We analyze quantum states formed as superpositions of an initial pure product state and its image under local unitary evolution, using two measurement-based measures of superposition size: one based on the optimal quantum binary distinguishability of the branches of the superposition and another based on the ratio of the maximal quantum Fisher information of the superposition to that of its branches, i.e., the relative metrological usefulness of the superposition. A general formula for the effective sizes of these states according to the branch-distinguishability measure is obtained and applied to superposition states of
N quantum harmonic oscillators composed of Gaussian branches. Considering optimal distinguishability of pure states on a time-evolution path leads naturally to a notion of distinguishability time that generalizes the well-known orthogonalization times of Mandelstam and Tamm and Margolus and Levitin. We further show that the distinguishability time provides a compact operational expression for the superposition size measure based on the relative quantum Fisher information. By restricting the maximization procedure in the definition of this measure to an appropriate algebra of observables, we show that the superposition size of, e.g., NOON states and hierarchical cat states, can scale linearly with the number of elementary particles comprising the superposition state, implying precision scaling inversely with the total number of photons when these states are employed as probes in quantum parameter estimation of a 1-local Hamiltonian in this algebra.
We extend several measurement-based definitions of effective superposition size to coherent state superpositions with branches composed of either single coherent states or tensor products of coherent states. These measures of superposition size depend on determining the maximal quantum distinguishability of certain states associated with the superposition state: e.g., in one measure, the maximal distinguishability of the branches of the superposition is considered as in quantum binary decision theory; in another measure, the maximal distinguishability of the initial superposition and its image after a one-parameter evolution generated by a local Hermitian operator is of interest. The scaling of the size of the superposition with the number of modes and mode intensity (i.e., photon number) is compared to the scaling of certain geometric properties of the Wigner function of the superposition and also to the superposition size estimated experimentally from decoherence. We also apply earlier comparison-based methods for determining macroscopic superposition size that require a reference Greenberger-Horne-Zeilinger (GHZ) state. The case of a hierarchical Schr\"odinger cat state with branches composed of smaller superpositions is also analyzed from a measurement-based perspective.
We analyze the conditional states in which one part of a split spin-squeezed state is left, upon performing a collective spin measurement on the other part. For appropriate measurement directions and outcomes, we see the possibility of obtaining states with high quantum Fisher information, even reaching the Heisenberg limit. This allows us to propose a metrological protocol that can outperform standard approaches, for example, in a situation where the number of particles in the probe is bounded. The robustness of this protocol is investigated by considering realistic forms of noise present in cold-atom experiments, such as particle number fluctuations and imperfect detection. Ultimately, we show how this measurement-based state-preparation approach can allow for the conditional (i.e., heralded) preparation of spin Schr\"odinger's cat states even when the initial state before splitting is only mildly squeezed.
Schrödinger’s cat condenses the quantum measurement problem into an amplification chain: a microscopic alternative becomes correlated with distinguishable records, while linear quantum dynamics retains the records in an entangled state. The problem is not a puzzle about an animal with incompatible predicates. It concerns the relation between unitary dynamics, reduced-state classicality, and observing a outcome. This article develops the cat problem from the density-operator description of measurement, deriving the role of partial tracing and environmental decoherence, and distinguishing a decohered improper mixture from a proper classical ensemble. It examines pointer-state selection, Lindblad dynamics, quantum Darwinism, and responses to the outcome problem: Everettian branching, consistent histories, Bohmian mechanics, and objective-collapse models. Experimental realizations in trapped ions, cavity quantum electrodynamics, and mechanical resonators demonstrate superpositions of distinguishable alternatives. Bosonic cat codes show that macroscopic-superposition structure can be engineered as an information resource. The article identifies four weaknesses: conflating diagonal reduced states with collapse; treating non-equivalent “cat-size” measures as interchangeable; insufficient treatment of environmental noise in collapse tests; and an underdeveloped connection between measurement theory and error-corrected bosonic architectures.
In this paper, we examine measurement using the Q-based, objective-field model for quantum mechanics. Schrodinger considered a microscopic system prepared in a superposition of states which is then coupled to a macroscopic meter. We analyze the entangled meter and system, and measurements on it, by solving forward-backward stochastic differential equations for real amplitudes
x(t) and
p(t) that correspond to the phase-space variables of the Q function of the system at a time
t. We model the system and meter as single-mode fields, and measurement of
x^ by amplification of the amplitude
x(t). Our conclusion is that the outcome for the measurement is determined at (or by) the time
tm, when the coupling to the meter is complete, the meter states being macroscopically distinguishable. There is consistency with macroscopic realism. By evaluating the distribution of the amplitudes
x and
p postselected on a given outcome of the meter, we show how the
Q-based model represents a more complete description of quantum mechanics: The variances associated with amplitudes
x and
p are too narrow to comply with the uncertainty principle, ruling out that the distribution represents a quantum state. We conclude that the collapse of the wavefunction occurs as a two-stage process: First there is an amplification that creates branches of amplitudes
x(t) of the meter, associated with distinct eigenvalues. The outcome of measurement is determined by
x(t) once amplified, explaining Born's rule. Second, the distribution that determines the final collapse is the state inferred for the system conditioned on the outcome of the meter: information is lost about the meter, in particular, about the complementary variable
p.
We study quantum states produced by optimal phase covariant quantum cloners. We argue that cloned quantum superpositions are not macroscopic superpositions in the spirit of Schrödinger's cat, despite their large particle number. This is indicated by calculating several measures for macroscopic superpositions from the literature, as well as by investigating the distinguishability of the two superposed cloned states. The latter rapidly diminishes when considering imperfect detectors or noisy states and does not increase with the system size. In contrast, we find that cloned quantum states themselves are macroscopic, in the sense of both proposed measures and their usefulness in quantum metrology with an optimal scaling in system size. We investigate the applicability of cloned states for parameter estimation in the presence of different kinds of noise.