategy is individual ($S_{I}$)/classical ($S_{C}$)/quantum ($S_{Q}$). The unidentified parts in the comb (area in the dashed boundary) do not influence the protocol. Fig. (\ref{['sequential']}) specifies the three sequential strategies ($S_{N,1\rightarrow 2}$)/$S_{C,1\rightarrow 2}$)/$S_{Q,1\rightarrow 2}$), which correspond to T/ C/ Q type of memory respectively. Fig. (\ref{['DI']}) illustrates Alice's MP channel (Bob's is analogous). In our protocol, the parts inside the gray dashed box do not require characterization.
• Figure 2: The procedure of identification using correlations $P_{J_1,J_2,K_1,K_2}$. In step 1, the markovian conditions in \ref{['HI4']} are classified into 3 levels. The hypothesis testing starts from level 1. If the condition in a lower level is rejected, we check the conditions of the next level, until a condition is accepted. If a nontrivial memory is detected in step 1, we go to step 2 to check if it is a classical or a quantum memory by checking nonlocality.
• Figure 3: Experimental setup. Red paths and light-gray dashed lines represent photon propagation directions of Alice $\rightarrow$ Bob and Bob $\rightarrow$ Alice, respectively. Detailed configurations for different strategies are provided in the Appendix \ref{['ApdxD']}.
• Figure E1: Maximal values among all CHSH inequalities in Eq. (16) for each setting. A misidentification occurs only for the strategy $S_{Q,1\to 2}$ at setting 3. The error bars are derived from Poissonian counting statistics.

Theorems & Definitions (7)

• Theorem 1
• Theorem 2
• Lemma 3
• proof
• Lemma 4
• proof
• proof : Proof of Theorem \ref{['TH5']}