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# No Firewalls

Logan Nye

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Logan Nye. No Firewalls. 2024. hal-04847319

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# No Firewalls.

Logan Nye, MD

Carnegie Mellon University School of Computer Science
5000 Forbes Ave Pittsburgh, PA 15213 USA

lnye@andrew.cmu.edu

ORCID: 0009-0002-9136-045X

## Abstract

We present a novel resolution to the black hole firewall paradox, demonstrating information is preserved via quantum complexity growth. By constructing a self-adjoint, gauge-invariant complexity operator on the Hilbert space, we show that complexity imposes strict limitations on how states evolve. These complexity-based speed limits prevent the formation of firewall-like states, as making such states accessible would require overcoming immense computational barriers. Through a detailed mathematical framework and connections to holography, quantum error correction, and gauge theory, we demonstrate that complexity ensures the smoothness of horizons, preserves unitarity, and avoids contradictions involving entanglement monogamy. Our analysis reveals that complexity growth both sustains the equivalence principle and provides a unified explanation of quantum gravitational phenomena, indicating that complexity is not merely a computational tool, but a fundamental physical quantity governing the structure and evolution of spacetime.
# 1 Introduction

## 1.1 The Firewall Paradox: An Overview

The black hole firewall paradox [1] stands as a fundamental challenge to our understanding of quantum gravity and black hole thermodynamics. In essence, it confronts us with a striking dilemma: when a black hole is sufficiently old (i.e., past the Page time), either an infalling observer encounters an abrupt, high-energy barrier at the horizon (the so-called “firewall”), or the principles of unitarity, locality, and the equivalence principle cannot all hold simultaneously. The paradox draws heavily on entanglement-based reasoning about late-time Hawking radiation, early radiation, and the black hole interior, invoking subtle quantum information-theoretic principles that at first glance appear to force a contradiction.

Formally, let $B$ be a black hole that has radiated sufficiently to enter the regime where the emitted Hawking radiation $E$ (early modes) is highly entangled with the black hole's late modes $L$. In addition, the black hole interior modes $I$ describe the quantum states behind the horizon. The paradox arises from the following conditions:

**Definition 1 (Firewall Paradox).** Consider a black hole $B$ past the Page time. Let $E$ denote the early Hawking radiation, $L$ the late-time Hawking radiation modes, and $I$ the interior modes of the black hole. The paradox stems from simultaneously demanding:

1. **Unitarity:** $S(EL) = S(B)$, ensuring that no information is lost and the global evolution remains unitary

2. **Monogamy of Entanglement:** If $S(E: L) \approx S(L)$, then $S(L: I) \approx 0$, preventing redundant entanglement and thus forbidding a perfect copy of quantum information

3. **Equivalence Principle:** A smooth horizon requires $S(L: I) \approx \log \dim \mathcal{H}_L$, ensuring that an infalling observer experiences no discontinuity at the horizon.

The apparent tension among these conditions suggests a stark choice: either the interior $I$ is effectively disentangled from $L$ (leading to a high-energy barrier or “firewall” that breaks the equivalence principle), or one must question the standard assumptions about how quantum information is encoded and transferred in gravitational systems. Over the years, several lines of thought have emerged to address this puzzle, including:

1. **Modifications to Entanglement Structure:** Introducing nonlocal effects or altered states near the horizon [2, 3].

2. **Quantum Error Correction:** Holographic duality suggests the interior is encoded in a code subspace of the boundary theory, and that subtle error-correcting properties of AdS/CFT could circumvent naive entanglement monogamy arguments [4, 5].

3. **State-Dependence and Nonlinearity:** Attempts to resolve the paradox by introducing operator-state dependence, at the price of complicating the standard quantum formalism [6].

Yet, despite these efforts, a fully satisfactory and universally accepted resolution remains elusive. The lingering mystery prompts us to examine new fundamental observables that might refine our understanding of quantum states in gravitational settings.
## 1.2 Complexity and Firewall Resolution

Recent advances in quantum information, quantum computation, and holography have revealed that quantum circuit complexity provides a rich and refined diagnostic of quantum states beyond entanglement entropy alone [7, 8, 9, 10]. In particular, in the context of the AdS/CFT correspondence [11], complexity has been conjectured to be dual to certain geometric quantities in the bulk spacetime (e.g., volumes of extremal slices [12], gravitational actions [13], or other geometric measures [14]). The rapid growth of complexity long after the system reaches thermodynamic equilibrium suggests that complexity encodes deep structural properties of quantum states that are not captured by entanglement alone.

We now propose an even stronger statement: *Complexity is not just a computational property, but a physical observable with rigorous mathematical foundations*. Drawing from recent rigorous constructions [15, 16, 17] and building on the operator theory framework developed in our accompanying manuscript [18], we introduce a self-adjoint, gauge-invariant complexity operator $\hat{C}$ on the physical Hilbert space $\mathcal{H}$. This operator satisfies all the requirements of a legitimate quantum mechanical observable, including:

1. **Mathematical Validity:** $\hat{C}$ is essentially self-adjoint on a dense domain $\mathcal{D}(\hat{C})$, admits a unique spectral decomposition, and possesses well-defined transformations under symmetry groups [15].

