The entangling power of non-entangling channels

TL;DR

The paper investigates how quantum processes that cannot create entanglement outright can nevertheless amplify existing entanglement through postselection. It introduces stochastically non-entangling maps and proves they do not increase the Schmidt number, while showing that some non-entangling maps gain entangling power via probabilistic postselection. A convex-roof construction yields a channel Schmidt number that quantifies probabilistic entanglement generation, and the dual map criterion ties non-entangling channels to witness-preserving duals. Additionally, the authors derive Bell-like inequalities that detect entangling dynamics and apply the framework to measurement channels and random-unitary channels, offering a comprehensive operational picture of entangling strength in quantum channels.

Abstract

There are processes that cannot generate entanglement but may, nevertheless, amplify entanglement already present in a system. Here, we show that a non-entangling operation can increase the Schmidt number of a quantum state only if it can generate entanglement with some non-zero probability. This is in stark contrast to the case where the parties of a quantum network are only able to control their joint state by local operations and classical communication (LOCC). There, being able to apply operations probabilistically (stochastic LOCC) does not increase the Schmidt number. Our findings show that certain non-entangling operations become entangling when selecting on specific measurement outcomes. This naturally leads us to the class of stochastically non-entangling maps, being those that cannot generate entanglement even probabilistically. Intrigued by this finding, we devise a Schmidt number for quantum channels that quantifies whether a channel can generate entanglement probabilistically. Moreover, we show that a channel is non-entangling if and only if its dual map is witness-preserving -- it takes entanglement witnesses to witnesses. Based on this finding, we derive Bell-like inequalities whose violation signals that a process generates entanglement.

Figures (4)

• Figure 1: Hierarchy of quantum channels. The set of channels (completely positive maps) comprises non-entangling maps, stochastically non-entangling maps $\mathcal{N}_\mathrm{s}$, as well as (stochastic) LOCC and separable maps. All subsets are proper subsets.
• Figure 2: Separable states $\mathcal{S}$ (dark gray) as a subset of quantum states (light gray). An optimal witness $W$ corresponds to a hyperplane tangent to $\mathcal{S}$. The dual of a non-entangling map $\Lambda$ gives a witness $\Lambda^*(W)$ which is not finer than $W$. The dual of an entangling channel $\Phi$ can yield a non-witness $\Phi^*(W)$.
• Figure 3: Entanglement test for the witness $\Lambda^*(V)$, with the measurement channel $\Lambda$ in Eq. \ref{['eq:meas-ex']}. The plot shows the minimum of $\braket{\Lambda^*(V)}_{\sigma}$ as a function of the probabilities $p$ and $q$.
• Figure 4: Entanglement test for the witness $\Lambda^*(W)$ with the random unitary channel $\Lambda$ in Eq. \ref{['eq:ex-rand-uni']}. The plot shows the minimum of $\braket{\Lambda^*(W)}_{\sigma}$ as a function of the probabilities $p$ and $q$.

Theorems & Definitions (18)

• Example 1
• Definition 1
• Proposition 1
• proof
• Theorem 1
• proof
• Definition 2
• Theorem 2
• proof
• Definition 3