Entanglement-Breaking Index in Quantum Channels

• Entanglement-breaking index is defined as the minimum number of channel iterations needed to completely remove entanglement between a quantum system and any ancilla.
• It utilizes properties such as the separability of the Choi matrix and Kraus representations, with bounds derived from spectral gap analysis and Jordan block sizes.
• Variants including direct, unitary-filtered, and general-filtered indices provide nuanced insights into channel decoherence and support investigations like the PPT-squared conjecture.

The entanglement-breaking index (EB index) quantifies the resilience of quantum channels to compositional noise by measuring the minimum number of channel iterations needed to destroy all entanglement between the system and any ancilla. Specifically, for a quantum channel Φ acting on $M_d$, the EB index is the smallest integer $n$ such that $\Phi^n$ is entanglement breaking—that is, for every $d'$ and all bipartite input states, $(\Phi^n \otimes \mathrm{Id}_{d'})(\rho)$ is separable. The associated structure and asymptotics form a core part of modern quantum information theory and have implications for noisy channel dynamics, separability criteria, and conjectures such as PPT-squared.

1. Definition and Foundational Properties

A channel $\Phi: M_d \to M_d$ is entanglement breaking (EB) if $(\Phi \otimes \mathrm{Id})(\rho)$ is separable for every ancilla and arbitrary bipartite $\rho$. For a completely positive trace-preserving (CPTP) map, the EB property is equivalently characterized by the separability of its Choi matrix $$C_\Phi = \sum_{i,j=1}^{d} E_{ij} \otimes \Phi(E_{ij})$$ or the existence of a Kraus-Holevo form $\Phi(X) = \sum_k \mathrm{Tr}(X R_k) Q_k$ with $n$0 and $n$1.

The EB index, also known as the index of separability, is

$n$2

with $n$3 if no such $n$4 exists. This definition extends to the continuous time case, where for a quantum dynamical semigroup $n$5, the EB time is

$n$6

The EB index serves as a metric for the time-scale or sequential steps required to erase all entanglement under iterated dynamics (Lami et al., 2014, Rahaman et al., 2018, Hanson et al., 2019, Sakuldee et al., 2022).

2. Classes of Indices and Filtering Variants

Beyond direct iteration, Lami & Giovannetti introduced a lattice of EB indices, reflecting the possibility of interspersed filtering operations between channel applications:

• Direct Index $n$7: Minimal $n$8 with $n$9.
• Unitary-filtered Index $\Phi^n$0: Minimal $\Phi^n$1 such that for all $\Phi^n$2, $\Phi^n$3.
• General-filtered Index $\Phi^n$4: Minimal $\Phi^n$5 such that for all CPTP $\Phi^n$6, $\Phi^n$7.

It holds that $\Phi^n$8, and these indices may differ, particularly for $\Phi^n$9 (Lami et al., 2014). For the depolarizing channel, all three indices coincide.

3. Structural Results and Density

For unital PPT channels—channels that are both completely positive and co-completely positive (i.e., composition with transpose is CP)—every such channel has finite EB index: repeated application drives the system to an EB regime after finitely many rounds. This is shown by decomposing the channel into components over abelian C*-subalgebras stabilized by the multiplicative domain, leveraging results such as Wolf's theorem for primitive channels and the Gurvits–Barnum separable ball criterion.

Furthermore, the class of unital channels with finite EB index is norm-dense (even cb-norm dense) within the set of all unital channels: every unital channel can be approximated arbitrarily closely by channels that become EB after a finite number of iterations (Rahaman et al., 2018).

4. Quantitative Bounds: Explicit and Spectral Estimates

For primitive, faithful quantum channels (those admitting a full-rank invariant state), explicit upper bounds for the EB index can be computed. The convergence rate to the decohered, fixed-phase map is governed by the spectral gap $d'$0 of a generalized generator and the size $d'$1 of the largest Jordan block. The structural result states:

$d'$2

with $d'$3 the invariant state (Hanson et al., 2019).