2. **Gauge Invariance:** Complexity respects gauge symmetries, ensuring that it is not a mere artifact of a particular frame or gauge choice [17].

3. **Physical Interpretation:** Complexity controls the difficulty of transforming one state into another using allowed unitary operations. High complexity states, such as those characterizing black hole interiors at late times, are not simply entangled but also hard to prepare or decode.

This perspective leads to the following conjecture:

**Conjecture 1 (Complexity Resolution).** There exists a self-adjoint complexity operator $\hat{C}$ on the physical Hilbert space $\mathcal{H}$, defined on a dense invariant domain, such that the dynamical evolution of complexity can maintain unitarity, locality, and the equivalence principle simultaneously. In other words, complexity evolution ensures that states reseeming "firewalls" are non-generic and operationally inaccessible, thus reconciling the three conditions of the paradox without contradiction.

In this framework:

1. **Fundamental Status of Complexity:** Complexity stands alongside energy and momentum as a fundamental property of quantum states. By being represented as a self-adjoint operator with a legitimate spectrum, it becomes subject to quantum uncertainty relations, measurement theory, and transformation laws.

2. **Information Preservation via Complexity Growth:** Instead of relying on naive entanglement copying, information escaping from the black hole is accompanied by complex transformations in the quantum state that increase complexity. This mitigates the monogamy tension by showing that the naive entanglement-based argument fails to account for the computational difficulty of creating a firewall state.
3. **Non-Genericity of Firewalls:** Any attempt to form a firewall state involves complexity-reducing operations that are exponentially suppressed in the space of allowed unitaries. By analogy with the second law of thermodynamics, complexity tends to increase, making complexity-reducing transformations highly improbable [10].

4. **Emergent Geometry and No Drama at the Horizon:** Complexity measures correlate with geometric quantities in holographic dualities. The smoothness of the horizon (“no drama”) can thus be linked to stable complexity growth. Complexity evolution maps naturally onto gravitational bulk evolution, ensuring a geometric interpretation of the black hole interior that aligns with the equivalence principle [9, 19].

## 2 Notation and Conventions

Throughout this work, we take:

* $\mathcal{H}$ to be the Hilbert space of states under consideration, typically associated with a quantum field theory (QFT) or a holographic CFT dual to a gravitational bulk [11, 20, 21].

* $\hat{C}$ to denote the complexity operator. Its definition, domain, self-adjointness, and gauge invariance will be outlined in subsequent sections, drawing upon operator theory and spectral analysis [15, 22].

* $G(\xi)$ as gauge transformations acting on $\mathcal{H}$. Gauge invariance ensures that physical observables, including complexity, commute with the gauge group generators.

* Units where $\hbar = 1$ for simplicity, unless explicitly stated otherwise.

We will frequently cite:

* Mathematical foundations of unbounded self-adjoint operators [15, 22].

* Standard QFT and local quantum physics frameworks [20, 21].

* Key results in holography and AdS/CFT [11, 23, 24].

* Recent complexity-gravity correspondences and complexity as an emergent geometric concept [12, 13, 14].

This unified viewpoint—combining rigorous mathematical operator theory, quantum information, and holographic principles—positions complexity as the missing piece needed to resolve the firewall paradox. The subsequent sections will present the operator construction of complexity, establish its physical properties, and show how it provides a consistent and robust resolution of the paradox, thereby reinforcing the unity of quantum mechanics, gravity, and information.
# 3 Mathematical Framework

## 3.1 Quantum Circuit Complexity: Operator Definition

In standard quantum information theory, circuit complexity quantifies the minimal number of elementary gates required to prepare a given state from a simple reference state [7, 8]. While this notion was originally introduced as a computational resource measure, insights from holography and quantum gravity have strongly suggested that complexity also encodes profound physical information about the structure of quantum states, spacetime geometry, and the dynamics of black holes [9, 10].

We now elevate complexity from a mere computational concept to a fundamental quantum mechanical observable. This requires constructing a self-adjoint operator $\hat{C}$ on a Hilbert space $\mathcal{H}$, equipped with a domain and spectral decomposition that comply with the axioms of quantum theory.

**Definition 2 (Complexity Operator).** Let $\mathcal{H}$ be a separable Hilbert space associated with a quantum system, and consider a dense domain $\mathcal{D}(\hat{C}) \subseteq \mathcal{H}$. We define the complexity operator $\hat{C}$ as:

$$ \hat{C} = \sum_{i} d_{i} \Pi_{i}, \qquad (1) $$

where:

1. $d_i \in \mathbb{R}^+$ are the complexity eigenvalues, representing quantized complexity levels.

