bibkey: siudzinska2020pauligeometry
authors: Katarzyna Siudzińska
year: 2020
title: "Geometry of generalized Pauli channels"
doi: 10.1103/PhysRevA.101.062323
url: https://arxiv.org/abs/2002.04657v2
claim: "The paper studies the geometry of generalized Pauli channels built from mutually unbiased bases, with the Hilbert-Schmidt line and volume elements of their Choi states. For N = d + 1 bases it computes, for d = 2, 3, 4, 5, the volumes of completely positive maps relative to a box of positivity estimates, of channels with nonnegative eigenvalues relative to all channels, and of entanglement-breaking channels relative to those, and conjectures closed formulas for all d."
strata_touched:
  - D5/S3/QuantumChannels/GeneralizedPauliVolumeRatios
license: citation-only
triage: anchor

Siudzińska, Geometry of generalized Pauli channels

K. Siudzińska, arXiv:2002.04657v2 [quant-ph]; Physical Review A 101(6) (2020), 062323. Quotations are from the v2 TeX source.

For $N=d+1$ mutually unbiased bases the channel eigenvalues are ${\lambda }_1,\dots ,{\lambda }_{d+1}$, with volume element $\mathrm{d}V={\left(\frac{\sqrt{d-1}}{d}\right)}^{d+1}{\prod }_{\alpha =1}^{d+1}\mathrm{d}{\lambda }_{\alpha }$. The positivity estimate is

$$-\frac{1}{d-1}\le {\lambda }_{\alpha }\le 1.$$

Appendix A gives the complete positivity region:

$$-\frac{1}{d-1}\le \sum _{\alpha =1}^{d+1}{\lambda }_{\alpha }\le 1+d\underset{\alpha }{\min }{\lambda }_{\alpha }.$$

The paper defines the channels with ${\lambda }_{\alpha }\ge 0$ as those “achievable with the time-local generators”, with volume $V_G(d)$, and states:

If ${\lambda }_{\alpha }\ge 0$, then ${\sum }_{\alpha =1}^{d+1}{\lambda }_{\alpha }\le 1$ is the necessary and sufficient condition for breaking the entanglement

After exact computations for $d=2,3,4,5$ it states three conjectures:

and conjecture that the general formula reads $$\frac{V_{CP}(d)}{V_P(d)}=\frac{d}{(d+1)!}.$$

and conjecture that the general formula reads $$\frac{V_G(d)}{V_{CP}(d)}=\frac{d^2-1}{d^2}{\left(\frac{d-1}{d}\right)}^d.$$

from which we conjecture that $$\frac{V_{EB}(d)}{V_G(d)}=\frac{1}{d+1}.$$

Verified locator

Source: https://arxiv.org/abs/2002.04657v2

• DOI: https://doi.org/10.1103/PhysRevA.101.062323 (Physical Review A 101(6), 062323; published 18 June 2020; title, author, journal, volume, issue and article number from Crossref).
• URL: https://arxiv.org/abs/2002.04657v2. The e-print is a gzipped tar archive; its member GGPC_volume5.tex has 569 lines and md5 fd8c2bae5bceeba366de934044c2363f. Locations:
  • Section “Relative volumes of quantum channels”, l. 180; the positivity estimate and $V_P(d)$, l. 210–228;
  • Subsection “$N=d+1$ and $N=d$ mutually unbiased bases”, l. 231; the three conjectures, l. 243–246, l. 276–280, l. 302–305;
  • Appendix A, l. 359–364.
• The arXiv record lists v1 (11 February 2020) and v2 (15 April 2020), with the journal reference “Phys. Rev. A 101, 062323 (2020)” (arXiv API metadata).