bibkey: liuqiudahlstenvedral2025lighttouch
authors: Xiangjing Liu; Yixian Qiu; Oscar Dahlsten; Vlatko Vedral
year: 2025
title: "Quantum causal inference with extremely light touch"
doi: 10.1038/s41534-024-00956-0
url: https://arxiv.org/abs/2303.10544v4
claim: "For coarse-grained measurements at two times, the pseudo-density matrix of a channel with Choi-Jamiolkowski matrix M is the Jordan product R = (M rho + rho M)/2 with rho = rho_1 tensor identity, and the PDM negativity f(R) = ||R||_tr - 1 vanishes for every input when the bipartite channel does not signal from B to A. The paper conjectures that every channel signalling from B to A admits a pure product input state with f(R_{B1A2}) > 0, and proves this for two-qubit unitary evolutions."
strata_touched:
  - D5/S3/Quantum/QuantumChannels/SignallingPureProductNegativity
license: citation-only
triage: anchor

Liu, Qiu, Dahlsten and Vedral, Quantum causal inference with extremely light touch

X. Liu, Y. Qiu, O. Dahlsten and V. Vedral, arXiv:2303.10544v4 [quant-ph]; npj Quantum Information 11, 54 (2025). Quotations are from the v4 TeX source, with the source’s macros \Tr, \mathbbm{1} and \map (trace, identity and calligraphic map) kept as written.

The causal monotone, Eq. (eq: causmono):

$$f(R):=||R|{|}_{tr}-1=\mathrm{Tr}\sqrt{R{R}^{†}}-1.$$ If $R$ has negativity, $f(R)>0$.

The Choi-Jamiolkowski matrix, Eq. (eq: CHOI):

$$M_{12}:=\sum _{i,j=0}^{2^n-1}|i⟩{⟨j|}^T\otimes {\mathcal{M}}_{2|1}\left(|i⟩⟨j|\right),$$

Theorem thm: 2timesPDM, the two-time pseudo-density matrix under coarse-grained measurements:

$$R_{12}=\frac{1}{2}(M_{12}\rho +\rho M_{12}),$$ where $\rho :={\rho }_1\otimes \text{mathbbm}{1}_2$.

Theorem thm:nullneg (null PDM negativity for semicausal channels):

If a quantum channel $\mathcal{P}$ does not allow signaling from $B$ to $A$, then, for any state ${\rho }_{A_1{B}_1}$ at time $t_1$, the PDM $R_{B_1{A}_2}$ is positive semidefinite and the PDM negativity $f(R_{B_1{A}_2})=0$.

The conjecture, after that theorem:

We moreover conjecture that if the channel is signalling from B to A, we can always find a pure {\em product} input state such that $f(R_{B_1{A}_2})>0$. We prove this conjecture for a quite general case of 2-qubit unitary evolutions in the Supplementary Information.

Verified locator

Source: https://arxiv.org/abs/2303.10544v4

• DOI: https://doi.org/10.1038/s41534-024-00956-0 (npj Quantum Information 11(1), article 54; published 29 March 2025; title, authors, journal, volume, issue and article number from Crossref).
• URL: https://arxiv.org/abs/2303.10544v4. The e-print member arxiv.tex has 852 lines and md5 fa6102d805431c16f176e32895ce216c. Locations:
  • Eq. eq: causmono, l. 143–145;
  • Eq. eq: CHOI, l. 161–164;
  • Theorem thm: 2timesPDM, l. 166–173;
  • the semicausal channels, l. 216;
  • Theorem thm:nullneg, l. 226–228;
  • the conjecture, l. 231.
• The arXiv record lists four versions (v1 posted 19 March 2023) and the journal reference “npj Quantum Information 11, 54 (2025)” (arXiv API metadata).