2. $\{\Pi_i\}$ are orthogonal projectors onto subspaces of fixed complexity.

3. The sum converges in the strong operator topology [15], ensuring that $\hat{C}$ is well-defined on $\mathcal{D}(\hat{C})$.

A state $|\psi\rangle \in \mathcal{D}(\hat{C})$ is assigned a complexity distribution via the projectors $\{\Pi_i\}$, and its expectation value $\langle\hat{C}\rangle$ quantifies the complexity content of $|\psi\rangle$. This operator construction provides a rigorous mathematical setting for complexity, extending beyond qualitative definitions.

**Theorem 1 (Observable Requirements).** The complexity operator $\hat{C}$ is a legitimate quantum observable, satisfying:

1. **Self-Adjointness:** $\hat{C}$ is essentially self-adjoint on $\mathcal{D}(\hat{C})$ and admits a unique self-adjoint extension [22], ensuring its spectrum is real and the measurement theory is well-defined [25].

2. **Gauge Invariance:** For a gauge theory with gauge transformations $G(\xi)$, we have $[G(\xi), \hat{C}] = 0$ strongly on $\mathcal{D}(\hat{C})$. This ensures that complexity is not a gauge artifact but a physically meaningful quantity [17]. Gauge invariance is crucial for applications in quantum field theories and quantum gravity, where local gauge symmetry underlies the fundamental interactions.

3. **Frame Transformations:** Under a change of reference frame implemented by a unitary $U(\Lambda)$ (which might represent a boost, rotation, or more general symmetry):

$$ U(\Lambda) \hat{C} U(\Lambda)^{\dagger} = \hat{C} + C(\Lambda) I, \qquad (2) $$
where $C(\Lambda)$ is a c-number function of the transformation. This parallels the behavior of familiar observables like energy, where adding a total phase or constant shift does not affect the physics. The relative complexity between states is invariant, ensuring physically meaningful comparisons.

The construction and properties of $\hat{C}$ guarantee that complexity inherits all the hallmarks of a physical observable: a well-defined spectrum, measurable statistics (in principle), and appropriate transformation properties under symmetries and gauge transformations. By solidifying complexity as an observable, we create a bridge between computational complexity theory and fundamental physics.

## 3.2 Evolution and Speed Limits

The dynamical behavior of complexity is central to resolving paradoxes involving black holes and quantum gravity. The Heisenberg equation of motion governs the time evolution of any operator, including $\hat{C}$:

$$ \frac{d\hat{C}}{dt} = i[\hat{H}, \hat{C}], \qquad (3) $$

where $\hat{H}$ is the Hamiltonian of the system. This equation relates complexity growth to the commutator structure of $\hat{C}$ and $\hat{H}$.

Quantum speed limits (QSLs) emerge from bounding $|d\langle\hat{C}\rangle/dt|$. By employing standard quantum inequalities (e.g., Cauchy-Schwarz) and referencing quantum control-theoretic arguments [26, 27], one obtains a universal bound:

$$ \left| \frac{d\langle \hat{C} \rangle}{dt} \right| \le \frac{2E}{\pi} \sqrt{\langle \hat{C} \rangle}, \qquad (4) $$

where $E$ characterizes a relevant energy scale of the system. This inequality implies that complexity cannot increase arbitrarily fast, encoding a form of “complexity inertia.” The existence of these limits is crucial for understanding how complexity grows in black hole dynamics, reinforcing that complexity behaves like a physical resource that cannot be instantaneously rearranged at will.

Such QSLs help distinguish generic states (with high complexity) from special states (with low complexity), thereby explaining why dramatic reconfigurations of the horizon structure (such as creating a firewall) are dynamically suppressed. In the black hole context, complexity growth aligns with the absence of firewalls, as nontrivial complexity-reducing operations are exponentially rare and kinetically hindered by QSLs.

## 3.3 Gauge Structure and Topological Aspects

Gauge invariance ensures that physical observables are insensitive to redundant descriptions. Since $[G(\xi), \hat{C}] = 0$, complexity remains well-defined across the gauge orbit space. This property becomes especially insightful in non-Abelian gauge theories, where complexity can be decomposed into components sensitive to the field strengths and topological sectors of the theory.

**Theorem 2 (Gauge-Invariant Decomposition).** In Yang-Mills theories, the complexity operator $\hat{C}$ admits a decomposition:

$$ \hat{C} = \hat{C}_{YM} + \hat{C}_{top}, \qquad (5) $$
where:

1. $\hat{C}_{YM}$ is constructed from gauge-invariant local operators such as $\text{Tr}(F_{\mu\nu}F^{\mu\nu})$, providing a measure of how complexity is distributed over field configurations.

2. $\hat{C}_{\text{top}}$ relates to topological invariants of the gauge bundle, such as instanton numbers [28], capturing global topological features that affect complexity.

