*The concept of heat death captivated Victorian-era physicists, serving as an early illustration of the intricate connection between everyday physics and profound cosmological themes. Analogous to dropping ice cubes into water, disrupting equilibrium, the subsequent melting and chilling reach a common temperature, albeit without a sense of progress. This principle, recognized by 19th-century thermodynamics pioneers, extends to the entire universe. After the stars exhaust their luminosity, remnants like gas, dust, and stellar corpses will culminate in equilibrium, leading to what Hermann von Helmholtz termed a “state of eternal rest” in 1854.*

*Despite the endurance of this foundational idea in modern cosmology, recent contemplations among physicists suggest that a seemingly heat-dead universe harbors more intrigue than initially perceived. Prompted by questions about black holes, a realm typically overshadowed by more prominent cosmic enigmas, researchers are revisiting assumptions about universal evolution. Standard understandings suggest black holes persistently change beyond the expected equilibrium, prompting a reevaluation of broader evolutionary processes. Physicist Brian Swingle from Brandeis University notes the newfound interest, acknowledging the transformative role of black holes in reshaping our understanding of equilibrium and universal dynamics.*

*Even after an ice cube melts and achieves equilibrium with the liquid, physicists traditionally declare the system’s evolution concluded. However, at the quantum level, life persists beyond the concept of heat death, revealing weird and fascinating phenomena. Xie Chen, a theoretical physicist at the California Institute of Technology, emphasizes that while particle and energy distributions may equilibrate in a quantum system, the complexity extends far beyond apparent equilibrium. Chen, along with researchers like Brian Swingle, suggests that if an equilibrated system appears mundane, it’s a matter of perspective.*

*The focus has shifted from directly observable quantities to highly delocalized ones, demanding innovative measures for tracking. Circuit complexity, a concept originating in computer science, has been adapted, or some argue, misappropriated, to quantify intricate patterns in quantum systems. This approach intertwines various scientific domains, encompassing not only black holes but also quantum chaos, topological phases of matter, cryptography, quantum computers, and the potential for even more potent machines. The exploration of quantum complexity opens a gateway to understanding the richness embedded in seemingly equilibrated systems, transcending conventional perspectives and unraveling the intricacies of the quantum realm.*

*A new paradox: black holes challenge conventional understanding.*

*In the mid-20th century, black holes intrigued scientists due to their singularity, a point at their core where falling matter becomes infinitely compacted, gravity escalates limitlessly, and the known laws of physics break down. Stephen Hawking’s insights in the 1970s introduced the information paradox related to the black hole’s perimeter or horizon, fueling the quest for a unified theory of physics. Leonard Susskind, in 2014, uncovered a fresh enigma: the black hole’s interior volume.*

*While externally resembling a massive black ball that grows when matter falls in, its interior defies conventional spatial volume formulas. Within a black hole, the challenge arises from defining spatial volume in a space-time continuum, where no unique method exists due to the warping effect. Susskind proposed a slicing process maximizing spatial volume at each moment, akin to the shortest distance across the hole, leading to perpetual growth of volume within the deforming space-time, challenging established principles and prompting a deeper exploration of black hole dynamics.*

*The perplexing growth of the black hole’s interior volume, seemingly violating thermodynamic laws akin to a glass of water reaching equilibrium, prompted Leonard Susskind to employ lateral thinking. Utilizing the AdS/CFT duality, a strategy positing two mathematically equivalent views of fundamental physics—one with gravity and one without—Susskind linked the strongly gravitating black hole to a nongravitational but intensely quantum system, specifically a thermal state of quantum fields resembling a hot plasma of nuclear particles.*

*Despite the apparent dissimilarity between a black hole and a hot plasma, the duality’s power lies in connecting seemingly unrelated entities. The temperature of the black hole, its interaction with dropped material, and the ripple effect in the plasma all find equivalence through this intriguing concept. The duality exchanges gravitational oddities for quantum theory complexities, allowing Susskind to explore how the black hole evolves in a manner seemingly contradictory to its equivalent plasma state. This approach led him to propose a property unrelated to plasmas or black holes,challenging conventional notions within the realm of physics.*

*The Meaning of Complexity in Black Hole Physics.*

*Leonard Susskind introduced a novel concept, circuit complexity, to decipher the mysteries of black hole evolution. Drawing inspiration from electronic circuits, the idea explores the intricate operations within black holes, akin to the logical and arithmetic functions in traditional computers. In this quantum realm, qubits serve as the fundamental units of information, manipulated by gates to perform diverse operations. The adoption of circuit complexity sheds light on the dynamic nature of black holes, providing a unique lens to understand their behavior and challenging conventional notions of equilibrium.*

