Exact renormalization group flow for matrix product density operators (2410.22696v1)
Abstract: Matrix product density operator (MPDO) provides an efficient tensor network representation of mixed states on one-dimensional quantum many-body systems. We study a real-space renormalization group transformation of MPDOs represented by a circuit of local quantum channels. We require that the renormalization group flow is exact, in the sense that it exactly preserves the correlation between the coarse-grained sites and is therefore invertible by another circuit of local quantum channels. Unlike matrix product states (MPS), which always have a well-defined isometric renormalization transformation, we show that general MPDOs do not necessarily admit a converging exact renormalization group flow. We then introduce a subclass of MPDOs with a well-defined renormalization group flow, and show the structure of the MPDOs in the subclass as a representation of a pre-bialgebra structure. As a result, such MPDOs obey generalized symmetry represented by matrix product operator algebras associated with the pre-bialgebra. We also discuss implications with the classification of mixed-state quantum phases.
- S. R. White, Phys. Rev. Lett. 69, 2863 (1992).
- S. Östlund and S. Rommer, Phys. Rev. Lett. 75, 3537 (1995).
- M. B. Hastings, J. Stat. Mech.: Theory Exp. 2007, P08024 (2007).
- M. B. Hastings and X.-G. Wen, Phys. Rev. B 72, 045141 (2005).
- M. B. Hastings, Phys. Rev. Lett. 107, 210501 (2011).
- R. König and F. Pastawski, Phys. Rev. B 90 (2014), 10.1103/physrevb.90.045101.
- A. Coser and D. Pérez-García, Quantum 3, 174 (2019).
- M. B. Hastings, Phys. Rev. B 73, 085115 (2006).
- M. Koashi and N. Imoto, Phys. Rev. A 66, 022318 (2002).
- K. Kato, arXiv:2309.07434 [quant-ph] .
- R. A. Fisher, Philos. Trans. R. Soc. Lond. A 222, 309 (1922).
- T. M. Cover and J. A. Thomas, Elements of Information Theory 2nd Edition (Wiley Series in Telecommunications and Signal Processing) (Wiley-Interscience, 2006).
- D. Petz, Commun. Math. Phys. 105, 123 (1986).
- D. Petz, Q. J. Math. 39, 97 (1988).
- B. Schumacher, Phys. Rev. A 51, 2738 (1995).
- M. Mosonyi and D. Petz, Lett. Math. Phys. 68, 19 (2004).
- A. Jenčová and D. Petz, Commun. Math. Phys. 263, 259 (2006).
- When we speak about the minimal sufficient subalgebra for a partition ABΛL𝐴𝐵subscriptΛ𝐿AB~{}\Lambda_{L}italic_A italic_B roman_Λ start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT, we implicitly assume that ℋBsubscriptℋ𝐵{\mathcal{H}}_{B}caligraphic_H start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT is restricted to supp(ρB)suppsubscript𝜌𝐵{\rm supp}(\rho_{B})roman_supp ( italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) so that ρB>0subscript𝜌𝐵0\rho_{B}>0italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT > 0.
- M. S. Leifer and D. Poulin, Ann. Phys. (N. Y.) 323, 1899 (2008).
- W. Brown and D. Poulin, arXiv:1206.0755 [quant-ph] .
- D. Perez-Garcia, private communication (2024).
- M. Levin and C. P. Nave, Phys. Rev. Lett. 99 (2007), 10.1103/physrevlett.99.120601.
Collections
Sign up for free to add this paper to one or more collections.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.