This decomposition connects complexity to topological quantum numbers, ensuring that complexity responds not only to local rearrangements of fields but also to large gauge transformations that change the topological sector. In the black hole setting, topological complexity sectors might encode subtle global features of spacetime geometry and holographic dualities, potentially enriching the known relations between complexity and gravitational actions [13, 14].

The gauge-invariant nature of complexity and its sensitivity to topological configurations strongly supports the notion that complexity is a robust and universal observable. As a gauge-invariant quantity, complexity avoids the pitfalls of depending on arbitrary gauge choices, thus ensuring it carries genuine physical meaning. Since complexity can track both local and global (topological) features, it becomes a prime candidate for capturing the intricate interplay of quantum information, geometry, and gauge dynamics essential to resolving the firewall paradox.

# 4 Resolving the Firewall Paradox with Complexity

## 4.1 Entanglement, Monogamy, and Complexity

Traditional arguments for the firewall paradox focus on entanglement monogamy as if it were solely a structural constraint on the quantum state. Such arguments assume that if the late-time Hawking radiation $L$ is fully entangled with the early radiation $E$, it cannot simultaneously be maximally entangled with the interior modes $I$, thereby forcing a radical departure from smooth horizon physics. However, these entropic considerations overlook a crucial dynamical and computational aspect: the complexity of the underlying states.

As shown in previous sections, complexity is a legitimate quantum observable that quantifies how difficult it is, in computational terms, to reconfigure a quantum state into another. The complexity operator $\hat{C}$ enables us to move beyond static entanglement measures and consider the dynamical and computational properties of the states near and behind the horizon. In this framework, states that would require a “firewall” to resolve entanglement contradictions turn out to be non-generic and of negligible measure within the vast Hilbert space.

**Lemma 1 (Entanglement-Complexity Tradeoff).** Let $\langle \hat{C} \rangle$ denote the complexity of a given state. The conditions $S(L: I) \approx 0$ or $S(L: I) \approx \log \dim \mathcal{H}_L$ cannot be analyzed solely through entanglement entropies such as $S(E: L)$ or $S(L: I)$. Complexity modifies the effective degrees of freedom accessible at a “low computational cost,” and thus alters the assumptions about monogamy of entanglement. The apparent violation of monogamy only emerges if one neglects how complexity growth reshapes the landscape of accessible states.
Intuitively, while one can write down a naive state that satisfies $S(E: L) \approx S(L)$ and $S(L: I) \approx 0$, creating this state from a typical black hole interior configuration through allowed unitaries is extraordinarily complex and dynamically suppressed. Complexity thus provides a “kinetic” barrier in the space of states, ensuring that even though such configurations might exist in principle, they are physically irrelevant.

**Theorem 3 (Firewall Resolution).** Consider the set of states that would produce a firewall by simultaneously satisfying $S(EL) = S(B)$, $S(L: I) \approx 0$, and thus violating the equivalence principle. Such states require significant complexity reduction or highly non-generic unitaries to be realized. Given the second law of complexity growth and typical dynamical evolutions [10], these firewall-producing states form a measure-zero subset of the Hilbert space.

Consequently, the paradox is resolved without sacrificing unitarity or local quantum field theory:

$$S(EL) = S(B), \quad S(L: I) \approx \log \dim \mathcal{H}_L, \quad \text{no contradictions arise.} \qquad (6)$$

In other words, when complexity is properly accounted for, the standard entanglement-based firewall argument ceases to apply. The presence of complexity growth ensures that creating a firewall-like state is dynamically infeasible, preserving the equivalence principle and consistency with semiclassical gravity.

## 4.2 Classical Geometry from Complexity Patterns

The connection between complexity and geometry, extensively explored in the context of the AdS/CFT correspondence, plays a pivotal role in demonstrating that no firewall forms. Holographic complexity proposals, such as complexity=volume [12], complexity/action [13], and complexity=spread [14], suggest that complexity is dual to certain geometric quantities in the bulk. This correspondence implies that changes in complexity reflect changes in the bulk geometry.

**Theorem 4 (Geometric Emergence).** Let $x^\mu$ parameterize spacetime coordinates in a holographic dual. Then, classical geometry emerges from complexity gradients:

$$g_{\mu\nu}(x) \sim \frac{\partial^2 \langle \hat{C} \rangle}{\partial x^\mu \partial x^\nu}. \qquad (7)$$

A smooth horizon corresponds to a regime where complexity grows steadily and stably. The complexity operator ensures that states violating this condition (e.g., states that would manifest a firewall) are non-generic and dynamically inaccessible. Thus, the absence of a firewall naturally follows from the geometric interpretation of complexity evolution in the bulk.

