6-Photon Interferometric Setup

• The paper introduces a 6-photon interferometric setup that extends Wigner’s friend scenarios by embedding quantum observers within fully unitary and decohering dynamics.
• It applies sequential measurement unitaries and environment-induced broadcasting to quantitatively capture information redundancy and the transition to classical objective outcomes.
• Numerical results demonstrate the exponential decay of quantum coherences and establish objectivity thresholds, offering insights for scalable quantum measurement implementations.

The Extended Wigner’s Friend (EWF) scenario generalizes the original Wigner's friend thought experiment by embedding quantum observers (“Friends”) and their environments within a broader, fully unitary dynamics explicitly coupled to decohering environments. This approach leverages the formalism of quantum Darwinism (QD) to quantitatively analyze the emergence of classicality, the objectivity of measurement records, and the viability of Wigner-friend-type paradoxes when measurement outcomes become redundantly encoded in environment fragments. The following sections give a detailed, technically rigorous account of this embedding and its consequences, as established in "Emergence of Classicality in Wigner's Friend Scenarios" (Rivlin et al., 28 Jul 2025).

1. Hilbert Space Architecture and Initial State

A bipartite EWF scenario comprises two "wings" (e.g., Alice/Charlie and Bob/Debbie). Each wing is structured hierarchically:

• System ($S_1$, $S_2$): $\mathcal{H}_i \simeq \mathbb{C}^2$ (qubits).
• Friend macroscopic register ($F_1$, $F_2$): multipartite, $\mathcal{H}_{F_i}=(\mathbb{C}^2)^{\otimes N_F}$, modeling $N_F$ qubits per friend; dimension $d_F=2^{N_F}$.
• Laboratory Environment ($E_1$, $E_2$): again, $S_2$0-qubit registers, $S_2$1, with $S_2$2.

The initial global state before any dynamics is a direct product:

$S_2$3

The system-system entangled state is:

$S_2$4

This architecture ensures the ability to encode, broadcast, and protect measurement results at varying levels of macroscopicity.

2. Unitary Measurement and Decoherence Dynamics

Measurement processes and decoherence are described by sequential, explicit unitaries:

Premeasurement (System–Friend coupling):

$S_2$5

which implements a von Neumann-type entanglement of the system’s pointer basis to the friend-register.

Friend–Environment broadcasting (Decoherence):

$S_2$6

for each environmental qubit $S_2$7 ($S_2$8), implementing dephasing and imprints of the measurement result onto the environment. The full environment interaction $S_2$9 realizes a spectrum broadcast structure (SBS) in the long-time limit, compatible with quantum Darwinism.

The total Hamiltonian governing joint evolution takes the block-diagonal form:

$\mathcal{H}_i \simeq \mathbb{C}^2$0

drawn from the Gaussian Unitary Ensemble to model chaotic, generic interactions. After equilibration ("pinching" in the pointer basis), the state becomes

$\mathcal{H}_i \simeq \mathbb{C}^2$1

3. Quantum Darwinism: Information Redundancy and Objectivity

After measurement and decoherence, the structure of objectivity is assessed using QD’s mutual information and redundancy metrics:

• Mutual information for system–fragment pairs:

$\mathcal{H}_i \simeq \mathbb{C}^2$2

with $\mathcal{H}_i \simeq \mathbb{C}^2$3 the von Neumann entropy.

• Classical objectivity is achieved if for many disjoint environment fragments $\mathcal{H}_i \simeq \mathbb{C}^2$4, the mutual information plateaus at the system’s classical entropy:

$\mathcal{H}_i \simeq \mathbb{C}^2$5

• Redundancy $\mathcal{H}_i \simeq \mathbb{C}^2$6 quantifies how many such fragments independently reveal the classical record up to error $\mathcal{H}_i \simeq \mathbb{C}^2$7:

$\mathcal{H}_i \simeq \mathbb{C}^2$8

Objectivity requires $\mathcal{H}_i \simeq \mathbb{C}^2$9, enabling multiple independent observers to access the same pointer record without disturbing the system.

4. Numerical Results: Emergence of Classicality

Simulation outcomes quantify the scaling of objectivity and the obfuscation of quantum effects:

• Indistinguishability decay: The overlap $F_1$0 decays exponentially with friend-register size, $F_1$1, and the Helstrom error $F_1$2 decays as $F_1$3.
• Objectivity thresholds: Plateau behavior in $F_1$4 and large $F_1$5 emerge already for modest memory+environment sizes ($F_1$6–$F_1$7 qubits), and $F_1$8 scales linearly with environment size $F_1$9 for fixed $F_2$0.
• Robustness: Variance of Hamiltonian parameters (chaotic/universal ensemble) does not affect qualitative outcomes; classicality emerges universally upon scaling up memory and environment fragments.

