Fundamental trade-off relation in probabilistic entanglement generation

TL;DR

This work studies entanglement generation between two non-interacting qubits by coherently superposing local processes realized via a quantum switch. It derives a universal bound $P_{ ext{succ}}(1+\mathcal{C})\le 1$ linking the post-selection success probability with the generated entanglement, and shows a piecewise maximum for $\mathcal{C}$ depending on $P_{ ext{succ}}$. The analysis predicts a quasi-deterministic mode where both post-selection branches can yield maximal entanglement, and demonstrates this with explicit Hamiltonian dynamics and a concrete protocol achieving maximally entangled states under symmetry conditions. The results generalize beyond ICO to coherent-control schemes and establish a fundamental resource bound for entanglement generation via superposed non-interacting processes, with practical implications for photonic and matter-based platforms and future extensions to robustness under noise and weak interactions.

Abstract

We investigate the generation of entanglement between two non-interacting systems by synthesizing a new quantum process from the superposition of distinct processes characterized by local-only operations. Our analysis leads to the derivation of a universal trade-off relation, $P_{\text{succ}}(1+\mathcal{C})\le1$, that fundamentally bounds the success probability ($P_{\text{succ}}$) and the generated entanglement (concurrence $\mathcal{C}$). The derivation of this trade-off relation is inspired by indefinite causal order, but applies for a broader class of quantum processes. Next, we show that the mathematical structure of this bound predicts the existence of a "quasi-deterministic" mode of operation, a surprising phenomenon which we then confirm with concrete entanglement generation protocols, where a maximally entangled state is guaranteed to be produced. In this mode of operation, both outcomes of the post-selection measurement on the auxiliary control system result in a maximally entangled state of the target system. Furthermore, we demonstrate how this general principle can be realized using a quantum switch, which leverages an indefinite causal order as a physical resource, and explore the rich variety of dynamical behaviors governed by the universal trade-off. Our results establish a general principle for entanglement generation with superposition of quantum processes and introduce a novel way of controlling entanglement generation.

Figures (7)

• Figure 1: Schematic of the different architectures discussed in the work for entanglement generation. (a) When the control system (here a qubit) is in the state $\ket{0}^C$, the target qubits, $\mathcal{Q}^A$ and $\mathcal{Q}^B$, undergo local process $\mathcal{M}$ followed by process $\mathcal{N}$. (b) When the control system is in the orthogonal state $\ket{1}^C$, the reverse causal order, $\mathcal{N}$ followed by $\mathcal{M}$, is applied. (c) The quantum switch realizing ICO. If the target qubits are initialized in a separable state. The control system, prepared in a superposition, coherently controls the order of operations. A final measurement on the control system performs the post-selection. The resulting state of the target qubits can be entangled even though $\mathcal{Q}^A$ and $\mathcal{Q}^B$ do not directly interact. (d) A simplified diagram illustrating how the superposition of two distinct, non-interacting evolution paths gives rise to coherent control schemes.
• Figure 2: Entanglement distribution between the two post-selection branches for three different Hamiltonian dynamics. The top row shows the $(\mathcal{C}_-, \mathcal{C}_+)$ relationship generated by sampling the evolution at random times up to time $t_\text{short}=25$. The bottom row shows the corresponding short-time evolution of $\mathcal{C}_-$ (blue) and $\mathcal{C}_+$ (yellow). The dynamics clearly exhibit distinct patterns, especially the "quasi-deterministic" feature in the middle. This intriguing feature in (b) where two branches of evolution reach maximal concurrence simultaneously at a specific time, is highlighted by the star mark. The parameters for (a), (b) and (c) correspond to the Case 1, 2, and 3 defined in the text.
• Figure 3: Long-time state space coverage in the $(\mathcal{C}_-, \mathcal{C}_+)$ plane for six Hamiltonian systems. Each one shows the distribution of points generated by uniformly sampling the evolution at 2500 of random times up to $T_\text{max}=1500$, illustrating the distinct long-time behavior of each system. The parameters for (a) to (f) are selected as $\omega_A=1.0, \omega_B=1.0, c_{N,B}^X=1.0$ being fixed, while $c_{M,A}^X$ takes the values of $0.1, 0.2, 0.6, 1.0, 2.0, 4.0$, respectively.
• Figure 4: (a) Short-time dynamical trajectories in the $(\mathcal{C}_-, P_{\text{succ}})$ plane for the three exemplary Hamiltonian dynamics (A, B, C) defined as: $A:\omega_A=1.0, \omega_B=1.0, c_{M,A}^X=1.0, c_{N,B}^X=1.0$, up to time $t_\text{short}=20$, $B:\omega_A=1.0, \omega_B=5.0, c_{M,A}^X=1.0, c_{N,B}^X=1.0$, up to time $t_\text{short}=5$, and $C:\omega_A=1.0, \omega_B=1.0, c_{M,A}^X=1.0, c_{N,B}^X=1.5, c_{N,B}^Y=1.0$, up to time $t_\text{short}=5$. These confirms they evolve within the allowed region. (b) Demonstration of the trade-off boundary. Each point represents the outcome of a single trial with Haar-random local unitary operators. The distribution densely populates the entire allowed region and no point violates the theoretical boundary, that is the dashed blue line.
• Figure 5: Long-time state space coverage in the $(\mathcal{C}_-, P_{\text{succ}})$ plane for the three examples in Fig. \ref{['fig:universality']} (a). Each panel shows the distribution of points generated by sampling the evolution at random times up to $T_\text{max}=9000$, with 7000 uniform randomly sampled points. The distinct shapes of the distributions illustrate the long-time behavior of each system within the universal boundary of the dashed blue line. The Hamiltonian parameters for panel (a), (b), and (c) are defined as: $A:\omega_A=1.0, \omega_B=1.0, c_{M,A}^X=1.0, c_{N,B}^X=1.0$, $B:\omega_A=1.0, \omega_B=5.0, c_{M,A}^X=1.0, c_{N,B}^X=1.0$, and $C:\omega_A=1.0, \omega_B=1.0, c_{M,A}^X=1.0, c_{N,B}^X=1.5, c_{N,B}^Y=1.0$.