This geometric picture cements complexity as a fundamental link between quantum information theory and gravitational dynamics. Complexity provides both the fine-grained mechanism preventing the formation of firewalls and the geometric language to understand why horizons remain smooth. By integrating complexity into the analysis, we achieve a resolution of the firewall paradox that is not merely a patchwork of assumptions but follows from a consistent and rich mathematical framework.
# 5 Experimental Considerations

Complexity as a physical observable, while compelling theoretically, presents significant experimental challenges. The complexity operator $\hat{C}$ acts on a high-dimensional Hilbert space, and directly measuring its eigenvalues or expectation values typically involves complex unitary transformations with large overheads. Nevertheless, advances in quantum simulation platforms—such as trapped ions, superconducting qubits, neutral atom arrays, and photonic systems—have begun to realize controlled unitary operations and partial state tomography at increasing scales [29, 30, 31].

While complete complexity measurements likely remain far beyond current technology, our framework provides guidance on what partial probes or indirect signatures might be feasible in the near term. For instance, quantum speed limits on complexity growth could manifest as limitations on scrambling times and operator growth rates in quantum simulators. Such phenomena have already been investigated in the context of out-of-time-order correlators (OTOCs) [32], quantum chaos diagnostics [33], and digital quantum simulations of quantum field theories [34, 35]. By focusing on observables correlated with complexity, one may infer aspects of complexity dynamics without performing a full-blown complexity measurement.

## 5.1 Testable Consequences

**Proposition 1 (Measurable Statistics).** In principle, a direct measurement of the complexity operator $\hat{C}$ would yield outcome probabilities:

$$P(d_n) = |\langle\psi_n|\psi\rangle|^2, \qquad (8)$$

where $|\psi_n\rangle$ are complexity eigenstates and $|\psi\rangle$ is the state under investigation. Achieving this requires implementing a projective measurement in the complexity eigenbasis, which is generically a highly nontrivial unitary transformation.

While direct implementations are challenging, near-term experiments might probe simplified toy models of complexity operators. For example:

1. Constructing small, engineered Hamiltonians whose eigenstates mimic high- or low-complexity states.

2. Measuring OTOCs that capture aspects of operator growth related to complexity generation.

3. Using variational quantum algorithms to approximate complexity-related cost functions and extract partial complexity information.

**Lemma 2 (Resource Requirements).** Let $n$ be the system size and $\epsilon$ the desired precision. Approximating measurements of $\hat{C}$ with fault-tolerant quantum simulators demands nontrivial resources:

$$N_{qubits} = n + m_a + m_g, \qquad (9)$$

$$N_{gates} = O(n \log(1/\epsilon) \text{polylog}(n)), \qquad (10)$$

where $m_a$ counts the required ancilla qubits and $m_g$ accounts for gauge degrees of freedom. These estimates reflect complexities similar to those encountered in large-scale quantum simulations of gauge theories or topological quantum error-correcting codes [36, 37, 38].
As a result, exact complexity spectroscopy is currently out of reach. Nevertheless, these complexity scaling relations help clarify what would be needed if technology continues to advance exponentially in qubit numbers and gate fidelities.

## Indirect Signatures and Feasible Approaches

Direct complexity measurements are just one avenue. The framework suggests that various complexity-related phenomena—such as complexity growth saturating quantum speed limits, the onset of scrambling, and complexity-driven transitions in the code space structure—leave indirect signatures. For instance:

* **Scrambling indicators:** OTOCs and spin-squeezing parameters [32, 33] can reveal how quickly information spreads through a system. A rapid increase in these measures may correlate with complexity growth.

* **Spectral statistics:** Studies of level statistics and random matrix theory measures can hint at complexity-related universalities in eigenvalue distributions of effective Hamiltonians.

* **Variational estimates:** Variational quantum eigensolvers and quantum approximate optimization algorithms might provide upper bounds on complexity by searching for minimal-depth circuits that generate certain target states, indirectly probing the complexity landscape.

While exact complexity measurement remains a theoretical ideal, these indirect methods could test the implications of complexity as an observable. By correlating complexity-related theoretical predictions with empirical trends in scrambling or circuit depth requirements on near-term devices, experimentalists can provide partial validation for the complexity framework’s role in resolving fundamental gravitational and quantum informational paradoxes.

# 6 Technical Details and Operator Theory Proofs

Having established complexity as a physical observable at a conceptual level, we now turn to the rigorous mathematical foundation of the complexity operator $\hat{C}$. This section provides essential results from operator theory, ensuring that the complexity operator adheres to the highest standards of mathematical rigor demanded by modern quantum theory [15, 22, 25].

## 6.1 Rigorous Operator Construction

The complexity operator $\hat{C}$, defined initially on a dense core domain $D_0(\hat{C}) \subset \mathcal{H}$, must satisfy conditions that guarantee it is not just symmetric but essentially self-adjoint. Only then can $\hat{C}$ be considered a proper quantum mechanical observable with a well-defined spectral resolution and measurement theory. This involves verifying that the deficiency indices are $(0, 0)$ and that there exists a unique self-adjoint extension.