For channels generated by Lindblad operators (continuous semigroups), quantum speed limit (QSL) arguments produce spectral lower bounds:

$d'$4

where $d'$5 are the nonzero eigenvalues (on the traceless subspace) of the Lindblad generator $d'$6 (Sakuldee et al., 2022). This approach generalizes Mandelstam–Tamm-type bounds to nonunitary dynamics.

5. Examples and Exact Calculations

Channels of interest include:

• Depolarizing channel: For $d'$7, the EB index is

$d'$8

and no filter can improve on this (Lami et al., 2014).

• Werner-type channels in $d'$9: For suitable parameter values, $(\Phi^n \otimes \mathrm{Id}_{d'})(\rho)$0, $(\Phi^n \otimes \mathrm{Id}_{d'})(\rho)$1, and $(\Phi^n \otimes \mathrm{Id}_{d'})(\rho)$2, demonstrating the unbounded filtering gap in high dimension.
• Non-PPT Schur channels: There exist non-PPT channels E with finite $(\Phi^n \otimes \mathrm{Id}_{d'})(\rho)$3, illustrating that PPT is not necessary for eventual EB status (Rahaman et al., 2018).
• Qubit amplitude-damping: For the corresponding Lindblad generator with rate $(\Phi^n \otimes \mathrm{Id}_{d'})(\rho)$4, $(\Phi^n \otimes \mathrm{Id}_{d'})(\rho)$5, and direct calculation shows the actual breaking time is $(\Phi^n \otimes \mathrm{Id}_{d'})(\rho)$6 (Sakuldee et al., 2022).

Channel Type	Direct Index $(\Phi^n \otimes \mathrm{Id}_{d'})(\rho)$7	Filtering Gap
Depolarizing	$(\Phi^n \otimes \mathrm{Id}_{d'})(\rho)$8	None ($(\Phi^n \otimes \mathrm{Id}_{d'})(\rho)$9)
Werner ($\Phi: M_d \to M_d$0)	$\Phi: M_d \to M_d$1	Unitary $\Phi: M_d \to M_d$2, General $\Phi: M_d \to M_d$3
Schur (non-PPT)	Finite ($\Phi: M_d \to M_d$4-dependent)	Possible
Amplitude-damping	See above	N/A

6. Asymptotic Theory and Structural Decomposition

A channel is asymptotically entanglement breaking if a subsequence of its powers converges (point-weakly) to an EB channel. For unital channels, this occurs precisely when the stabilized multiplicative domain is abelian. All unital irreducible channels are asymptotically EB. However, examples exist (e.g., certain channels on $\Phi: M_d \to M_d$5) where the EB index is infinite but the map is asymptotically EB, i.e., no finite power is EB, but the sequence converges to an EB map in the limit (Rahaman et al., 2018).

7. Connections to PPT-Squared Conjecture and Open Problems

The PPT-squared conjecture states that for every PPT channel $\Phi: M_d \to M_d$6, the composition $\Phi: M_d \to M_d$7 is EB. Structural results show that for unital or faithful PPT channels, after finitely many self-compositions, the channel becomes EB, supporting the conjecture up to a bounded exponent. The general case reduces to the study of faithful primitive PPT channels. Open research directions include removing dimension-dependent growth in upper bounds, tightening constants in robustness-based estimates, and fully resolving the PPT-squared conjecture's sharpness for all PPT channels (Rahaman et al., 2018, Hanson et al., 2019).

8. Filtering Protocols Beyond Local Channels

Generalized protocols—incorporating local channels, LOCC, separable CPTP maps, and multistage adaptive processes—induce distinct integer indices $\Phi: M_d \to M_d$8. In various scenarios, such as qubit depolarizing noise, even the most general protocol (local multistage separable with probabilistic success) cannot outperform the direct index:

$\Phi: M_d \to M_d$9

implying the optimality of the direct iterative sequence (Lami et al., 2014).

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Fundamental references include Lami & Giovannetti (Lami et al., 2014), Rahaman–Jaques–Paulsen (Rahaman et al., 2018), and further structural and bound-oriented studies by Sutter–Ticozzi–Wolf (Hanson et al., 2019) and Facchi et al. (Sakuldee et al., 2022).