Unambiguous arbitrary high-dimensional Bell states analyzer via indefinite causal order

Abstract

High-dimensional quantum systems greatly outperform their two-dimensional counterparts in channel capacity, quantum complexity and efficiency, quantum communication security, etc. Bell-state analyzer (BSA) is a crucial prerequisite for a number of quantum communication protocols. We propose an approach for completely and deterministically distinguishing a set of arbitrary $d$-dimensional ($d \geq 3$) Bell states via indefinite causal order (ICO). In previous schemes, bit and phase information are discriminated in succession. Exploiting the gravitational ICO as the sole resource, we propose some high-dimensional BSA schemes. Independent of the dimensions, a set of generalized Bell states are completely and deterministically discriminated by adjusting the form of the embedded local single-qudit gates within ICO switch and measuring each qudit in the $\{|0\rangle, |1\rangle, \cdots, |d-1\rangle\}$ basis. Notably, in our high-dimensional BSA process, the indefinite causal structure is not consumed. Hence a completely nondestructive high-dimensional BSA can be achieved by iterating the indefinite causal structure process for two rounds.

Figures (6)

• Figure 1: Quantum control of temporal order for constructing gravitational ICO 3-switch $\mathcal{S}^3$. (a) Alice's clock is located near the mass object with $r_A=R_1$ and $r_B=R_1+h$. Events $B_1$, $B_2$, and $A$ will occur in succession, i.e., $M_{B_1 \rightarrow B_2 \rightarrow A}$. (b) Alice's clock is located near the mass object with $r_A=R_2>R_1$ and $r_B=R_2+h$. Events $B_1$, $A$, and $B_2$ will occur in succession, i.e., $M_{B_1 \rightarrow A \rightarrow B_2}$. (c) Bob's clock is located near the mass object with $r_B=R_1$ and $r_A=R_1+h$. Events $A$, $B_1$, and $B_2$ will occur in succession, i.e., $M_{A \rightarrow B_1 \rightarrow B_2}$. $r_A$ ($r_B$) is the spatial distance between the mass and Alice (Bob).
• Figure 2: The gravitational ICO 3-switch $\mathcal{S}^3$. $U^3_{A}$, $U^3_{B_1}$, $U^3_{B_2}$, $U^3_{B_1|A}$, $U^3_{B_2|A}$, $U^3_{A|B_1}$, $U^3_{B_1|A}$, and $U^3_{B_2|A}$ are the local single-qutrit unitary operations.
• Figure 3: Quantum control of temporal order for gravitational ICO 4-switch $\mathcal{S}^4$. (a) The space-time geometry is $M_{B_1 \rightarrow B_2 \rightarrow B_3 \rightarrow A}$. (b) The space-time geometry is $M_{B_1 \rightarrow B_2 \rightarrow A \rightarrow B_3}$. (c) The space-time geometry is $M_{B_1 \rightarrow A \rightarrow B_2 \rightarrow B_3}$. (d) The space-time geometry is $M_{A \rightarrow B_1 \rightarrow B_2\rightarrow B_3}$.
• Figure 4: The gravitational ICO 4-switch $\mathcal{S}^4$. $U^4_{A}$, $U^4_{B_1}$, $U^4_{B_2}$, $U^4_{B_3}$, $U^4_{B_1|A}$, $U^4_{B_2|A}$, $U^4_{B_3|A}$, $U^4_{A|B_1}$, $U^4_{A|B_2}$, and $U^4_{A|B_3}$ are the embedded local single-ququart unitary operations.
• Figure 5: Quantum control of temporal order for gravitational ICO $d$-switch $\mathcal{S}^d$. (a) Alice's clock is located near the mass object. (b) Bob's clock is located near the mass object.