On the characterization of partially entanglement breaking and annihilating channels

=== DRAFT ===
Technical Summary: On the Characterization of Partially Entanglement Breaking and Annihilating Channels

Problem Statement
Quantum entanglement is a fundamental resource for quantum information processing tasks such as teleportation, cryptography, and channel discrimination. While the presence of entanglement is often binary (separable vs. entangled), the dimensionality of entanglement, quantified by the Schmidt number (SN), is crucial for high-dimensional protocols. States with higher Schmidt numbers offer distinct advantages in tasks like channel discrimination and quantum key distribution. However, quantum channels (noise) can degrade these resources by reducing the Schmidt number of transmitted states.

While "entanglement breaking" (EB) channels are well-characterized as those that completely disentangle a subsystem, there exists a broader class of channels that reduce the Schmidt number without necessarily destroying all entanglement. These are termed "partially entanglement breaking" (PEBT) channels. Furthermore, a distinction exists between channels that reduce entanglement across a bipartition (A|B) and those that destroy entanglement within a subsystem (B) of a larger composite system. The latter are known as "entanglement annihilating" (EA) channels. The paper addresses the need to characterize these intermediate classes—specifically $r$-partially entanglement breaking ($r$-PEBT) and $r$-partially entanglement annihilating ($r$-PEAT) channels—and to develop methods for identifying channels that preserve high-dimensional entanglement (non-resource breaking channels).

Methodology
The authors employ a resource-theoretic framework combined with operator-theoretic tools to analyze quantum channels acting on $d$-dimensional Hilbert spaces ($C^d$).

1. Definitions and Characterization:

   • $r$-PEBT: A channel $S$ is $r$-PEBT if it reduces the Schmidt number of any input state $\rho$ such that $SN[(id_A \otimes S)\rho] \leq r$.
   • $r$-PEAT: A channel $E$ acting on a subsystem $B$ is $r$-partially entanglement annihilating if $SN[E(\rho_B)] \leq r$ for all $\rho_B$. The authors distinguish between local (tensor product of individual channels) and non-local $r$-PEAT channels.
   • Choi-Jamiołkowski Isomorphism: The authors utilize the Choi matrix ($C_S$) to characterize these channels. A key result from prior literature (Chruściński et al.) is used: $S \in r\text{-PEBT}$ if and only if $SN(C_S) \leq r$.

2. Topological Analysis:

   • The paper establishes that the set of $r$-PEBT channels is both convex and compact. This is proven by showing that convex combinations of $r$-PEBT channels remain in the set and that the set contains all its limit points.
   • Similar topological properties (convexity and compactness) are established for the set of non-local $r$-PEAT channels.

3. Detection Criteria:

   • Witness Operators: To detect channels that are not $r$-PEBT (i.e., those preserving $SN > r$), the authors propose a witness-based approach. They define a scalar functional based on the diamond norm distance to the set of $r$-PEBT channels.
   • Spectral Conditions: A sufficient condition is derived using $r$-positive but not $(r+1)$-positive maps ($\Lambda$). If a channel $S$ satisfies $\text{Tr}[(id_A \otimes \Lambda)(C_S)]^2 \leq \frac{1}{d^2-1}$, then $S \in r\text{-PEBT}$.
   • Absolute Schmidt Number: For $r$-PEAT, the paper introduces the concept of "absolute Schmidt number channels" (channels where the output SN remains $\leq r$ under any non-local unitary). A sufficient condition based on the eigenvalue spectrum of the output state is provided for covariant channels.

4. Case Studies:

   • The authors apply these criteria to qutrit depolarizing and dephasing channels to determine precise parameter regimes where these channels fail to be 2-PEBT or 2-locally r-PEAT.

Key Contributions and Results

1. Characterization of $r$-PEBT:

   • Proved that the set of $r$-PEBT channels is convex and compact, enabling the use of witness operators for detection.
   • Demonstrated that the series concatenation of two $r$-PEBT channels remains $r$-PEBT.
   • Showed that the set is not closed under tensor products for $r > 1$ (unlike standard EB channels where $r=1$). Specifically, if $S_1, S_2 \in r\text{-PEBT}$, then $S_1 \otimes S_2$ may not be in $r\text{-PEBT}$ (though it is in $r^2$-PEBT).

2. Introduction of $r$-PEAT:

   • Defined and characterized non-local $r$-PEAT channels.
   • Proved that the series concatenation of non-local $r$-PEAT channels remains non-local $r$-PEAT.
   • Demonstrated that the order of concatenation matters: if $E$ is non-local $r$-PEAT and $F_B$ is an arbitrary channel, $E \circ F_B$ is non-local $r$-PEAT, but $F_B \circ E$ may not be (as $F_B$ could increase the Schmidt number).

3. Detection of Non-Resource Breaking Channels:

   • Provided explicit parameter ranges for qutrit depolarizing and dephasing channels where they act as non-partially entanglement breaking (preserving $SN=3$).
   • For the qutrit depolarizing channel $S_3(\rho) = p\rho + \frac{1-p}{3}I$, it is 2-PEBT if $0 \leq p \leq 5/8$ and non-2-PEBT (preserving $SN=3$) if $5/8 < p \leq 1$.
   • For the qutrit dephasing channel, it is 2-PEBT if $0 \leq v \leq 1/2$ and non-2-PEBT if $1/2 < v \leq 1$.

4. Set Relations:

   • Established the hierarchy: $EB \subset 1\text{-PEBT} \subseteq 2\text{-PEBT} \subseteq \dots \subseteq r\text{-PEBT}$.
   • Showed that $r\text{-PEBT} \subset 2\text{-local } r\text{-PEAT}$ (every $r$-PEBT channel is a 2-locally $r$-PEAT channel).
   • Demonstrated that the converse is false: $2\text{-local } r\text{-PEAT} \not\subset r\text{-PEBT}$.
   • Proved that $r\text{-PEBT}$ and non-local $r$-PEAT have a non-empty intersection, but neither is a subset of the other.

Significance and Claims
The paper claims to provide a comprehensive theoretical framework for understanding how quantum channels degrade the dimensionality of entanglement, moving beyond the binary view of entanglement breaking. By characterizing the topological properties (convexity and compactness) of these channel sets, the authors enable the development of efficient detection methods (witnesses) to identify channels that preserve high-dimensional entanglement resources.

The introduction of "partially entanglement annihilating channels" extends the resource theory to scenarios where entanglement is destroyed within a subsystem but not necessarily across the system. The authors emphasize that identifying "non-resource breaking" channels is imperative for reliable quantum information processing. The work lays down prescriptions to identify such channels, specifically through the derived sufficient conditions and parameter regimes for standard noise models.

The authors conclude modestly, noting that while they have provided a topological characterization, obtaining a Choi-Kraus type representation for $r$-PEAT channels remains an open problem. They also highlight that the lack of closure under tensor products for $r$-PEBT ($r>1$) motivates further investigation into whether similar properties hold for $r$-PEAT. The paper does not propose new experimental setups but rather offers the theoretical tools necessary to analyze channel capacities and resource preservation in high-dimensional systems.