Abstract
We study holographic subregion complexity, and its possible connection to purification complexity suggested recently by Ag\'on et al. In particular, we study the conjecture that subregion complexity is the purification complexity by considering holographic purifications of a holographic mixed state. We argue that these include states with any amount of coarse-graining consistent with being a purification of the mixed state in question, corresponding holographically to different choices of the cutoff surface. We find that within the complexity = volume and complexity = spacetime volume conjectures, the subregion complexity is equal to the holographic purification complexity. For complexity = action, the subregion complexity seems to provide an upper bound on the holographic purification complexity, though we show cases where this bound is not saturated. One such example is provided by black holes with a large genus behind the horizon, which were studied by Fu et al. As such, one must conclude that these offending geometries are not holographic, that CA must be modified, or else that holographic subregion complexity in CA is not dual to the purification complexity of the corresponding reduced state.
Highlights
In the past several years, quantum complexity has entered into discussions of quantum gravity and holography, starting with a discussion of complexity and the firewall paradox in [1] and later in [2]
Conjecture [3,4] which speculates that the volume of a maximal spatial slice is dual to the quantum circuit complexity of the dual quantum state living on the intersection of that spatial slice with the boundary
By minimization over cutoffs we argue on general grounds that the purification complexity is bounded above by the subregion complexity in all of complexity 1⁄4 volume (CV), complexity 1⁄4 action (CA), and CV2.0
Summary
In the past several years, quantum complexity has entered into discussions of quantum gravity and holography, starting with a discussion of complexity and the firewall paradox in [1] and later in [2]. Beginning with the entanglement wedge dual to the mixed state in question, all holographic geometries which geodesically complete the wedge provide a family of purifications in their boundary dual states. In the absence of a superadditivity property, we must worry that the subregion complexity is not truly minimal among all holographic purifications To examine this possibility, we consider geodesic completions of the one-sided Bañados-Teitelboim-Zanelli (BTZ) geometry dual to a thermal state. We note that for any geodesic completion of the entanglement wedge dual to our mixed state, there will be one purification which corresponds to a cutoff skirting just outside the entanglement wedge along the Hubeny-Rangamani-Takayanagi (HRT) surface This purification will have a complexity equal (up to a possible boundary like the term discussed at the end of the section) to the subregion complexity. We further find that one must impose a limit on the cutoffs considered
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