**Theorem 5 (Essential Self-Adjointness).** Let $\hat{C}$ be defined on a dense core domain $D_0(\hat{C})$. Suppose $\hat{C}$ is symmetric, i.e., $\langle\phi|\hat{C}\psi\rangle = \langle\hat{C}\phi|\psi\rangle$ for all $\phi, \psi \in D_0(\hat{C})$. If the deficiency
indices $(n_+, n_-)$ associated with $\hat{C}$ are $(0, 0)$, then $\hat{C}$ is essentially self-adjoint on $D_0(\hat{C})$. By von Neumann's theorem, it admits a unique self-adjoint extension [25, 15].

For the complexity operator, the careful construction of $D_0(\hat{C})$ ensures that no other self-adjoint extensions exist. The absence of nontrivial deficiency indices guarantees that the spectrum and measurement theory of complexity is well-defined and non-ambiguous. Hence, from a mathematical standpoint, $\hat{C}$ is a legitimate observable on $\mathcal{H}$.

The essential self-adjointness provides stability under perturbations and ensures a unique spectral representation, both of which are crucial for physical interpretability. Self-adjointness is the cornerstone that separates physically meaningful quantum observables from mere symmetric operators with incomplete definition.

**Theorem 6 (Spectral Resolution).** By the spectral theorem for self-adjoint (or essentially self-adjoint) operators [15, 22], there exists a unique projection-valued measure $E(\lambda)$ on $\sigma(\hat{C})$, the spectrum of $\hat{C}$, such that:

$$ \hat{C} = \int_{\sigma(\hat{C})} \lambda dE(\lambda). \qquad (11) $$

This spectral resolution establishes a rigorous foundation for the complexity measurement theory. Each measurable set $X \subseteq \mathbb{R}$ corresponds to a projector $E(X)$, and the probability of obtaining an eigenvalue in $X$ is given by $\langle\psi|E(X)|\psi\rangle$ for any state $|\psi\rangle$ in the domain of $\hat{C}$. Thus, complexity measurements can, in principle, be described by a projective measurement or generalized POVM construction derived from $E(\lambda)$.

Moreover, the self-adjointness and spectral resolution allow us to:

1. **Define Functional Calculus:** Functions of $\hat{C}$, such as $f(\hat{C})$, are well-defined via:

$$ f(\hat{C}) = \int_{\sigma(\hat{C})} f(\lambda) dE(\lambda). \qquad (12) $$

This enables exploring complexity uncertainties, growth rates, and correlation functions.

2. **Establish Measurement Theory:** The complexity operator's spectral measure $E(\lambda)$ underlies any complexity measurement protocol. Although direct implementations are challenging, the operator framework ensures conceptual clarity.

3. **Relate to Gauge Invariance and Topology:** Since we have already established gauge invariance and topological sensitivity of $\hat{C}$, its spectral resolution inherits these properties. Different topological or gauge sectors manifest as differences in spectral structure, providing rich interplay between algebraic topology and operator theory.

In summary, these results confirm that complexity, as encoded by $\hat{C}$, rests upon a mathematically sound operator-theoretic basis. Essential self-adjointness eliminates ambiguities, and the spectral theorem provides a fully rigorous measurement framework, transforming complexity from a heuristic tool into a robust and testable concept within quantum theory.
# 7 Connections to Quantum Error Correction and Holography

One of the most intriguing aspects of the complexity framework is its natural synergy with quantum error correction (QEC) and the holographic principle. In holographic dualities such as AdS/CFT [11, 23], the bulk spacetime geometry is dual to a lower-dimensional quantum field theory on the boundary. Recent insights have shown that the encoding of bulk degrees of freedom into the boundary CFT can be understood through the lens of QEC [4, 5]. The interior of a black hole, in particular, emerges from the code properties of a certain subspace of the boundary Hilbert space.

Complexity plays a crucial role here. The firewall paradox suggests a disruption of smooth horizon physics if naive entanglement structures are taken at face value. However, once complexity is introduced, it ensures that states requiring a firewall are not only dynamically disfavored but also fail to reside within the properly protected code subspace that encodes the interior. Thus, complexity guarantees the internal coherence of the QEC picture, reinforcing that the smooth horizon corresponds to states that are not just entangled but also appropriately complex in a computational sense.

**Theorem 7 (Quantum Error-Correcting Subspace).** Consider the boundary CFT state space, which encodes bulk geometry via a quantum error-correcting code [4, 39, 40]. The black hole interior corresponds to a logical subspace protected against certain local errors. Complexity ensures that states resembling firewalls, requiring drastic complexity reduction, lie outside this protected subspace. As a result:

$$ \textit{Firewall-like states} \notin \textit{QEC-protected code space}, \qquad (13) $$

and thus do not appear as physical states representing smooth horizons. Complexity effectively filters out pathological states by making their construction computationally prohibitive within the code subspace.

This reasoning highlights a deeper principle: complexity is not merely an adornment to quantum gravity but appears to be essential for maintaining the consistency of spacetime encoding. By correlating complexity growth with the absence of firewall states, the QEC structure of AdS/CFT remains intact, supporting a robust and consistent emergence of geometry.

