Operational Meaning, Nonuniqueness, and Evidence Status

Holographic complexity is a family of conjectural dictionaries, not one observable with several calculational representations. Complexity equals volume (CV), complexity equals action (CA), Euclidean path-integral optimization, tensor-network size, Nielsen circuit geometry, and Krylov growth answer inequivalent questions unless an explicit map proves otherwise. Their shared qualitative successes are valuable diagnostics, but they do not presently select a unique operational boundary task.

Required background. Boundary Complexity Inputs: Tasks, Reference States, and Gate Sets fixes the data without which boundary complexity is undefined; Proposed Complexity Bounds and Their Counterexamples separates model successes from universal bounds.

Helpful background. Claim–Evidence Records, Replication, and Retraction Handling supplies the evidence discipline used here; QEC Evidence, Current Disputes, and Status is a parallel case in which attractive structures must remain logically distinct. Boundary definitions and alternatives are developed in Interacting-Field Complexity, Krylov and Operator-Growth Complexity, and Complexity with Symmetry, Gauge, and Locality Constraints.

What an operational dictionary would have to establish

Let a regulated boundary task be

$$\mathfrak T=(\rho_{\mathrm R},\rho_{\mathrm T},\mathcal G,F,\epsilon,\Lambda, \text{ancillas},\text{symmetry constraints}),$$

and let $\mathcal C_{\mathfrak T}$ be its optimum cost. A bulk proposal $X$ supplies a different functional $B_X[g,\phi;\delta,\kappa_X]$, where $\delta$ is a radial regulator and $\kappa_X$ collects normalization and boundary-term choices. Operational equivalence would require more than correlated growth. At minimum one needs a state-independent map

$$(\mathfrak T,\Lambda)\longleftrightarrow(\delta,\kappa_X), \qquad \mathcal C_{\mathfrak T}=B_X+o(N^p)$$

on a declared code or state class, with the power $p$, error norm, order of the large-$N$ and cutoff limits, and finite subtraction fixed before comparison. The map must predict normalization, survive changes of representative within the same physical task, and fail under a stated falsifier. No CV, CA, path-integral, or tensor-network proposal currently meets this standard for a generic interacting holographic CFT.

The distinction is structural. CV extremizes a codimension-one volume and introduces an arbitrary length $\ell_{\mathrm V}$ Stanford and Susskind 2014, §§ 2–3. CA evaluates a codimension-zero gravitational action on a Wheeler–DeWitt patch, including null joints and normalization-restoring terms Brown et al. 2016, §§ II–III. Path-integral optimization minimizes a chosen cost over Weyl-equivalent Euclidean preparations and gives a Liouville action in two-dimensional examples Caputa et al. 2017, pp. 1–4. A tensor network counts tensors, bonds, or weighted local maps only after a network class and equivalence moves are specified; MERA’s relation to hyperbolic geometry is an organizing analogy rather than an equality to CV or CA Swingle 2012, §§ I–III. A Nielsen circuit distance is well-defined only after its reference, generators, penalties, and tolerance are supplied Nielsen et al. 2006, §§ II–IV.

First application: one thermofield-double family, separate tasks

Take the two-sided thermofield-double family

$$|\mathrm{TFD}(t_L,t_R)\rangle =e^{-i(H_Lt_L+H_Rt_R)}|\mathrm{TFD}(0,0)\rangle$$

dual, at leading large $N$ and strong coupling, to an eternal AdS black hole. Fix the boundary UV cutoff, Hamiltonian normalization, temperature, and the combination $t=t_L+t_R$. The proposals can now be compared without declaring them equivalent:

• Circuit task. Prepare $|\mathrm{TFD}(t_L,t_R)\rangle$ to trace-distance tolerance $\epsilon$ from a specified product or Gaussian reference using a declared local gate set. This is the operational optimization problem. Its regulator, gate penalties, and reference dependence are physical parts of the definition.
• CV proxy. Extremize the bulk slice anchored at $(t_L,t_R)$ and divide by $G_N\ell_{\mathrm V}$. The late-time maximal volume is linear in $t$, but its coefficient changes with $\ell_{\mathrm V}$; the ultraviolet divergence depends on the radial cutoff.
• CA proxy. Evaluate the fully regulated Wheeler–DeWitt action and divide by $\pi$. It also grows linearly at late time for classical stationary black holes, while its transient behavior, null-boundary terms, and singularity sensitivity differ from CV.
• Euclidean path task. Optimize a Euclidean strip or half-cylinder that prepares the initial thermal purification. This directly addresses preparation geometry at $t=0$; extending it to Lorentzian time evolution requires additional choices and is not automatically the same task as circuit evolution.
• Tensor-network task. Choose a network architecture and minimize a declared count or weighted cost for representing the same regulated state. Bond dimensions can encode entanglement scaling, but refinement, tensor identities, and gauge moves change raw counts unless quotient rules are fixed.

