bibkey: burniston2020necessary
authors: John Burniston; Michael Grabowecky; Carlo Maria Scandolo; Giulio Chiribella; Gilad Gour
year: 2020
title: "Necessary and sufficient conditions on measurements of quantum channels"
doi: 10.1098/rspa.2019.0832
url: https://arxiv.org/abs/1904.09161v5
claim: "In finite dimensions, a CPP, CPTNI-preserving supermap is completable to a superchannel exactly when it is completely CPTNI-preserving; a jointly normalized super-instrument has a common CPTP preprocessor and a CPTNI postprocessing instrument."
strata_touched: []
license: citation-only
triage: anchor

Probabilistic channel transformations and their common normalization

The source is arXiv:1904.09161v5, published in Proceedings of the Royal Society A 476, 20190832. The definitions and results below use the v5 numbering. They are literature results, with the stated specializations; no new theorem, priority or Lean verification is claimed.

Finite-dimensional hypotheses and completion

Write $A=(A_0,A_1)$, with $A_0$ the input and $A_1$ the output of the inserted map, and similarly $B=(B_0,B_1)$. All Hilbert spaces are finite-dimensional. A supermap ${\Theta }^{A\to B}$ is a linear map from linear maps $A_0\to A_1$ to linear maps $B_0\to B_1$.

• CPP means completely CP-preserving: for every ancillary pair $R=(R_0,R_1)$ and every bipartite CP map ${\mathcal{E}}^{RA}$, $({\mathrm{id}}^R\otimes \Theta )[{\mathcal{E}}^{RA}]$ is CP (§2, Eq. (2.1)).

• CPTNI-preserving means CPP and $\mathrm{Tr}\Theta [\mathcal{N}](\rho )\le 1$ for every CPTP ${\mathcal{N}}^{A_0\to A_1}$ and every density matrix ${\rho }^{B_0}$ (§2, Eq. (2.2)). For CPP supermaps, testing CPTP inserted maps is equivalent to sending every CPTNI inserted map to a CPTNI map (Appendix A).

• Completely CPTNI-preserving additionally requires

  $$\mathrm{Tr}({\mathrm{id}}^R\otimes \Theta )[{\mathcal{N}}^{RA}]({\rho }^{R_0{B}_0})\le 1$$

  for every ancillary pair $R$, bipartite CPTP map ${\mathcal{N}}^{RA}$ and density matrix ${\rho }^{R_0{B}_0}$ (§3, Eq. (3.1)). The inserted channel and the test input may have joint correlations.

Theorem 4.1, proved as Theorem D.1 in Appendix D, states that a CPTNI-preserving $\Theta$ admits another CPTNI-preserving ${\Theta }_{\mathrm{f}}$ with $\Theta +{\Theta }_{\mathrm{f}}$ a superchannel if and only if $\Theta$ is completely CPTNI-preserving. A superchannel is a CPP supermap taking CPTP maps to CPTP maps; Appendix A explains why this deterministic condition already holds completely.

For a finite collection of CPP branches $\{{\Theta }_x\}$, apply the completion criterion to the aggregate ${\Theta }_{\mathrm{S}}={\sum }_x{\Theta }_x$. If this aggregate is completely CPTNI-preserving, a complementary failure branch completes the collection to one super-instrument. Conversely, a subcollection of an actual super-instrument has a completely CPTNI-preserving aggregate. Separate completability of each branch does not establish joint normalization: for example, two copies of a superchannel are each completable but their sum has trace two on normalized tests.

For an already jointly normalized finite family ${\sum }_x{\Theta }_x=\Phi$, with $\Phi$ a superchannel, §5, Eq. (5.1), and Proposition E.1 give a finite memory $E$ and

$${\Theta }_x[\mathcal{N}]={\Gamma }_{\mathrm{post},x}\circ (\mathcal{N}\otimes {\mathrm{id}}_E)\circ {\Gamma }_{\mathrm{pre}},$$

where ${\Gamma }_{\mathrm{pre}}:B_0\to A_{0}E$ is CPTP, each ${\Gamma }_{\mathrm{post},x}:A_{1}E\to B_1$ is CPTNI, and ${\sum }_x{\Gamma }_{\mathrm{post},x}$ is CPTP. The same preprocessor works for every outcome of this one instrument, including its completion branch. The formula does not identify separately chosen preprocessors of unrelated branches.

Ordinary preservation is insufficient

Appendix B.1 uses the unnormalized, input-first Choi operator

$$J_{\mathcal{N}}=({\mathrm{id}}_{A_0}\otimes \mathcal{N})(|\Omega ⟩⟨\Omega |),|\Omega ⟩=\sum _a|a,a⟩.$$

Thus $\mathcal{N}$ is CPTP exactly when $J_{\mathcal{N}}⪰0$ and ${\mathrm{Tr}}_{A_1}{J}_{\mathcal{N}}=I_{A_0}$; its trace is $\dim A_0$, not one. For a supermap Choi operator $J_{\Theta }^{AB}$, CPP is $J_{\Theta }⪰0$. Put $Q={\mathrm{Tr}}_{B_1}{J}_{\Theta }$. Equations (4.1)–(4.2) compare

$$\begin{array}{rl}\mathrm{Tr}[Q(J_{\mathcal{N}}\otimes {\rho }^{B_0}{)}^{\mathsf{T}}]& \le 1,\\ \mathrm{Tr}[Q{M}^{\mathsf{T}}]& \le 1\text{for every }M⪰0,{\mathrm{Tr}}_{A_1}M=I_{A_0}\otimes {\rho }^{B_0}.\end{array}$$

The first is ordinary CPTNI preservation; the second is complete CPTNI preservation. In these tests $\mathcal{N}$ ranges over CPTP maps and ${\rho }^{B_0}$ over density matrices. Every transpose here is the full matrix transpose in the fixed product basis.

