Intractability of Witnessing Entangled Measurements Device Independently

Abstract

Protocols have been previously proposed to certify the presence of an entangled measurement in a fully device-independent manner. Here, I provide models for these protocols in which the claimed measurement is not entangled, and demonstrate it is always possible to displace entanglement from measurements to measured states for a general class of device-independent scenarios. This indicates that no black-box measurement scenario requires entangled measurements to replicate its behavior, which is relevant to our fundamental understanding of this phenomenon and how to witness it.

Figures (6)

• Figure 1: The scheme of Rabelo et al, in which Charlie ($\mathsf C = \mathsf {C_A}$ and $\mathsf {C_B}$) either a) performs two parallel CHSH-Bell tests with $\mathsf A$ and $\mathsf B$, or b) a Bell state measurement (BSM) which swaps an entangled state to $\mathsf A$ and $\mathsf B$.
• Figure 2: Circuit diagram for measurement setting $C_3$ in the Rabelo et al. scheme, corresponding to Fig. \ref{['f:rabeloscheme']} (b). As the Bell state measurement at $\mathsf C = (\mathsf{C_A},\mathsf{C_B})$ (BSM, outer box) can be implemented with a unitary $U$ (inner box, see also equation \ref{['e:Udef']}) followed by two independent computational basis measurements (meters), we can replace it with an un-entangled measurement by absorbing $U$ into the measured state.
• Figure 3: Spacetime layout for the scheme of Bancal et al.bancal:2018. In various rounds, either $\mathsf {C_A}$ and $\mathsf {C_B}$ perform measurements self-testing Bell states shared with $\mathsf A$ and $\mathsf B$ (dashed lines), or they physically send their qubits to $\mathsf C$ for an entangled BSM.
• Figure 4: A scheme replicating the behavior of Figure \ref{['f:bancalscheme']} in which only classical information is transported and no quantum measurement is performed at event $\mathsf C$. The table indicates the state of $\mathsf A$ and $\mathsf B$ conditioned on the BSM outcomes.
• Figure 5: Example of a general spacetime-ordered device-independent scenario. For an arbitrary collection of spacetime events at which measurements occur (vertices $\mathsf{A}-\mathsf{F}$), associate to each maximal-length timelike path a Hilbert space $\mathcal{H}_i$ indicating a possible route taken by quantum information. This includes zero-length paths at events spacelike separated from all others. Here the resultant state space is $\mathcal{H}_1\otimes \mathcal{H}_2 \otimes \mathcal{H}_3 \otimes \mathcal{H}_4$; a measurement at $\mathsf C$ for example acts on $\mathcal{H}_2 \otimes \mathcal{H}_3$, while $\mathsf B$ acts only on $\mathcal{H}_2$.