Macroscopic Cat States in Many-Electron Systems

• Macroscopic cat states are quantum superpositions of distinct electronic configurations that probe the quantum-to-classical transition.
• W-cat states demonstrate high resistance to particle loss and local depolarizing noise, maintaining entanglement (up to ~44% noise tolerance).
• Metrics like logarithmic negativity and Bell inequality violations quantify their coherence and nonlocality, critical for quantum metrology and computing.

A macroscopic cat state is a quantum superposition of states that are macroscopically distinct, such as all electrons in a system collectively occupying different sectors of Hilbert space, or exhibiting distinct values of a large magnetization or current. In the many-electron context, these states reflect entanglement across large numbers of particles and are of central interest for probing the quantum-to-classical transition, testing foundational questions of realism and locality, and designing robust quantum information devices. The following sections outline key advances in constructing, protecting, and diagnosing macroscopic cat states in many-electron systems, drawing on theoretical models, experimental protocols, and measures of quantum coherence and robustness.

1. Structure and Definition of Macroscopic Cat States

In many-electron systems, a macroscopic cat state generically takes the form of a superposition between two (or more) macroscopically distinct electronic configurations. A prominent instance is the W-cat state, defined as an entangled state between a microscopic qubit and an N-electron macroscopic sector:

$$|H_C\rangle_{\mu\,A_1\dots A_N} = \frac{1}{\sqrt{2}}\left(|0\rangle_\mu \otimes |W_N\rangle_{A_1\dots A_N} + |1\rangle_\mu \otimes |0^{\otimes N}\rangle_{A_1\dots A_N}\right)$$

where $|W_N\rangle$ denotes the N-particle W state (a uniform superposition over all states with one excitation), and $|0^{\otimes N}\rangle$ is the collective vacuum. The states of the macroscopic sector are chosen so that one (for instance $|W_N\rangle$) violates a multipartite Bell inequality—with critical visibility $p_N^{crit} = N/[(\sqrt{2} - 1)\cdot2^{N-1} + N]$—while the other ($|0^{\otimes N}\rangle$) does not (Mishra et al., 2012). This nonlocal distinction is central to the macroscopic nature of the superposition.

More generally, macroscopic cat states include states in which collective observables (such as total spin, current, or magnetization) have sharply separated expectation values in each component, or where electron occupation is macroscopically different in momentum or real space.

2. Robustness against Decoherence and Particle Loss

Robustness to environmental noise is a critical property for macroscopic cat states in realistic many-electron systems. The W-cat state exhibits unique resistance to two main classes of errors:

• Particle loss: If $m$ electrons are lost from the system, the remaining state (microscopic qubit + N–m macroscopic electrons) preserves a significant fraction of its entanglement. The smallest eigenvalue (from partial transposition with respect to the microscopic part) is $\lambda_- = -\frac{1}{2}(1 - m/N)$, and the logarithmic negativity—a measure of entanglement—is $E_N = \log_2(2 - m/N)$. For large $N$ and $m \ll N$, entanglement remains high.
• Local depolarizing noise: When each electron undergoes depolarization ($D_p$ channel: $$|i\rangle\langle j| \to \frac{p}{2}I + (1-p)|i\rangle\langle j|$$), the degradation of entanglement in the W-cat state is significantly slower than in GHZ states. For $N=10$, the W-cat state retains entanglement up to $\sim44\%$ noise, compared to $\sim28\%$ for GHZ (Mishra et al., 2012).

Combined models of particle loss and decoherence (with analytic and numerical treatment of composite eigenvalues) confirm the persistence of substantial entanglement in experimentally relevant regimes, indicating practical viability for scalable quantum computation and metrology.