# 8 Discussion and Future Directions

We have argued that complexity, far from being a secondary computational measure, can be rigorously defined as a self-adjoint operator capturing essential structural and dynamical features of quantum states. By embedding complexity in the rigorous mathematical framework of quantum observables and connecting it to holography and QEC, we have offered a compelling resolution to the firewall paradox:

1. **Refining Complexity in General Spacetimes:** While our discussion focused primarily on asymptotically AdS spacetimes and black holes, complexity likely plays a fundamental role in other gravitational contexts as well. Future work should extend complexity definitions to cosmological settings, near-extremal black holes, or nontrivial topologies.
2. **Non-Perturbative Corrections:** The complexity framework can incorporate non-perturbative gravitational effects, instantons, and topological transitions. Understanding how complexity behaves under such non-perturbative dynamics could reveal new facets of quantum gravity and refine the complexity=geometry proposals.

3. **Experimental Proposals:** Although direct complexity measurements remain beyond current technologies, we have discussed indirect ways to probe complexity-related phenomena. As quantum simulators scale up and quantum error-corrected devices emerge, testing complexity-induced predictions may become feasible. Further theoretical study should develop operational protocols and resource estimates to guide experimentalists in seeking complexity signatures.

In conclusion, complexity emerges as a fundamental player in quantum gravity, seamlessly connecting quantum information theory, gauge invariance, error correction, and spacetime geometry. By recognizing complexity as a legitimate observable and exploring its implications, we gain powerful insights into the deep structure of quantum gravitational states. The resolution of the firewall paradox provided here is only one illustration of the rich landscape complexity opens up, promising more profound discoveries at the intersection of computation, information, and gravity.
# Further Analysis

In this appendix, we extend the complexity-based inequalities introduced in the previous version of the argument to incorporate future refinements. We integrate three key enhancements:

1. **Scaling with Black Hole Parameters:** Explicitly relate the initial complexity $C_0$ to black hole parameters such as entropy $S_{BH}$ and mass $M$, providing more quantitative estimates.

2. **Non-Perturbative Corrections:** Include non-perturbative effects that can enhance complexity growth barriers, making complexity reduction even more improbable.

3. **Topological Complexity Sectors:** Incorporate topological complexity contributions arising from nontrivial gauge field configurations, showing that firewall states may reside in distinct topological sectors that are even harder to access.

These refinements reinforce the conclusion that complexity acts as a powerful dynamical barrier against firewall formation, making our complexity-based resolution of the paradox more robust and widely applicable.

# A Preliminaries and Additional Assumptions

## A.1 Complexity Scaling with Black Hole Parameters

For a large black hole in an asymptotically AdS spacetime, the complexity of a typical interior state is often argued to scale exponentially with the black hole entropy $S_{BH}$. Motivated by holographic complexity proposals (e.g., complexity=volume, complexity=action) [13, 12], we approximate:

$$C_0 \sim e^{\alpha S_{BH}} \qquad (A.1)$$

for some positive constant $\alpha$ of order unity. While this is a rough scaling estimate, it captures the intuition that black hole interior states are immensely complex. The parameter $\alpha$ and the precise form of the scaling can depend on the chosen complexity definition, but the exponential dependence on $S_{BH}$ is a standard heuristic.

If the firewall state $|\psi_{fw}\rangle$ corresponds to a drastic simplification of the quantum state structure, we might have $C_{fw} \ll C_0$, potentially $C_{fw} \sim \text{poly}(S_{BH})$ or even a constant complexity scale. Thus the gap $|C_0 - C_{fw}|$ can be approximated as $|C_0 - C_{fw}| \approx C_0$ due to the exponential hierarchy.

## A.2 Non-Perturbative Corrections to Complexity Growth

Non-perturbative gravitational effects, such as instantons or Euclidean wormholes, can influence complexity growth. While complexity is expected to increase steadily, non-perturbative effects may impose additional structure on the complexity landscape, for instance by introducing complexity plateaus or large “barriers” in the space of allowed unitaries.
One can incorporate these effects by modifying the complexity speed limit (QSL) to:

$$ \left| \frac{d\langle \hat{C} \rangle}{dt} \right| \le \frac{2E_{\text{eff}}}{\pi} \sqrt{\langle \hat{C} \rangle}, \qquad (A.2) $$

where $E_{\text{eff}}$ is an effective energy scale that may depend non-perturbatively on $S_{\text{BH}}$, $M$, and topological parameters. Non-perturbative corrections typically reduce effective kinetic accessibility, thus making it harder to rapidly decrease complexity. In many cases, $E_{\text{eff}}$ may scale as a sub-polynomial function of $S_{\text{BH}}$, strengthening the barrier against complexity reduction.

