Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Petz recovery map
open-in-new
<h2 id="definition">Definition</h2> <p>Suppose we have a quantum state which is described by a density operator σ {\displaystyle \sigma } and a quantum channel E {\displaystyle {\mathcal {E}}} , the Petz recovery map is defined as<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a><a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p> P σ , E ( ρ ) = σ 1 / 2 E † ( E ( σ ) − 1 / 2 ρ E ( σ ) − 1 / 2 ) σ 1 / 2 . {\displaystyle {\mathcal {P}}_{\sigma ,{\mathcal {E}}}(\rho )=\sigma ^{1/2}{\mathcal {E}}^{\dagger }({\mathcal {E}}(\sigma )^{-1/2}\rho {\mathcal {E}}(\sigma )^{-1/2})\sigma ^{1/2}.} <p>Notice that E † {\displaystyle {\mathcal {E}}^{\dagger }} is the Hilbert-Schmidt adjoint of E {\displaystyle {\mathcal {E}}} . </p><p>The Petz map has been generalized in various ways in the field of quantum information theory.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a><a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p> <h2 id="properties-of-the-petz-recovery-map">Properties of the Petz recovery map</h2> <p>A crucial property of the Petz recovery map is its ability to function as a quantum channel in certain cases, making it an essential tool in quantum information theory. </p> <ol><li>The Petz recovery map is a <a href="/facts/Completely_positive_map/GJGjwqgh">completely positive map</a>, since (i) sandwiching by the positive semi-definite operator E ( σ ) − 1 / 2 ( ⋅ ) E ( σ ) − 1 / 2 {\displaystyle {\mathcal {E}}(\sigma )^{-1/2}(\cdot ){\mathcal {E}}(\sigma )^{-1/2}} is completely positive; (ii) E † {\displaystyle {\mathcal {E}}^{\dagger }} is also completely positive when E {\displaystyle {\mathcal {E}}} is completely positive; and (iii) sandwiching by the positive semi-definite operator σ 1 / 2 ( ⋅ ) σ 1 / 2 {\displaystyle \sigma ^{1/2}(\cdot )\sigma ^{1/2}} is completely positive.</li> <li>It's also clear that P σ , E {\displaystyle {\mathcal {P}}_{\sigma ,{\mathcal {E}}}} is is trace non-increasing, since</li></ol> <p> Tr [ P σ , N ( X ) ] = Tr [ σ 1 2 E † ( E ( σ ) − 1 2 X E ( σ ) − 1 2 ) σ 1 2 ] = Tr [ σ E † ( E ( σ ) − 1 2 X E ( σ ) − 1 2 ) ] = Tr [ E ( σ ) E ( σ ) − 1 2 X E ( σ ) − 1 2 ] = Tr [ E ( σ ) − 1 2 E ( σ ) E ( σ ) − 1 2 X ] = Tr [ Π E ( σ ) X ] ≤ Tr [ X ] {\displaystyle {\begin{aligned}\operatorname {Tr} \left[{\mathcal {P}}_{\sigma ,{\mathcal {N}}}(X)\right]&=\operatorname {Tr} \left[\sigma ^{\frac {1}{2}}{\mathcal {E}}^{\dagger }\left({\mathcal {E}}(\sigma )^{-{\frac {1}{2}}}X{\mathcal {E}}(\sigma )^{-{\frac {1}{2}}}\right)\sigma ^{\frac {1}{2}}\right]\\&=\operatorname {Tr} \left[\sigma {\mathcal {E}}^{\dagger }\left({\mathcal {E}}(\sigma )^{-{\frac {1}{2}}}X{\mathcal {E}}(\sigma )^{-{\frac {1}{2}}}\right)\right]\\&=\operatorname {Tr} \left[{\mathcal {E}}(\sigma ){\mathcal {E}}(\sigma )^{-{\frac {1}{2}}}X{\mathcal {E}}(\sigma )^{-{\frac {1}{2}}}\right]\\&=\operatorname {Tr} \left[{\mathcal {E}}(\sigma )^{-{\frac {1}{2}}}{\mathcal {E}}(\sigma ){\mathcal {E}}(\sigma )^{-{\frac {1}{2}}}X\right]\\&=\operatorname {Tr} \left[\Pi _{{\mathcal {E}}(\sigma )}X\right]\\&\leq \operatorname {Tr} [X]\end{aligned}}} </p><p>From 1 and 2, when E ( σ ) {\displaystyle {\mathcal {E}}(\sigma )} is invertible, the Petz recovery map P σ , E {\displaystyle {\mathcal {P}}_{\sigma ,{\mathcal {E}}}} is a quantum channel, viz., a completely positive trace-preserving (CPTP) map. </p> <h2 id="further-reading">Further reading</h2> <ul><li><a href="https://mcgreevy.physics.ucsd.edu/w23/final-papers/2023W-213-Santoso-Owen.pdf">Brief Overview of the Petz recovery map and its applications</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Petz, Dénes (1986-03-01). "Sufficient subalgebras and the relative entropy of states of a von Neumann algebra" (PDF). Communications in Mathematical Physics. 