5. Residual Quantum Effects: Surviving Wigner–Friend Paradoxes

Despite the emergence of objectivity, subtle quantum coherence effects between $F_2$1 and $F_2$2 can survive in small labs:

• Errors for reading out the pointer value from $F_2$3 alone, $F_2$4, fall rapidly ($F_2$5) as $F_2$6 increase.
• Discrepancy for non-pointer measurements $F_2$7 on $F_2$8, $F_2$9, remains substantial ($\mathcal{H}_{F_i}=(\mathbb{C}^2)^{\otimes N_F}$0) for small registers. This directly quantifies the presence of residual quantum coherences accessible to a sufficiently powerful “superobserver.”
• As the total register size increases, $\mathcal{H}_{F_i}=(\mathbb{C}^2)^{\otimes N_F}$1 decays on the same exponential scale as $\mathcal{H}_{F_i}=(\mathbb{C}^2)^{\otimes N_F}$2, leading to the practical vanishing of WF-type discrepancies in the macroscopic limit.

6. Comparison of Simple, Extended, and Quantum Darwinism Models

• Simple WF (no $\mathcal{H}_{F_i}=(\mathbb{C}^2)^{\otimes N_F}$3): maximal paradoxes survive, as coherences are unprotected.
• WF + unspecific $\mathcal{H}_{F_i}=(\mathbb{C}^2)^{\otimes N_F}$4 (tracing out): artificial enforcement of agreement ($\mathcal{H}_{F_i}=(\mathbb{C}^2)^{\otimes N_F}$5), unjustified unless $\mathcal{H}_{F_i}=(\mathbb{C}^2)^{\otimes N_F}$6 truly inaccessible.
• WF + explicit QD: genuine paradoxical effects persist, size-limited by the joint register size; only when pointer records are redundantly broadcasted does classical objectivity emerge and paradoxes become unobservable.
• Extended WF (Local Friendliness): multipartite WF inequalities (e.g., CHSH-type) can only be violated for small labs; plateaus in redundancy and vanishing $\mathcal{H}_{F_i}=(\mathbb{C}^2)^{\otimes N_F}$7 preclude any violation once memory+environment size exceeds a modest threshold ($\mathcal{H}_{F_i}=(\mathbb{C}^2)^{\otimes N_F}$8).

7. Implications: Absolute Events and the “Classical Limit”

In this explicit, unitary QD framework, the transition from quantum “relative facts” to classical absolute events is quantitatively controlled:

• For small quantum apparatuses, “superobserver” measurements can still access quantum coherences and produce disagreements with naive classical assignments—a vivid manifestation of quantum relativity of outcomes.
• As soon as measurement data is redundantly and robustly broadcast to the environment, any observer accessing a distinct fragment must agree with all others on the pointer record—the operational signature of classicality and the objectivity of events.
• Thus, the classical notion of an “absolute event” is not a fundamental axiom but an emergent property of the spectrum broadcast structure reached in the thermodynamic limit, as modeled and confirmed by QD metrics and numerical simulation.

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Summary Table: Objectivity and WF Effects as a Function of Register Size

Total Register Size (N_F + N_E)	Overlap Tr[ρ_F^0 ρ_F^1]	Helstrom Error δ_HD	Redundancy R_δ	Paradox Grade (Δ)	Objectivity
2–3 qubits	~0.5	~0.4	~1	~0.4	No
5–8 qubits	≲0.05	≲0.05	≳4	0.05–0.1	Emerging
>10 qubits	≲0.001	≲10⁻³	≫10	≲10⁻³	Yes

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Conclusion:

The unitary quantum Darwinism framework for extended Wigner’s friend scenarios demonstrates that the emergence of classicality—a regime where pointer records become objective and WF-type paradoxes evaporate—is governed by the scaling of information redundancy via decoherence and the broadcasting of records, rather than by any fundamental principle of absolute events. Genuine quantum paradoxes remain only in sufficiently small laboratories with unprotected coherence; classical objectivity is a consequence of environmental-induced decoherence in the macroscopic limit (Rivlin et al., 28 Jul 2025).