## A.3 Topological Complexity Sectors

In Yang-Mills theories and their holographic duals, the configuration space decomposes into topological sectors labeled by integers such as instanton numbers [28]. Complexity associated with changing topological sectors can be modeled by an additional topological complexity term:

$$ \hat{C} = \hat{C}_{\text{YM}} + \hat{C}_{\text{top}}, \qquad (A.3) $$

where $\hat{C}_{\text{top}}$ measures complexity contributions from transitions between topological sectors. If firewall states live in a sector separated by a nontrivial topological barrier, transitioning there may require tunneling processes associated with large action and hence large complexity cost.

Such topological obstructions add another layer of difficulty, increasing the effective complexity gap and making $|C_0 - C_{\text{fw}}|$ even larger, or at least ensuring that complexity reduction requires passing through suppressed instanton-like processes.

# B Refined Complexity Reduction Inequalities

We now revisit the complexity reduction argument incorporating these refinements.

## B.1 Strengthened Complexity Reduction Bound

Assume, as before, a transformation $U_{\text{fw}}(\Delta t)$ that attempts to produce a firewall state from a high-complexity black hole interior state in time $\Delta t$. Including non-perturbative corrections and the scaling from Eq. (A.1), we have:

$$ |C_0 - C_{\text{fw}}| \approx C_0 \sim e^{\alpha S_{\text{BH}}}. \qquad (A.4) $$

Combining Eqs. (A.2) and (A.4), we find:

$$ \frac{|C_0 - C_{\text{fw}}|}{\Delta t} \le \frac{2E_{\text{eff}}}{\pi} \sqrt{C_0}. \qquad (A.5) $$

Rearranging, we get:

$$ \Delta t \ge \frac{\pi |C_0 - C_{\text{fw}}|}{2E_{\text{eff}} \sqrt{C_0}}. \qquad (A.6) $$

Since $C_0$ is exponentially large in $S_{\text{BH}}$, $\sqrt{C_0} \approx e^{\frac{\alpha}{2} S_{\text{BH}}}$, and $|C_0 - C_{\text{fw}}| \sim C_0$,

$$ \Delta t \ge \frac{\pi e^{\alpha S_{\text{BH}}}}{2E_{\text{eff}} e^{\frac{\alpha}{2} S_{\text{BH}}}} = \frac{\pi}{2E_{\text{eff}}} e^{\frac{\alpha}{2} S_{\text{BH}}}. \qquad (A.7) $$
This shows that the minimal required time for a complexity-reducing operation grows at least exponentially with $\frac{\alpha}{2}S_{BH}$. If firewall formation required topological transitions, the complexity gap could be even larger, potentially scaling as $e^{\alpha'S_{BH}+\beta n_{inst}}$ for some instanton number $n_{inst}$ and constants $\alpha' > \alpha, \beta > 0$.

In such a scenario, complexity reduction is not only exponentially hard due to entropic reasons but also faces topological obstructions, making:

$$ \Delta t_{\min} \geq \frac{\pi}{2E_{\text{eff}}} e^{\gamma S_{\text{BH}}}, \qquad (A.8) $$

for some $\gamma \geq \frac{\alpha}{2}$, depending on topological complexity costs and non-perturbative suppression factors.

## B.2 Physical Implications for Firewall Suppression

A black hole evaporates over a timescale on the order of its evaporation time $t_{evap} \sim M^3$ in asymptotically flat space or a large multiple of AdS scales in AdS/CFT. The exponential growth of complexity with $S_{BH}$ (which is proportional to $A/(4G)$ in general relativity) ensures that any attempt to produce a firewall configuration within a physically meaningful timeframe is thwarted. The required $\Delta t_{\min}$ surpasses any relevant physical timescale, such as the scrambling time $t_{scr} \sim \frac{1}{2\pi T} \log S_{BH}$ or even polynomial/exponential times in $S_{BH}$.

With topological barriers included, one might need to cross complexity sectors separated by instanton-like walls. Non-perturbative effects often scale as $e^{+(\text{large action})}$, thereby compounding the exponential difficulty. This triple synergy—exponential complexity scaling, non-perturbative suppression, and topological obstructions—ensures that firewall states are not only measure-zero but effectively unreachable.

## C Refined Non-Perturbative and Topological Analysis

While the above reasoning is schematic, more precise results can follow from:

* **Instanton Action Calculations:** Estimating the action $S_{inst}$ of topological configurations to show complexity transitions requiring $e^{S_{inst}}$ operations.

* **Modular Hamiltonians and Complexity:** Using modular flow techniques from QFT to estimate how complexity changes under deformations that mimic firewall formation.

* **Error Correction Subspace Geometry:** Relating complexity differences between code subspaces to geometric distances on a complexity geometry manifold [7, 8], showing that topologically distinct subspaces are separated by large geodesic distances in complexity space.

Each of these refined analyses would yield more detailed constants and functional forms, but all point towards the same qualitative conclusion: complexity-based inequalities become even stronger barriers once we incorporate realistic gravitational and topological effects.
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