105 (1): 123–131. Bibcode:1986CMaPh.105..123P. doi:10.1007/BF01212345. ISSN 1432-0916. S2CID 18836173. <a href="https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-105/issue-1/Sufficient-subalgebras-and-the-relative-entropy-of-states-of-a/cmp/1104115260.pdf" target="_blank">https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-105/issue-1/Sufficient-subalgebras-and-the-relative-entropy-of-states-of-a/cmp/1104115260.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Leifer, M. S.; Spekkens, Robert W. (2013-11-27). "Towards a formulation of quantum theory as a causally neutral theory of Bayesian inference". Physical Review A. 88 (5): 052130. arXiv:1107.5849. Bibcode:2013PhRvA..88e2130L. doi:10.1103/PhysRevA.88.052130. S2CID 43563970. <a href="https://link.aps.org/doi/10.1103/PhysRevA.88.052130" target="_blank">https://link.aps.org/doi/10.1103/PhysRevA.88.052130</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Furuya, Keiichiro; Lashkari, Nima; Ouseph, Shoy (2022-01-27). "Real-space RG, error correction and Petz map". Journal of High Energy Physics. 2022 (1): 170. arXiv:2012.14001. Bibcode:2022JHEP...01..170F. doi:10.1007/JHEP01(2022)170. ISSN 1029-8479. <a href="https://doi.org/10.1007/JHEP01(2022)170" target="_blank">https://doi.org/10.1007/JHEP01(2022)170</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Chen, Chi-Fang; Penington, Geoffrey; Salton, Grant (2020-01-28). "Entanglement wedge reconstruction using the Petz map". Journal of High Energy Physics. 2020 (1): 168. arXiv:1902.02844. Bibcode:2020JHEP...01..168C. doi:10.1007/JHEP01(2020)168. ISSN 1029-8479. <a href="https://doi.org/10.1007/JHEP01(2020)168" target="_blank">https://doi.org/10.1007/JHEP01(2020)168</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Cotler, Jordan; Hayden, Patrick; Penington, Geoffrey; Salton, Grant; Swingle, Brian; Walter, Michael (2019-07-24). "Entanglement Wedge Reconstruction via Universal Recovery Channels". Physical Review X. 9 (3): 031011. arXiv:1704.05839. Bibcode:2019PhRvX...9c1011C. doi:10.1103/PhysRevX.9.031011. <a href="https://link.aps.org/doi/10.1103/PhysRevX.9.031011" target="_blank">https://link.aps.org/doi/10.1103/PhysRevX.9.031011</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Petz, Dénes (1986-03-01). "Sufficient subalgebras and the relative entropy of states of a von Neumann algebra" (PDF). Communications in Mathematical Physics. 105 (1): 123–131. Bibcode:1986CMaPh.105..123P. doi:10.1007/BF01212345. ISSN 1432-0916. S2CID 18836173. <a href="https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-105/issue-1/Sufficient-subalgebras-and-the-relative-entropy-of-states-of-a/cmp/1104115260.pdf" target="_blank">https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-105/issue-1/Sufficient-subalgebras-and-the-relative-entropy-of-states-of-a/cmp/1104115260.pdf</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Khatri, Sumeet; Wilde, Mark M. (2024-02-11), Principles of Quantum Communication Theory: A Modern Approach, arXiv:2011.04672 <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Junge, Marius; Renner, Renato; Sutter, David; Wilde, Mark M.; Winter, Andreas (2016). Universal recoverability in quantum information. pp. 2494–2498. doi:10.1109/ISIT.2016.7541748. hdl:20.500.11850/120060. ISBN 978-1-5090-1806-2. S2CID 18189775. Retrieved 2024-01-03. <a href="978-1-5090-1806-2" target="_blank">978-1-5090-1806-2</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Cree, Sam; Sorce, Jonathan (2022-06-01). "Approximate Petz Recovery from the Geometry of Density Operators". Communications in Mathematical Physics. 392 (3): 907–919. arXiv:2108.10893. Bibcode:2022CMaPh.392..907C. doi:10.1007/s00220-022-04357-2. ISSN 1432-0916. S2CID 237292763. <a href="https://doi.org/10.1007/s00220-022-04357-2" target="_blank">https://doi.org/10.1007/s00220-022-04357-2</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> </ol>