Entanglement-breaking channel — measure-and-prepare maps: separable Choi matrix ⇔ destroys all entanglement

Statement

Definition. A quantum channel (CPTP map) $\Phi:\mathcal L(\mathcal H_A)\to\mathcal L(\mathcal H_B)$ is entanglement-breaking (EB) if, for *every* ancilla and every input state $\rho$ on $\mathcal H_A\otimes\mathcal H_R$, the output $$ (\Phi\otimes\mathrm{id}_R)(\rho)\ \text{is separable.} $$ That is: no matter how entangled the input is with a reference system, the channel's output is unentangled with that reference. (Source: Horodecki–Shor–Ruskai 2003, arXiv:quant-ph/0302031.)

Equivalent characterizations (Horodecki–Shor–Ruskai 2003). For a channel $\Phi$ the following are equivalent:

1. (Breaks all entanglement) $(\Phi\otimes\mathrm{id})(\rho)$ is separable for all bipartite $\rho$ — the definition. 2. (Separable Choi matrix) The Choi matrix $J(\Phi)=(\Phi\otimes\mathrm{id})(|\Omega\rangle\langle\Omega|)$ (see Choi–Jamiołkowski isomorphism — state–channel duality: a map is CP iff its Choi matrix is PSD) is a separable operator on $\mathcal H_B\otimes\mathcal H_{A'}$. It suffices to test the *single* maximally entangled input, because $|\Omega\rangle\langle\Omega|$ is the "most entangled" state — separability of its image forces separability of every image. 3. (Holevo / measure-and-prepare form) $\Phi$ can be written as $$ \Phi(\rho)=\sum_{k} R_k\,\operatorname{Tr}(F_k\,\rho),\qquad F_k\ge0,\ \sum_k F_k=\mathbb 1,\ R_k\ \text{density matrices,} $$ i.e. $\Phi$ measures the input with the POVM $\{F_k\}$ and, conditioned on outcome $k$, prepares a fixed state $R_k$. All quantum coherence between input and any reference is destroyed at the classical measurement step. One may always take the $R_k$ to be pure and the $F_k$ rank-one. (Source: HSR 2003, "canonical form introduced by Holevo," WebFetch-confirmed.)

The chain (1) ⇔ (2) ⇔ (3) is the content of the HSR paper; "entanglement-breaking" and "measure-and-prepare" are synonyms.

Facts

- **Its role in PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4: EB is the target conclusion.** The PPT-squared conjecture states that the composition $\Phi_2\circ\Phi_1$ of any two PPT channels (PPT map — a channel that is CP and completely co-positive: Choi matrix PSD with positive partial transpose) is entanglement-breaking. So "EB" is exactly the property one must certify for the composite — and by characterization (2) that means proving $J(\Phi_2\circ\Phi_1)$ is *separable*. Certifying separability is the hard direction; refuting it (finding one channel whose Choi matrix is entangled) is cheap via an Entanglement witness — a Hermitian W with Tr(Wσ)≥0 on all separable states but Tr(Wρ)<0 on some entangled ρ, which is why a counterexample would be verification-light even though the search for it is believed intractable (Jin 2020: "the genuine part of the separability problem"). - Strict place in the hierarchy: EB ⊊ PPT ⊊ CP. Every EB channel is PPT (its Choi matrix is separable, hence has positive partial transpose), so $\mathrm{EB}\subseteq\mathrm{PPT}$. The inclusion is strict: there exist PPT channels that are not EB — precisely those whose Choi matrix is PPT but *entangled* (a bound-entangled / Bound entanglement — entangled states that are PPT, hence undistillable; the rare structured regime where a PPT² counterexample must live Choi state). The gap $\mathrm{PPT}\setminus\mathrm{EB}$ is exactly the region where PPT² has content. (Source: WebSearch — "there are PPT channels that are not entanglement breaking.") - EB is a two-sided ideal under composition. If either $\Phi_1$ or $\Phi_2$ is EB, then both $\Phi_2\circ\Phi_1$ and $\Phi_1\circ\Phi_2$ are EB. (Once entanglement with a reference is destroyed by a measure-and-prepare step, no later CPTP map can restore it.) This absorbing behavior is why PPT² is naturally phrased as "two PPT steps suffice to *reach* the EB ideal," and why the related "eventually EB" results (Kennedy–Manor–Paulsen; Rahaman–Jaques–Paulsen) ask only for *some* number of self-compositions. - Zero quantum capacity, additive classical capacity. EB channels have quantum capacity $Q=0$ (they cannot transmit quantum information — a measure-and-prepare relay is classical). Shor proved their minimum output entropy and Holevo capacity are additive; equivalently, tensoring an EB channel with anything cannot super-multiply the max output $p$-norm. This makes EB channels the canonical "classical relay" endpoints of channel theory. - Convexity and extreme points. The EB CPTP maps form a convex set. Its extreme points are the classical–quantum (CQ) maps for small dimension, but HSR proved that for input dimension $d>2$ there are additional extreme EB maps that are not extreme CQ maps — a subtlety that blocks naive extreme-point arguments for PPT². (Source: HSR 2003, WebFetch-confirmed.)

Related

- PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4 — "entanglement-breaking" is the exact conclusion the PPT-squared conjecture asserts for a composition of two PPT maps; this page defines the target. - PPT map — a channel that is CP and completely co-positive: Choi matrix PSD with positive partial transpose — the strictly larger class $\mathrm{EB}\subsetneq\mathrm{PPT}$; PPT² lives entirely in the gap between them. - Choi–Jamiołkowski isomorphism — state–channel duality: a map is CP iff its Choi matrix is PSD — supplies characterization (2): EB ⇔ Choi matrix separable, the matricial form used in every proof and search. - Entanglement witness — a Hermitian W with Tr(Wσ)≥0 on all separable states but Tr(Wρ)<0 on some entangled ρ — the cheap certificate that a candidate composite is *not* EB: a witness detecting entanglement in $J(\Phi_2\circ\Phi_1)$ would refute PPT². - Bound entanglement — entangled states that are PPT, hence undistillable; the rare structured regime where a PPT² counterexample must live — the PPT-but-entangled Choi states are exactly the ones separating PPT from EB; a PPT² counterexample would be a bound-entangled-Choi channel arising as a composition of two PPT maps. - Peres–Horodecki (PPT) criterion — partial transpose positivity is necessary for separability, and sufficient only in 2×2 and 2×3 — decides separability of a Choi matrix (hence EB-ness) exactly in the $2\times2$ and $2\times3$ cases, which is why PPT² is folklore-true for qubits.