In Eq. (2.4) and Appendix C all four systems are qubits. The example has

$$J_{\Theta }=I_{A_0}\otimes |{\psi }^{-}⟩⟨{\psi }^{-}{|}_{A_1{B}_0}\otimes I_{B_1}/2,|{\psi }^{-}⟩=(|01⟩-|10⟩)/\sqrt{2}.$$

It is positive and passes every product test in the first inequality. Taking $M=Q^{\mathsf{T}}$ gives the allowed marginal $I_{A_0}\otimes I_{B_0}/2$, but $\mathrm{Tr}(Q{M}^{\mathsf{T}})=2>1$. Hence it cannot be a branch of a physical channel super-instrument, despite being CPP and ordinarily CPTNI-preserving.

This distinction concerns transformations of maps. For an ordinary state instrument, CP branches ${\mathcal{E}}_x$ are jointly normalized when ${\sum }_x{\mathcal{E}}_x$ is CPTP. A CP trace-nonincreasing state map can be completed to such an instrument; it is not subject to an additional independent higher-order preservation test.

Scalar outputs, grouped slots and original success

With $B_0=B_1=\mathbb{C}$, Appendix B.1, Eq. (B.2), gives

$$p_x(\mathcal{N})=\mathrm{Tr}(J_{{\Theta }_x}{J}_{\mathcal{N}}^{\mathsf{T}})=\mathrm{Tr}(T_x{J}_{\mathcal{N}}),T_x=J_{{\Theta }_x}^{\mathsf{T}}⪰0.$$

For a normalized one-slot instrument the shared preparation has an input marginal $\sigma$, and ${\sum }_x{T}_x={\sigma }^{\mathsf{T}}\otimes I_{A_1}$. This is tester normalization, not the usual POVM normalization on the Choi space. It allows an arbitrary finite probe memory; it does not assume supplied normalized Choi states.

A parallel group of $N$ memoryless channel uses may be treated as one inserted channel ${\mathcal{N}}^{\otimes N}$ on the grouped input and output spaces. After a fixed tensor permutation, its Choi operator is $J_{\mathcal{N}}^{\otimes N}$, so the same pairing applies. This grouped-slot statement supplies no interleaved adaptive $N$-slot realization. The fixed-order adaptive application is given separately in the amplitude-damping problem dossier, using the existing quantum-network source.

The native state-conversion contract in VIII, §§248–250 supplies same-source states, not callable unknown channels. Its finite success branches are ordinary state maps. If the erased success map is $\mathcal{E}={\sum }_{h\in \mathrm{S}}{\mathcal{E}}_h$, then

$$s_{\mathcal{E}}(q)=\mathrm{Tr}\mathcal{E}({\rho }_q^{\otimes k}),{\rho }_{\mathrm{out},q}=\frac{\mathcal{E}({\rho }_q^{\otimes k})}{s_{\mathcal{E}}(q)},\mathrm{Pr}(\text{original success})=P_q^k{s}_{\mathcal{E}}(q).$$

The normalized expression is defined only for positive success. For an already allowed native map, an independent fixed coin retaining a fraction $c\in (0,1]$ multiplies the original success by $c$ and preserves this conditional output. Branch-dependent scaling can change the erased mixture; division by source-dependent success is conditioning, not a physical installation.

Neither the supermap completion theorem nor a state-map completion provides number-preserving LOCC realization, a preparation-to-channel conversion, an auxiliary resource or a polynomial original yield. The five-mode decoder, §§15–16 returns a classical state description modulo one common conjugation; it is not a channel-query, preparation or continuation interface. The simultaneous native approximation and yield assertion (248.4) remains unresolved.

Locator

• DOI: https://doi.org/10.1098/rspa.2019.0832
• Versioned source: https://arxiv.org/abs/1904.09161v5
• Original PDF: https://arxiv.org/pdf/1904.09161v5
• Definitions: §2, Eqs. (2.1)–(2.3), PDF p. 2; §3, Eq. (3.1), p. 3.
• Completion: §4, Theorem 4.1, pp. 3–4; Appendix D, Theorem D.1.
• Realization: §5, Eq. (5.1), p. 4; Appendix E, Proposition E.1 and Eqs. (E.3)–(E.4).
• Choi conventions and scalar specialization: Appendix B.1, Eqs. (B.1)–(B.3), pp. 10–11.
• Counterexample: §2, Eq. (2.4); §4, Eqs. (4.1)–(4.2); Appendix C, pp. 13–15.