Black Hole Protocol: Quantum Measurement & Memory

• Black Hole Protocol is a quantum procedure that uses black hole dynamics to encode, transfer, and recover information through entanglement and memory concepts.
• It employs negative entropy and dual measurement protocols to model black hole evaporation, linking holography with quantum measurement theory.
• Operator correspondence between external observables and internal memory pointers recasts black hole entropy as a quantifier of memory capacity.

A "Black Hole Protocol" refers to any precisely specified quantum or classical procedure that exploits the distinctive information-theoretic or dynamical properties of black holes for encoding, transferring, recovering, or probing information. These protocols systematically relate black hole physics to measurement theory, quantum information, and holography, enabling the extraction, reconstruction, or manipulation of black hole microstates, Hawking radiation, or interior information. Diverse realizations exist in the literature, ranging from Dirac-sea-inspired quantum memory models and toy teleportation schemes to rigorous AdS/CFT microstate detection algorithms. This article surveys the theoretical frameworks, mathematical formalism, key operators, and representative implementations drawn from contemporary arXiv research.

1. Quantum Measurement Protocols and Negative Entropy

The foundational insight connecting quantum measurement theory to black hole thermodynamics stems from the equivalence of single-system (Copenhagen) and entanglement-assisted (von Neumann) measurement schemes via the concept of negative entropy (Song, 2013).

• Copenhagen protocol (single system, S): A system S in pure state $|\psi\rangle$ is measured by an observable O; after measurement, S is projected onto an eigenstate $|\theta\rangle$ with no explicit auxiliary system.
• von Neumann protocol (system + memory, S+M): A memory M (pointer basis $|\theta\rangle_M$) becomes entangled with S via a unitary: $$U: |\theta\rangle_S \otimes |0\rangle_M \rightarrow |\theta\rangle_S \otimes |\theta\rangle_M$$. Tracing out M leaves S in a mixed state.

Critically, after entanglement, the randomness (uncertainty) of S increases ($\Delta H_{\mathrm{ran}}^S = +1$), while M's decreases ($\Delta H_{\mathrm{ran}}^M = -1$). In information entropy, $H_{\mathrm{info}} = -H_{\mathrm{ran}}$, so the conditional entropy $H(S|M)$ becomes negative—the memory M carries $-1$ bit (“antiqubit”) of information, analogous to the Dirac sea of negative energy. Cerf–Adami interpret this as a sea of negative information; measurement observables are "holes" in this sea.

The equivalence: The two-system entangled protocol with a memory antiqubit is operationally identical to a single-system protocol with negative information filling the vacuum.

2. Black Hole Evaporation as Quantum Measurement

This measurement equivalence has direct analogues in black hole evaporation (Song, 2013).

• Single-system (“Copenhagen-like”) Hawking evaporation: Only outside-horizon modes are considered; outgoing occupation numbers are measured by an observable $N^{\text{out}}$ via Bogoliubov transformations, with no manifest memory, reproducing thermal radiation statistics.
• Two-system (“von Neumann-like”) Unruh–Israel approach: The eternal black hole Hartle–Hawking vacuum is written as a maximally entangled sum over inside and outside microstates:

$$|\psi\rangle = \frac{1}{\sqrt{\Omega}} \sum_{\theta=0}^{\Omega-1} |\theta\rangle_{\text{out}} \otimes |\theta\rangle_{\text{in}}$$

Inside-horizon degrees of freedom serve as quantum memory.

By treating the inside Hilbert space as a Dirac sea of negative information, a "hole" represents a definite memory pointer $$Q_\theta^{\text{in}} = |\theta\rangle_{\text{in}} \langle \theta|$$.

3. Operator Correspondence: $O_\theta^{\text{out}} = Q_\theta^{\text{in}}$

A central result is the operator identity between the outside observer's measurement and the inside memory pointer (Song, 2013):

$O_\theta^{\text{out}} = Q_\theta^{\text{in}}$

Here,

• $O_\theta^{\text{out}}$ is a basis-changing observable acting on the outside modes, generated via unitary transformations on the outside Hilbert space, labelling distinct ways the black hole can be read.
• $Q_\theta^{\text{in}}$ labels the pointer state of the interior (memory).
• The identification asserts strict correspondence: "pointer $\theta$ inside" $\leftrightarrow$ "observable $\theta$ outside".

This operator equation underpins the entire protocol: choice of observable outside the horizon is dual to a pointer selection in the black hole’s internal entanglement structure.

4. Black Hole Entropy and Measurement Choices

The information-theoretic content of the black hole protocol is crystallized in the formula for the Bekenstein–Hawking entropy as a log-count of measurement or memory choices (Song, 2013):

$S_{\text{BH}} = \ln M$

where $M = \Omega$ is the number of distinguishable pointer states $|\theta\rangle_{\text{in}}$ or, equivalently, the number of measurable observables $O_\theta^{\text{out}}$. Entropy is thus recast as the logarithm of the number of equally probable measurement choices available to an outside observer, directly mirroring the statistical mechanical definition in terms of microstate counting.

5. Black Hole as Quantum Memory: Dirac-Sea Analogy and Protocol Steps

The Dirac-sea analogy is central to understanding the black hole as a quantum memory filled with negative information (Song, 2013):

1. Initialize: Eternal black hole in entangled microcanonical state across inside and outside.
2. Memory filling: Interior is interpreted as a Dirac sea of negative information (antiqubits), each "hole" ($|\theta\rangle_\text{in}$) representing an empty memory slot.
3. Pointer selection: Each interior label $\theta$ determines a memory pointer $Q_\theta^{\text{in}}$, identified with an observable $O_\theta^{\text{out}}$ via $O_\theta^{\text{out}} = Q_\theta^{\text{in}}$.
4. Thermalization: Tracing out the interior (forgetting the memory content) leaves the outside in a mixed, thermal state, replicating quantum measurement decoherence.
5. Entropy: The horizon entropy $S_{\text{BH}} = \ln M$ quantifies the number of possible pointer choices, recasting horizon entropy as a measure of quantum memory capacity.

This protocol demonstrates that black hole evaporation is formally indistinguishable from ordinary quantum measurement: unitary entanglement with a memory system, followed by loss of access to that memory (trace operation).

6. Protocol Implications and Generalizations

The Black Hole Protocol, as detailed above, shows that the quantum information processing carried out by black holes is no different than that in standard quantum theory—Hawking evaporation is a measurement-like process, with non-unitarity arising from the usual tracing over inaccessible degrees of freedom (Song, 2013). The identification $O_\theta^{\text{out}} = Q_\theta^{\text{in}}$ brings out the equivalence of entanglement-based and observable-based approaches to black hole radiation. The interpretation of negative entropy as the quantum analog of the Dirac sea provides a conceptual foundation for the black hole as a quantum memory device.

Extensions and related constructs can involve

• Negative entropy and "antiqubits" in generalized quantum settings,
• Quantum memory protocols in other objective-reduction (collapse) theories,
• Information-theoretic reconstructions of black hole entropy in contexts incorporating islands or refined measurement protocols.

The protocol thus anchors black hole information theory within the formal machinery of quantum measurement and memory, clarifying the statistical and operational meaning of black hole entropy and the unitary dynamics underpinning Hawking radiation (Song, 2013).