Observer Dependence without Contradiction

Observer dependence is consistent with quantum mechanics when it is implemented by a fixed restriction of one global state to an observer’s accessible algebra, or by compatible linear representations on declared code subspaces. It becomes a contradiction when the operator rule changes with an unknown state in a way that is nonlinear, gives different predictions on overlapping domains, or assigns different outcomes to two decompositions of the same density operator.

Required background. Quantum Information supplies channels, mixtures, code subspaces, Born probabilities, and distinguishability used in the consistency tests.

Helpful background. Semiclassical Gravity supplies observer and backreaction regimes; Quantum Fields in Curved Spacetime supplies infalling and exterior mode algebras; Causality and Global Structure supplies causal patches and horizon-crossing domains.

A valid change of description

Let $\mathcal A$ be a physical observable algebra with state $\omega$. Observer $O$ has access to a subalgebra $\mathcal A_O$ embedded by a fixed map $\iota_O:\mathcal A_O\to\mathcal A$. The observer’s state is the restriction

$$\omega_O(A)=\omega(\iota_O(A)).$$

In a finite-dimensional Schrödinger description, the same operation is a CPTP channel $\mathcal N_O$ from the global density operator to the observer’s records. Crucially, it is affine:

$$\mathcal N_O\!\left(p\rho+(1-p)\sigma\right) =p\mathcal N_O(\rho)+(1-p)\mathcal N_O(\sigma).$$

Partial trace, dephasing caused by an inaccessible phase reference, and restriction to a causal patch are examples when their physical algebras are fixed independently of the unknown state. Different observers can have different reduced states without the complete evolution violating unitarity.

A code-subspace representation can also be valid. A logical operator may have one exterior representative on code space $\mathcal C_1$ and another infalling representative on $\mathcal C_2$. Consistency requires each representation to be linear on its domain and the two to agree, to stated error, on $\mathcal C_1\cap\mathcal C_2$. The code domain and error are physical data, not labels chosen after seeing the state.

Operational complementarity

Black-hole complementarity restricts which interior and exterior observables one observer can compare. If the restriction follows from causal support, detector couplings, or bounded backreaction, it is a change in the accessible algebra. The global Born rule remains common to all observers. The stretched-horizon proposal gives the classic causal version Susskind, Thorlacius, and Uglum 1993, §§ 2–4.

This operational statement can prevent a duplication experiment. It does not by itself solve a contradiction formulated entirely within one common density operator. If early radiation $E$, late mode $B$, and interior partner $A$ are independent factors in that state, entropy inequalities apply whether or not one observer measures them. The AMPS argument uses precisely this distinction Almheiri et al. 2013, § 2.

Thus observer dependence can change the premises by denying a common factorization or limiting an operator’s domain. It cannot retain the same global factors and state while merely declaring an inconvenient inequality unobservable.

Linearity and overlap tests

Three tests expose a genuine change of description.

Mixture test. Predictions must depend on the density operator, not on a chosen ensemble decomposition, unless a classical ensemble label is included in the physical state. The affine equation above enforces this.

Superposition test. If $A_O$ is represented by an operator on a code space, its action on $\alpha\lvert\psi\rangle+\beta\lvert\phi\rangle$ must follow linearly from its action on the two states whenever the superposition remains in the domain.

Overlap test. Two observer reconstructions defined on overlapping code spaces must yield the same correlators and commutators on the overlap within one declared tolerance. Otherwise an allowed state has two incompatible Born probabilities.

Channel distance and entanglement fidelity make approximate versions quantitative Watrous 2018, Chapter 3. Agreement on a two-point function is too weak: many inequivalent operators share one correlator while differing on higher moments, commutators, or reference-entangled inputs.

