In quantum information theory, a mix of quantum mechanics and information theory, the Petz recovery map can be thought of a quantum analog of Bayes theorem. Proposed by Dénes Petz,[1] the Petz recovery map is a quantum channel associated with a given quantum channel and quantum state. This recovery map is designed in a manner that, when applied to an output state resulting from the given quantum channel acting on an input state, it enables the inference of the original input state. In essence, the Petz recovery map serves as a tool for reconstructing information about the initial quantum state from its transformed counterpart under the influence of the specified quantum channel.
The Petz recovery map finds applications in various domains, including quantum retrodiction,[2] quantum error correction,[3] and entanglement wedge reconstruction for black hole physics.[4][5]
Definition
editSuppose we have a quantum state which is described by a density operator and a quantum channel , the Petz recovery map is defined as[1][6]
Notice that is the Hilbert-Schmidt adjoint of .
The Petz map has been generalized in various ways in the field of quantum information theory.[7][8]
Properties of the Petz recovery map
editA 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.
- The Petz recovery map is a completely positive map, since (i) sandwiching by the positive semi-definite operator is completely positive; (ii) is also completely positive when is completely positive; and (iii) sandwiching by the positive semi-definite operator is completely positive.
- It's also clear that is is trace non-increasing, since
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74ee7a4a322f0de163594b8510bea5f0e77ce36a)
From 1 and 2, when is invertible, the Petz recovery map is a quantum channel, viz., a completely positive trace-preserving (CPTP) map.
References
edit- ^ a b 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.
- ^ 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.
- ^ 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. Bibcode:2022JHEP...01..170F. doi:10.1007/JHEP01(2022)170. ISSN 1029-8479.
- ^ 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.
- ^ 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.
- ^ Khatri, Sumeet; Wilde, Mark M. (2024-02-11), Principles of Quantum Communication Theory: A Modern Approach, arXiv:2011.04672
- ^ 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.
- ^ 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.