Four robust-looking signatures can then be recorded: a cutoff-extensive leading cost, a finite vacuum-subtracted formation cost, approximately linear late-time growth, and a switchback delay for early perturbations. These signatures are not unique. Many local circuits and coarse-grained tensor models have them; CV and CA can agree on them while differing in coefficients, transients, and quantum corrections. The comparison therefore supports a universality class of qualitative diagnostics, not a unique dictionary.

Evidence status through 10 August 2026

The primary calculations support a deliberately limited status statement.

CV. Classical black holes exhibit maximal-slice growth and switchback behavior. The prescription is geometrically clean at leading semiclassical order, but $\ell_{\mathrm V}$, regulator matching, and the absent boundary-task derivation prevent an absolute operational interpretation.

CA. Classical examples reproduce late growth and switchbacks with a fixed $I/\pi$ normalization. The complete null-boundary prescription is essential. Quantum BTZ supplies a direct adverse case: generalized CA is sensitive to the quantum-corrected singular region and lacks the smooth classical limit found for generalized CV Emparan, Frassino, and Sasieta 2022, §§ 4–6. Rotating quantum BTZ makes the answer depend further on horizon structure and limit order Chen, Liu, and Yu 2024, §§ 3–5. Other 2026 charged quantum-black-hole calculations find model-specific generalized Lloyd-type behavior Parihar and Punia 2026, §§ 4–6. The variation among examples is evidence against treating one finite-$G_N$ rule as settled.

Path-integral optimization. The Liouville extremum in two-dimensional CFT is an explicit and productive construction. Its cost functional, measure, allowed Weyl transformations, and relation to a circuit gate set remain inputs, so the result does not establish CV or CA.

Tensor networks. Exact finite-dimensional codes and optimized networks clarify entanglement organization and reconstruction. Network size depends on discretization, bond dimension, architecture, and allowed rewrites. A network model may reproduce geometric scaling without being the microscopic circuit complexity of the CFT.

Boundary alternatives. Circuit, operator, Krylov, path, and preparation complexities can be mathematically defined once their tasks are fixed, but they need not agree even parametrically. Matching one of them to a bulk proxy remains a separate conjecture.

Adversarial control: translate without refitting

Freeze one TFD state family and one circuit task. Calibrate at most one state-independent map between the boundary cutoff and radial cutoff, one normalization, and one finite counterterm scheme. Then demand simultaneous predictions for formation cost, late-time slope, a shockwave switchback, reference-state variation, and the weak-backreaction quantum-BTZ limit. Do not refit $\ell_{\mathrm V}$, the CA finite terms, the reference state, or the gate penalties between observables.

If CV and CA require different translations, they remain distinct proxies. If a finite counterterm erases a discrepancy only by depending on the state, it is not a fixed renormalization scheme. If quantum BTZ violates a claimed CA limit while the boundary task is unchanged, the claim fails on that family; changing the ultraviolet hypothesis may motivate a new proposal but does not annul the test.

The evidence ceiling is a set of proposal-specific semiclassical regularities and discriminating counterexamples. As of the stated cutoff there is no settled, regulator-complete, finite-$N$ theorem identifying CV, CA, path-integral optimization, or tensor count with a unique operational boundary complexity. Live priority, new quantum corrections, and comparative confidence belong in a dated research dossier; the durable handoff to quantum information is the precise task definition against which any future dictionary must be tested.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Brown, A. R., Roberts, D. A., Susskind, L., Swingle, B., and Zhao, Y. (2016). “Complexity, Action, and Black Holes.” Physical Review D 93, 086006. DOI.
• Caputa, P., Kundu, N., Miyaji, M., Takayanagi, T., and Watanabe, K. (2017). “Liouville Action as Path-Integral Complexity: From Continuous Tensor Networks to AdS/CFT.” Physical Review Letters 119, 071602. DOI.
• Chen, B., Liu, Y., and Yu, B. (2024). “Holographic Complexity of Rotating Quantum Black Holes.” Journal of High Energy Physics 2024(1), 055. DOI.
• Emparan, R., Frassino, A. M., and Sasieta, B. (2022). “Holographic Complexity in Quantum Black Holes.” Journal of High Energy Physics 2022(2), 204. DOI.
• Nielsen, M. A., Dowling, M. R., Gu, M., and Doherty, A. C. (2006). “Quantum Computation as Geometry.” Science 311, 1133–1135. DOI.
• Parihar, M. K., and Punia, A. (2026). “Holographic Complexity of Dyonic Quantum Black Holes.” Journal of High Energy Physics 2026(2), 248. DOI.
• Stanford, D., and Susskind, L. (2014). “Complexity and Shock Wave Geometries.” Physical Review D 90, 126007. DOI.
• Swingle, B. (2012). “Entanglement Renormalization and Holography.” Physical Review D 86, 065007. DOI.