*In the realm of quantum computing, qubits, whether particles, ions, or superconducting loops, embody the essence of information manipulation within circuits. This extends beyond the traditional computer architecture, with everyday systems, like air molecules in a room, envisioned as circuits with collisions representing gates. The concept of circuit complexity aligns with our intuitive understanding of complexity, reflecting the number of elementary gates required to replicate a specific state in a quantum system. This numerical perspective offers insights into the intricate dynamics of quantum processes, bridging the technical and everyday notions of complexity.*

*Applying the concept of circuit complexity to the quantum realm, Susskind proposed an intriguing solution to the black hole evolution paradox. In the realm of AdS/CFT duality, where hot plasmas are equivalent to black holes, Susskind argued that even after reaching thermal equilibrium, the quantum state of the plasma continues evolving, becoming progressively more complex.The reverberations within the plasma, though dissipating at a macroscopic level, persist at the quantum level, making the recreation of a plasma with identical ripples increasingly intricate. Drawing the parallel, Susskind asserted that the black hole’s volume is mathematically equivalent to the circuit complexity of the plasma. As circuit complexity perpetually expands, so does the volume, resolving the enigma of the perpetually growing black hole.*

*Unveiling the Circuit Complexity Paradox: Borrowed Without Permission.*

*Computer scientists initially recoiled at the proposal, as circuit complexity wasn’t intended to depict the evolution of physical systems. Its purpose was to gauge the intrinsic difficulty of computational tasks. Physicist Aram Harrow from MIT clarified that circuit complexity aims to capture instances where computations can be performed more efficiently. Take multiplication as an example—traditional long multiplication scales quadratically with the number of digits, but circuit complexity reveals it to be more efficient than the conventional grade school method suggests.*

*For computer scientists, the connection between circuit complexity and physics remained elusive. They viewed circuit complexity as a theoretical tool to assess algorithms, not a tangible aspect of physics. To illustrate, if presented with the seemingly random sequence 3, 1, 4, 1, 5, 9, it appears as the outcome of a complex algorithm. However, recognizing it as the beginning of π reveals a simpler algorithm at play. The challenge lies in distinguishing between apparent randomness and the underlying simplicity encoded in certain sequences.*

*Determining circuit complexity without the hint of a recognizable pattern would necessitate exhaustive trial and error—testing every conceivable circuit to identify the one reproducing the given digits. Merely finding one solution wouldn’t suffice; the goal is to locate every possible circuit and then select the absolute shortest. Adam Bouland from Stanford acknowledged the challenge, stating that grasping or estimating the complexity of such functions is inherently difficult. Even Scott Aaronson, a computer scientist at the University of Texas, Austin, who advocated for the relevance of complexity theory in fundamental physics, found himself in an ironic position when Lenny Susskind got involved, prompting him to apply the brakes on his enthusiasm.*

*Cracking the Quantum Code: A Challenge Yet to Be Surmounted.*

*While computer scientists acknowledged Susskind’s observation that both complexity and a black hole’s interior volume increase, they questioned the existence of a genuine connection. Skepticism lingered—either another quantity equated to interior volume, or the AdS/CFT duality was flawed, rendering the pursuit of such a quantity a futile endeavor.*

*In a thorough investigation, Bouland, Bill Fefferman from the University of Chicago, and Umesh Vazirani from the University of California, Berkeley, meticulously examined Susskind’s proposal, scrutinizing both facets of the holographic duality. On one side, they delved into the black hole and its purportedly equivalent interior volume; on the other, they explored the hot plasma counterpart.*

*Analyzing the black hole initially seemed straightforward, assuming that Susskind’s defined interior volume was a measurable quantity. However, the challenge emerged when considering practical measurements within a black hole. Astronauts entering a black hole at the speed of light toward their inevitable demise lack the opportunity to measure its entire interior volume. “They don’t have much time in there before they hit the singularity, so they can’t possibly feel the entire space,” explained Bouland.*

*Realizing that direct entry into a black hole wasn’t necessary, Bouland and colleagues opted for a simulated approach governed by gravity’s laws. This simulation involved astronauts within the black hole exchanging laser signals, revealing the interior’s volume. Despite initial concerns, the researchers found the black hole’s interior volume to be readily calculable, thanks to precise computer simulations offering a God’s-eye view.*