A decomposition counterexample

The maximally mixed qubit has two decompositions,

$$\frac{I}{2} =\frac12\bigl(\lvert0\rangle\langle0\rvert+\lvert1\rangle\langle1\rvert\bigr) =\frac12\bigl(\lvert+\rangle\langle+\rvert+\lvert-\rangle\langle-\rvert\bigr).$$

To exhibit an actual contradiction, suppose a proposed device outputs $\lvert0\rangle$ on either pure input $\lvert0\rangle$ or $\lvert1\rangle$, but outputs $\lvert1\rangle$ on either $\lvert+\rangle$ or $\lvert-\rangle$. Extending this rule by averaging over the preparation ensemble assigns $\lvert0\rangle\langle0\rvert$ to the first decomposition of $I/2$ and $\lvert1\rangle\langle1\rvert$ to the second. Measuring $Z$ on the output distinguishes them with certainty. No channel on the input qubit can do this because both inputs are the same density operator. This is preparation-context dependence, not ordinary observer dependence. Merely choosing a $Z$ or $X$ measurement on $I/2$ would not demonstrate the contradiction: either choice gives equally likely outcomes.

If the ensemble label is stored in an accessible classical register $C$, the two global states on qubit plus $C$ are different and a channel can prepare its output conditional on that register. The adverse check therefore pinpoints what must be added to make the rule physical.

Results and open questions as of 2026-08-19

It remains unresolved whether a complete black-hole interior can be represented with observer- or state-dependent rules that pass all mixture, superposition, and overlap tests. Competing positions differ over factorization, code-subspace size, and the permitted operator domain.

Evidence or formulation	Result	Scope and limitation
Fixed observer channel Watrous 2018, Chapters 2–3	Linear restriction produces consistent observer-dependent reduced states	Requires a state-independent channel and a declared global input algebra
Causal-patch complementarity Susskind, Thorlacius, and Uglum 1993, §§ 2–4	Operationally blocks one observer from comparing interior and recovered exterior records	Does not alone construct globally compatible operators for old-hole states
AMPS common-state test Almheiri et al. 2013, § 2	Entropy tension survives if $E$, $B$, and $A$ remain independent in one state	Changing factorization can evade the premise but requires a replacement algebra
Restricted code-subspace representation Harlow 2016, §§ 7.1–7.4	Allows multiple logical representations without independent clones	Overlap consistency, arbitrary superpositions, and nonperturbative accuracy remain model-dependent

The conservative position accepts observer-dependent access while demanding one state-independent global algebra. A code-subspace position permits representation dependence within controlled domains. Stronger state-dependent proposals allow the interior map to vary with a reference state, raising Born-rule and overlap questions. The alternatives must be judged on their domains rather than merged into one slogan.

The conclusions could change after a construction proves overlap consistency for a parametrically large state class, or after an explicit counterexample shows unavoidable Born-rule violation, or after an operational protocol distinguishes two proposed observer algebras without leaving the claimed semiclassical and backreaction regime.

Boundaries

The criteria do not require every observer to access the same records. They require all predictions to arise from linear quantum mechanics on a declared domain. Generic channel and code theory belongs to Quantum Information, mode algebras to Quantum Fields in Curved Spacetime, and explicit holographic interior reconstructions elsewhere.

Exercise

An operator rule gives different outcomes for the two decompositions of $I/2$ above. What minimal additional system can make the rule linear, and when does that repair fail?

Solution

Add an orthogonal classical register recording which preparation procedure was used. A controlled channel can then output $\lvert0\rangle$ or $\lvert1\rangle$ according to that register, and linearity holds on the enlarged density operator. The repair fails if the register is not physically present or accessible, because the two preparations then define the same qubit density operator and no channel on that qubit alone can distinguish their decompositions. It also fails if superpositions of register states are assigned nonlinear actions.

References

• Almheiri, Ahmed, Donald Marolf, Joseph Polchinski, and James Sully. 2013. “Black Holes: Complementarity or Firewalls?” Journal of High Energy Physics 2013 (02): 062. Open text.
• Harlow, Daniel. 2016. “Jerusalem Lectures on Black Holes and Quantum Information.” Reviews of Modern Physics 88: 015002. Open text.
• Susskind, Leonard, Lárus Thorlacius, and John Uglum. 1993. “The Stretched Horizon and Black Hole Complementarity.” Physical Review D 48: 3743–3761. Open text.
• Watrous, John. 2018. The Theory of Quantum Information. Cambridge: Cambridge University Press. DOI.