3. Metrics of Quantum Coherence and Entanglement

Quantum coherence in a macroscopic cat state is operationally captured by measures such as logarithmic negativity for bipartite decompositions:

$E_N(\rho) = \log_2\left[2N(\rho) + 1\right]$

where $N(\rho)$ is the sum of negative eigenvalues after partial transposition. High $E_N$ implies persistent quantum superposition between microscopic and macroscopic sectors—even under environmental interactions. For states with collective observables, macroscopic "indefiniteness" and Fisher information (classical and quantum) are employed to quantify the degree to which the state's vital status (e.g., alive vs. dead in the cat analogy) is truly quantum rather than classically mixed (Kelly et al., 2018). The ratio $r_q$ of quantum-to-total uncertainty serves as a benchmark: $r_q=1$ indicates that essentially all uncertainty is quantum, while $r_q \to 0$ reflects a transition to a statistical mixture.

For W-cat states and related constructions, off-diagonal coherence remains robust: the remaining quantum coherence after local noise and loss surpasses that found in other canonical macroscopic superpositions.

4. Bell Nonlocality and Macroscopic Distinction

A defining trait of certain cat states is that the macroscopic components are not only distinguishable via collective observables, but also possess fundamentally different nonlocal properties. In the W-cat construction, $|W_N\rangle$ violates a multipartite Bell inequality, while $|0^{\otimes N}\rangle$ does not. As $N$ increases, their critical visibilities diverge, so the macroscopic distinction is sharpened in the violation or satisfaction of local realism (Mishra et al., 2012). This dichotomy provides a rigorous framework for discussing quantum nonlocality at scale and illustrates that macroscopic distinctness need not correspond simply to orthogonality on collective observables, but can encode deep differences in fundamental quantum correlations.

The practical implication is that cat states with this structural property are diagnostically accessible via Bell-type measurements, providing unambiguous certification of macroscopic quantum behaviour.

5. Macroscopic Cat States in Quantum Information, Measurement, and Technology

Macroscopic cat states that are robust to particle loss and local noise offer substantial advantages for quantum information processing:

• Fault-tolerant encoding: The persistence of coherence and entanglement even with partial loss positions these states as strong candidates for physical qubits or logical encoding in large-scale quantum devices.
• Quantum measurement theory: The hybrid micro–macro entanglement informs foundational studies of the measurement problem, enabling precise tests of the quantum-classical boundary in multi-electron systems.
• Quantum metrology and networks: The ability to maintain entanglement and superposition across macroscopic electron sectors underpins advances in quantum-enhanced sensing and distributed entangled networks.

From a fundamental perspective, the existence and robustness of such states exemplify how quantum nonlocality and superposition can persist even as the number of constituents grows, challenging classical intuitions about the emergence of macroscopic realism.

6. Theoretical and Experimental Context

The findings for W-cat states contrast with canonical GHZ-type or Greenberger–Horne–Zeilinger states, where macroscopic distinction is often solely in collective observables and is highly sensitive to local noise and loss. The explicit construction of W-cat states with their specific noise-resilience features is informed by advances in quantum optics, atomic ensemble physics, and superconducting circuit architectures—sectors where these states can, in principle, be engineered and detected via parity, collective excitation counting, or quantum tomography.

Experimental protocols for verifying macroscopicity include entanglement witnesses (e.g., logarithmic negativity), Bell inequality measurements (suitable for large-$N$ sectors), and phase-space methods for distinguishable pointer states. Preservation of coherence under sequential loss and noise channels remains a key benchmark for scalability.

7. Significance for the Quantum-to-Classical Transition

Robust macroscopic cat states act as testbeds for investigating the interplay of decoherence, locality, and realism in many-body quantum systems. The observation that one can construct superpositions whose components are not only macroscopically distinct by measurement outcome but also fundamentally different in their nonlocal properties provides a stringent challenge to classical explanations. Analyses of resilience to decoherence show that quantum-mechanical predictions remain dominant until noise reaches critical (but quantifiable) thresholds.

This body of research—exemplified by the W-cat state—clarifies the conditions under which quantum effects can persist on macroscopic scales and shapes our understanding of practical limits in quantum computing and metrology, as well as the quantum-classical transition for mesoscopic and many-electron systems (Mishra et al., 2012).