*Shifting focus to the plasma, they likened it to a cryptographic block cipher, where reshuffling code characters mirrors particle churning in the plasma. This seemingly disparate analogy was mathematically demonstrated as equivalent. Moreover, decrypting a block cipher’s message aligned with inferring the circuit complexity of a quantum state.*

*However, combining the black hole’s straightforward volume calculation with the intricate nature of circuit complexity posed a challenge. The clash of apples-and-oranges emerged, as black hole volume calculation appeared manageable while circuit complexity defied easy computation. This dilemma disrupted the foundational principle of theoretical computer science, challenging the clear delineation between computational difficulty levels. Susskind’s conjecture now became a conundrum not just for physicists but also for computer scientists, as it jeopardized established distinctions in computational complexity.*

*Delving into the Complex Conundrums of Quantum Physics.*

*To resolve the paradox, researchers sought to maintain the integrity of computational difficulty classifications. The challenge was to introduce hidden complexity into seemingly straightforward volume calculations. Bouland and collaborators explored two possibilities. First, they considered the idea that simulating black holes might not be as easy as assumed, making interior volume calculations more complex. However, this proposition clashed with the foundational concept of a computer as a universal device capable of efficiently simulating any natural phenomenon.*

*The quantum extended Church-Turing thesis, regarded as a profound principle by computer scientists, underscores the generality of computers in recapitulating the reductionist structure of nature. While the violation of this thesis is not implausible, scientists acknowledge that, as seen with the original Church-Turing thesis, which required quantum computers to efficiently simulate quantum systems, the complexity landscape may hold surprises.*

*To unravel the paradox, researchers considered the possibility that the quantum theory of gravity, governing black holes, might surpass the computational capacity even of quantum computers. In this scenario, jumping into a black hole could provide insights that surpass the computational capabilities of simulations. This perspective envisions a black hole as a computational entity vastly more powerful than a quantum computer, akin to how a quantum computer surpasses a classical one.*

*However, theorists, including Susskind and Aaronson, lean towards the alternative view proposed by Bouland, Fefferman, and Vazirani. They suggest that the translation between black hole and plasma, or vice versa, poses the computational challenge. While individually analyzing the black hole or plasma may be computationally feasible, mapping properties between them could be exponentially hard, even for a quantum computer. This convoluted mapping, according to Aaronson, renders a hard problem persistently challenging, whether approached directly or through the AdS/CFT duality.*

*The AdS/CFT duality, a mind-bending concept for over 25 years, poses challenges in visualizing the equivalence of vastly different systems like a black hole and a hot plasma. Yet, the difficulty transcends human imagination; it’s deeply ingrained in the mathematics. Computer scientists, once hesitant about circuit complexity as a legitimate physical quantity due to its computational challenges, now align with Susskind.*

*The complexity’s inherent difficulty, if linked to the challenging black-hole-to-plasma translation, doesn’t undermine its validity. Instead, it reflects the expected struggle when dealing with equivalences between two sides. Susskind finds satisfaction in his critics evolving into allies, acknowledging the profound implications of this realization within the realm of physics.*

*The Dynamics of Space Travel.*

*Susskind’s conjecture faces a potential challenge regarding the growth rate of the circuit complexity of a hot plasma. While the expectation is for complexity to intuitively increase over time, reflecting the dynamic changes in the plasma, the issue arises when applying circuit complexity to uncontrolled random interactions in the plasma. Unlike the predictable logic operations of a computer algorithm, these random interactions make it uncertain how complexity will evolve.*

*The plasma might undergo numerous interactions, leading to a complex quantum state, but a subsequent interaction could abruptly reset it to a simpler state. The defined complexity is based on the number of interactions needed to reach the endpoint, not the total number experienced. This discrepancy is likened to exploring a neighborhood, where turning left or right at intersections can lead to unexpected outcomes, much like arriving at a nearby restaurant that appears distant due to relative positions rather than the actual walking distance.*

*Susskind’s initial argument against the plasma’s complexity abruptly resetting was based on the vastness of the space of possibilities. He deemed it highly improbable for the system to stumble into a simpler state given the mind-bogglingly large array of possibilities. However, translating this intuition into a rigorous argument proved challenging.*

