Aharonov, Anandan, Popescu, and Vaidman [Phys. Rev. Lett. 64, 2965 (1990)] showed that a pre- and postselected system coupled to an auxiliary quantum control can undergo an effective evolution generated by a superposition of unitary time evolutions, realizing an effective force or an effective time translation whose parameter lies far outside the convex range of the individual evolutions, including a gravitational time-translation machine with an anomalous proper-time shift. The equalities, however, are heuristic and state-dependent, and the required postselection succeeds with exponentially small probability precisely when the effective parameter is large. This problem asks for the missing quantitative theory of that construction: a complete characterization of the states and parameter regimes for which the effective-evolution equality holds to specified error, the achievable success probability and the possibility of high-probability implementations, the rigorous status of the claimed operator-level version for bounded Hamiltonians, and the experimental feasibility of any realization, including the gravitational variant. A 2026 continuation by Aharonov, Cohen, and Shushi (APS Open Science 1, 000049) gives construction-specific short-time error bounds and multistep success probabilities, but no general characterization exists, so the core remains open.

Background

For a closed quantum system, dynamics over a fixed interval is a single unitary operator, U(T)=e−iHTU(T)=e^{-iHT}, fixed by one Hamiltonian HH and one duration TT. Aharonov, Anandan, Popescu, and Vaidman asked in 1990 what happens if one coherently superposes such evolutions. Their construction couples the system of interest to an auxiliary ("external") system whose conserved observable AA enters the Hamiltonian, so that each eigenvalue aia_i drives a different evolution U(T,ai)U(T,a_i). Preparing the auxiliary system in a superposition of the eigenstates ∣ai⟩|a_i\rangle and then postselecting it — keeping only experimental runs whose final measurement yields a chosen superposition outcome — leaves the system, up to normalization, in the state ∑iciU(T,ai)∣Ψ⟩\sum_i c_i U(T,a_i)|\Psi\rangle, with complex weights cic_i fixed by the pre- and postselected auxiliary states and satisfying ∑ici=1\sum_i c_i=1. When this linear combination acts on suitable initial states ∣Ψ⟩|\Psi\rangle as a single unitary U′U', one says the system has undergone an effective evolution. Two applications were given. First, amplification of weak forces: for the Hamiltonian H=−xAH=-xA, the momentum-space wave function transforms as ∑iciΨ(p−aiT)\sum_i c_i\Psi(p-a_iT), which equals Ψ(p−a′T)\Psi(p-a'T) with an effective force a′=∑iciaia'=\sum_i c_i a_i whenever the higher moments also match, ∑iciaik≅(a′)k\sum_i c_i a_i^k\cong(a')^k for k<Kk<K; since the cic_i may be negative, a′a' can lie far outside the convex range of the aia_i. A spin-N/2N/2 realization, with pre- and postselected spin directions making an angle θ\theta, yields a′=1/cos⁡θa'=1/\cos\theta, but the postselection succeeds only with probability of order (cos⁡2θ)N(\cos^2\theta)^N (about 10−1410^{-14} for N=15N=15, θ=70∘\theta=70^\circ). Second, a quantum time-translation machine: superposing evolutions of different durations gives ∑icie−iHTi∣Ψ⟩≅e−iHT′∣Ψ⟩\sum_i c_i e^{-iHT_i}|\Psi\rangle\cong e^{-iHT'}|\Psi\rangle with T′=∑iciTiT'=\sum_i c_iT_i, which may even be negative (effective evolution toward the past); the proposed gedanken implementation puts the radius of a massive spherical shell in a quantum superposition, so that the proper time inside the shell, Ti=T(1−2GM/c2Ri)1/2T_i=T(1-2GM/c^2R_i)^{1/2}, is coherently controlled, and postselection of the shell's state yields an anomalous effective proper-time shift for the system inside.

The catch, acknowledged by the authors, is that all these equalities are approximate and state-dependent. The initial momentum-space wave function (force amplification) or energy distribution (time translation) must decrease sufficiently rapidly at large values so that the neglected higher-order moments are small, and the duration TT must be sufficiently short; no quantitative bounds are given, the largest classes of admissible states are not characterized, and the asserted operator-level upgrade of the equality for systems with bounded, confining Hamiltonians is stated without proof. Moreover, the postselection probability becomes exponentially small exactly when the effective parameter is far from the given range, and no physical mechanism for coherently controlling the radius of a macroscopic massive shell is described. The construction is a close relative of weak-value amplification, where pre- and postselection of an ensemble produces anomalously large shifts of a weakly coupled meter, and its conceptual descendants include the quantum switch (a quantum control operation that coherently superposes the two orderings in which two operations act on a target), quantum operations with indefinite time direction (formalisms superposing an operation with its time reverse), and quantum temporal superposition in quantum field theory (superpositions of a detector's or field's evolution over different spacetime histories).

Problem Statement

For the postselected-superposition construction of Aharonov, Anandan, Popescu, and Vaidman, in which a pre- and postselected auxiliary system applies the linear combination ∑iciUi\sum_i c_iU_i (with ∑ici=1\sum_i c_i=1) to a system state ∣Ψ⟩|\Psi\rangle and one seeks an effective single evolution ∑iciUi∣Ψ⟩≅U′∣Ψ⟩\sum_i c_iU_i|\Psi\rangle\cong U'|\Psi\rangle with an effective force a′a' or effective duration T′T' possibly far outside the convex range of the contributing parameters: (i) characterize, quantitatively and as completely as possible, the classes of initial states for which the effective-evolution equality holds to a specified error — giving explicit bounds in terms of the tails of the momentum or energy distribution, the moment-matching order, the coefficients {ci}\{c_i\}, and the maximal duration or amplification for which the approximation is valid — and determine rigorously whether, and for which Hamiltonians, the equality upgrades to an operator-level statement on the whole Hilbert space; (ii) determine the achievable success probability of postselected implementations as a function of the target effective parameter, decide whether the exponentially small probabilities for large effective parameters are unavoidable within this mechanism, and establish whether high-probability or deterministic implementations of the same effective evolutions exist; (iii) assess whether any physically realizable variant — in particular the gravitational time-translation machine based on coherently controlled massive shells — can be implemented with present or foreseeable experimental capabilities; and (iv) determine whether amplification of weak effects by superposition of evolutions can yield a genuine, noise-robust practical advantage at acceptable success rates and, if so, identify concrete systems and measurement protocols that realize it. These four questions are jointly posed and together quantify when, at what cost, and with what experimental reach the superposition of time evolutions realizes a well-defined effective dynamics; for the feasibility and practical-advantage questions (iii) and (iv), a determination — including a negative one, that no physically realizable variant is feasible or that no noise-robust practical advantage exists — settles those questions. This is the objective left open by the 1990 paper.

References

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