Agreement Theorem for Quantum Mechanics

• The Agreement Theorem for Quantum Mechanics is a formal result that extends Aumann’s classical notion by using Hilbert spaces and projective measurements.
• It employs density operators and conditional quantum states to determine when agents with a common prior must converge in their probability assignments.
• The theorem distinguishes quantum from postquantum regimes and shows how decoherence and classical recording can restore epistemic agreement among agents.

The Agreement Theorem for Quantum Mechanics formalizes constraints on epistemic consistency between agents who assign probabilities to quantum events, generalizing Aumann’s classical result that “rational agents cannot agree to disagree.” In quantum theory, the theorem explores when agents interacting with a common quantum system, possessing a common prior (density operator), can or cannot maintain differing posteriors or probability assignments in the presence of common knowledge or common certainty. Canonical formulations and extensions cover standard quantum measurement modeling, generalized probability frameworks, modal hierarchies, agent-relativist interpretations, and discrimination between quantum and post-quantum regimes.

1. Classical Motivation and Quantum Generalization

Aumann’s original Agreement Theorem asserts that rational agents, sharing a common prior over a classical probability space and possessing common knowledge of their posteriors, cannot assign different probabilities to any given event. This principle was foundational in decision theory’s treatment of common knowledge and epistemic reasoning. Quantum generalization necessitates substituting classical sample spaces and partitions with Hilbert spaces, density operators, and measurement-induced projections. In the quantum setting, agents (e.g., Alice and Bob) interact with a shared finite-dimensional Hilbert space $H$, where knowledge and belief updating is mediated via projective measurements and conditional quantum states (Leifer et al., 2022).

2. Formal Quantum Setting: States, Measurements, and Conditionals

Let $H$ denote a finite-dimensional Hilbert space. Agents share a prior quantum state modeled by a density operator

$\rho \ge 0, \quad \Tr[\rho] = 1$

Events (propositions) are encoded by orthogonal projection operators,

$P_E = P_E^* = P_E^2, \quad 0 \le P_E \le I$

Each agent’s information is accessed via a projective measurement $\{P_K^k\}_{k=1}^n$ with

$$\sum_k P_K^k = I, \quad P_K^k P_K^{k'} = \delta_{kk'} P_K^k$$

Upon observing outcome $k$, the agent's Lüders-updated state is

$\rho_{|k} = \frac{P_K^k \rho P_K^k}{\Tr[\rho P_K^k]}$

The quantum conditional probability for event $E$ is then

$p(E \mid k) = \Tr[\rho_{|k} P_E] = \frac{\Tr[\rho P_K^k P_E P_K^k]}{\Tr[\rho P_K^k]}$

Analogous constructions hold for scenarios involving multiple agents, possibly on product Hilbert spaces and with agent-specific measurement bases (Leifer et al., 2022, Díaz et al., 26 Nov 2025).

3. Quantum Agreement Theorem: Statement and Proof Structure

For agents indexed by $i$, each with measurement $\{P_i^k\}$, define their post-outcome assignment on event $E$: $q_i(k) = \frac{\Tr[\rho P_i^k P_E P_i^k]}{\Tr[\rho P_i^k]}$ The event “agent $i$’s posterior for $E$ is $q_i$” is captured by the projector

$P_i(q_i) = \sum_{k: q_i(k) = q_i} P_i^k$

Common knowledge of the agents’ posteriors is encoded by the common-knowledge projector

$P_{\mathrm{CK}} = \bigwedge_{i=1}^N P_i(q_i)$

or, in finite dimensions,

$$P_{\mathrm{CK}} = \text{projector onto } \bigcap_i \mathrm{Ran}(P_i(q_i))$$

If the system state lies in $\mathrm{Ran} P_{\mathrm{CK}}$ and has nonzero prior ($\Tr[\rho P_{\mathrm{CK}}] > 0$), the theorem asserts: $q_1 = q_2 = \cdots = q_N$ The proof exploits convex decompositions into minimal orthogonal subspaces, demonstrating that all convex components assign the same probability to $E$ by linearity of the Born rule (Leifer et al., 2022).

4. Modal Hierarchies, Commutativity, and Common Certainty

A broader formulation distinguishes modalities of “certainty” via modal projectors and operator recursion. For agents (e.g., Alice, Bob) pursuing property $E$, define certainty-projectors for assignments $q_A, q_B$: $A_0 = Q_A(E; q_A), \quad B_0 = Q_B(E; q_B)$ Recursively,

$$A_{n+1} = A_n C_A(B_n), \quad B_{n+1} = B_n C_B(A_n)$$

with stabilization $A_* = A_N, B_* = B_N$, and the joint projector $C_* = A_* B_*$. Under commutativity ($[P_A^i, P_B^j] = 0$, $[P_A^i, P_E] = 0$, $[P_B^j, P_E] = 0$), QM enforces equality: $q_A = q_B$ Non-commuting scenarios permit “common certainty of disagreement” (CCD), where $q_A \neq q_B$ and both modalities stabilize, exemplified in explicit entangled setups. CCD is a distinct quantum phenomenon, not found classically (Díaz et al., 26 Nov 2025).

5. Disagreement Bounds and Postquantum Contrasts

Quantum mechanics forbids absolute disagreement: it is impossible for one agent to be certain that an event $E$ occurs and simultaneously certain that another agent is certain that $\neg E$ occurs. Formally: $\Tr[Q_A(E;1) \, C_A(Q_B(E;0)) \, \rho] > 0$ cannot hold for any $\rho$. No branch supports joint full certainty of opposing outcomes. By contrast, postquantum no-signaling boxes (notably the Popescu-Rohrlich box) can realize 0-1 disagreement, violating Tsirelson’s bound and signaling a departure from quantum epistemic discipline (Díaz et al., 26 Nov 2025, Contreras-Tejada et al., 2021).

6. Agreement Restoration via Classical Recording and Decoherence

When measurement outcomes are irreversibly recorded in a classical register (pointer states), the extended Hilbert space model incorporates a commutative subalgebra. Decoherence translates quantum branches into classical certainty, restoring standard agreement results. The quantum agreement theorem’s classical limit is thus an emergent property of quantum theory augmented by decoherent measurement (Díaz et al., 26 Nov 2025).

7. Interpretations Across QBism, Relational QM, and Foundations

QBism interprets the theorem conditionally: quantum formalism does not necessitate mutual consistency of measurement outcomes or probabilities between agents, as state assignments and outcomes are personal. Agreement is possible when agents adopt matching assumptions about outcome statistics, but no formal requirement enforces it. Classical Aumann agreement asserts a necessity; the quantum and QBist forms describe a possibility, contingent on agent beliefs and formal structure (Schack, 2023). The theorem delineates the boundaries of epistemic consistency, observer-dependence, and the transition from quantum to classical intersubjectivity (Leifer et al., 2022, Díaz et al., 26 Nov 2025, Contreras-Tejada et al., 2021).

8. Extension to Generalized Probability Theories and Foundational Impact

The quantum proof generalizes to any convex-operational or generalized probabilistic theory (GPT), provided the existence of affine states, exclusive measurement effects, a total-probability update mechanism, and finite partition refinements. Agreement remains enforced wherever these abstract structures apply, establishing epistemic consistency as a unifying principle across probabilistic physical theories (Leifer et al., 2022). Foundational consequences include the codification of quantum-classical epistemic transitions, formal agent-relativity bounds, and discriminators for viable postquantum theories.