*One approach, taken by Fernando Brandão and his co-authors, explored what occurs when a system undergoes consecutive random interactions. This leads to the formation of a set known as a design, with states uniformly spread throughout the space of possibilities. Chaotic systems naturally generate a sequence of designs approximating a truly random distribution with increasing refinement. As randomness signifies maximal complexity, the system becomes more complex at a rate nearly matching the growth of the black hole interior.*

*Despite these insights, full proof remains elusive as debatable simplifications persist, and not all approaches align perfectly with the behavior of black holes. This leaves a conclusive demonstration on the theorists’ to-do list.*

*The emerging second law of quantum complexity.*

*Undeterred by the absence of a complete proof, Susskind and Brown proposed in 2018 that the continuous growth of complexity constitutes a new law of nature—the second law of quantum complexity, analogous to the second law of thermodynamics. Drawing a parallel with the second law of thermodynamics, which states that closed systems increase in entropy until reaching thermal equilibrium, Susskind and Brown asserted that complexity undergoes a similar trajectory. A system continues to increase in complexity for an extended period after achieving thermal equilibrium, eventually reaching “complexity equilibrium.” At this juncture, a quantum system exhaustively explores all possible states it can attain, signaling the end of progress.*

*This conceptualization of complexity’s eventual plateau prompted Susskind to reassess his original motivation for exploring circuit complexity—namely, the growth of black hole interiors. While general relativity predicts perpetual growth, the notion of complexity reaching equilibrium suggests an eventual cessation. This realization adds a new dimension to the existing reasons for suspecting that black holes must ultimately be described by a quantum theory of gravity, hinting at the limitations of general relativity.*

*In 2021, Iliesiu, Márk Mezei from Oxford University, and Gábor Sárosi from CERN delved into the implications of complexity equilibrium for black holes. Employing the path integral, a standard quantum physics method agnostic to the specific quantum theory of gravity, they discovered that quantum effects accumulate akin to barnacles on a ship’s hull, ultimately halting the growth of the black hole’s interior. This marks a significant milestone in black hole evolution, introducing an additional stage unrelated to known events such as the final evaporation and disappearance of the object. The change in the black hole’s interior geometry at this point represents a novel facet in the understanding of black hole dynamics.*

*Decoding the Quantum Journey: Understanding the Five Stages of Quantum Systems.*

*The insights gleaned from black holes extend beyond these celestial objects, revealing a broader principle about matter. The unfolding narrative encompasses the entire life cycle of quantum systems, a category that encompasses most entities, including the entire universe. The envisioned life cycle comprises five distinct stages. **The first stage is initialization, characterized by the system’s simplicity—initially just a collection of particles or building blocks, each acting independently.*

*Subsequently, the system enters thermalization. Particles engage in random motion and collisions, gradually achieving thermal equilibrium. This dynamic process intertwines the particles through quantum entanglement, a phenomenon Susskind terms “scrambling.” Information disseminates throughout the system to the extent that it is no longer localized, akin to the far-reaching impact a butterfly’s wings can have on global weather patterns. Nick Hunter-Jones, a theoretical physicist at the University of Texas, Austin, likens it to operators that, initially localized, become distributed across the entire system in a butterfly-effect-like manner.*

*Following thermalization and the establishment of thermal equilibrium, the quantum system progresses to the stage of complexification. Despite being in equilibrium, the system continues to evolve, growing more intricate in a manner that traditional measures like entropy may not readily capture. Theorists turn to circuit complexity, a tool unveiling the increasingly elaborate connections among entangled particles. This stage, significantly lasting exponentially longer than thermalization, provides a microscopic view into the system’s entanglement structure.*

*The subsequent phase is complexity equilibrium, where the system’s complexity reaches a peak. Although changes persist, the system no longer exhibits a sense of directed evolution and instead meanders among equal states of maximal complexity. **The final stage, recurrence, sees the system returning to its original simple state, a highly improbable event unfolding over an exceedingly long time—an exponential of an exponential. This entire life cycle repeats indefinitely.*

*In essence, quantum systems reaching thermal equilibrium resemble the happy couples in romantic comedies. The film often concludes with marriage, portraying it as the endpoint of the characters’ love story. However, in reality, this marks just the beginning, as the quantum systems continue their perpetual cycle of evolution.*

*For more information about this topic, https://www.quantamagazine.org/in-new-paradox-black-holes-appear-to-evade-heat-death-20230606/?fbclid=IwAR0qc7V2FX0zx9UrWSYACY3M3EEAKDZ_Ub-So1YSU-COjQH3-qnp8hhBeiU*