bibkey: carmeliheinosaaritoigo2025postponing
authors: Claudio Carmeli; Teiko Heinosaari; Alessandro Toigo
year: 2025
title: "Postponing the choice: advantage of deferred measurements in quantum information processing"
doi: 10.1088/1367-2630/adf87c
url: https://arxiv.org/abs/2502.19871v1
claim: "The paper determines the compatibility region of a noisy sharp meter Q_s and a depolarizing channel I_t, describes the joint instruments attaining its boundary, shows that on the upper boundary the instrument tilde-J^(t) is the unique W-covariant joint instrument, and asks whether other joint instruments exist there."
strata_touched:
  - D5/S3/Quantum/Measurement/DeferredInstrumentUniqueness
license: citation-only
triage: anchor

Postponing the choice: advantage of deferred measurements in quantum information processing

C. Carmeli, T. Heinosaari and A. Toigo, arXiv:2502.19871v1 (2025-02-27); New J. Phys. 27, 094501 (2025). Quotations are from the v1 TeX source.

The noisy meters (eq:QsPt):

\Qo_s(x) = s,\kb{\varphi_x}{\varphi_x} + (1-s)\tfrac{1}{d}\id ,,\qquad \Po_t(y) = t,\kb{\psi_y}{\psi_y} + (1-t)\tfrac{1}{d}\id,,

The depolarizing channel (eq:Ir):

I_r = r I + (1-r)\tfrac{1}{d}\id{\rm tr}

The coefficients (eq:ab_bis):

a_k(r) = \frac{1}{\sqrt{d^k}}\left[\sqrt{r(d^k-1)+1} - \sqrt{1-r}\right],,\qquad b(r) = \sqrt{1-r},.

Joint instruments (eq:joint_J):

\ii^\dag(\cdot,\id) = \Qo_s,,\qquad \sum_x \ii(x,\cdot) = I_t

An instrument $\text{ii}$ which satisfies \eqref{eq:joint_J} is a {\em joint instrument} for ${\text{Qo}}_s$ and $I_t$

The compatibility region (thm:main_neg, eq:QI_comp_neg), upper boundary:

d^2 s \leq (d^2 - 2) (1 - t) + 2\sqrt{(1 - d^2) t^2 + (d^2 - 2) t + 1}

The upper-boundary instrument (eq:tildeJ):

\tilde{\ii}^{(t)}(x,\rhoo) = \frac{1}{d} \big{ b(t)^2 \tr{(\rhoo\Qo(x))} \id + a_2(t)b(t) (\rhoo\Qo(x) + \Qo(x)\rhoo) + a_2(t)^2 \rhoo \big}

The question:

We conclude that, when the point $(s,t)$ attains the second equality in \eqref{eq:QI_comp_neg}, the instrument ${\widetilde{\text{ii}}}^{(t)}$ is actually the unique $W$-covariant joint instrument for ${\text{Qo}}_s$ and $I_t$. It is an open question whether in this case there exist other (necessarily not $W$-covariant) joint instruments.

Verified locator

Source: https://arxiv.org/abs/2502.19871v1

• URL: https://arxiv.org/abs/2502.19871v1 (source CHT_postponing_choice.tex, md5 260d15a3cfb6a4b474b99d950e675b89):
  • the noisy meters, l. 622–623;
  • the depolarizing channel, l. 656–657;
  • the coefficients, l. 738–739;
  • joint instruments, l. 813–816;
  • the compatibility theorem, l. 1039–1046;
  • the instrument, l. 1198–1203;
  • the question, l. 1532.
• DOI: 10.1088/1367-2630/adf87c (https://doi.org/10.1088/1367-2630/adf87c).