Minimal Length from a Bounded Group Velocity in Quantum Mechanics

Bijan Kumar Bagchi, Andreas Fring

Francisco Ancelmo Pinheiro Ferreira

Francisco Ancelmo Pinheiro Ferreira

Fundamental kinematic constraints in condensed matter systems, such as Lieb–Robinson bounds, sound velocities, and finite bandwidths, impose strict limits on quasiparticle group velocities. We show that these ubiquitous bounds naturally lead to an emergent minimal position uncertainty, derived entirely within standard quantum mechanics without deforming the canonical commutation relations. Using a variational Sturm–Liouville formulation in momentum space, we establish a universal relation that links spatial resolu- tion limits directly to measurable system parameters. Beyond variance-based measures, we further characterize this emergent scale using Shannon entropy, Fisher information, and quantum Fisher information, thereby revealing its information-theoretic fingerprint. Representative estimates for semiconduc- tors and ultracold atomic systems yield minimal lengths on the nanometer scale, many orders of magnitude above the Planck length but fundamental within effective descriptions. Our results provide a unified perspective that connects abstract localization limits with tangible condensed matter con- straints, opening new avenues for experimental and metrological exploration of minimal-length phenomena.

Michael A. Nielsen

What is the minimal size quantum circuit required to exactly implement a specified n-qubit unitary operation, U, without the use of ancilla qubits? We show that a lower bound on the minimal size is provided by the length of the minimal geodesic between U and the identity, I, where length is defined by a suitable Finsler metric on the manifold SU(2^n). The geodesic curves on these manifolds have the striking property that once an initial position and velocity are set, the remainder of the geodesic is completely determined by a second order differential equation known as the geodesic equation. This is in contrast with the usual case in circuit design, either classical or quantum, where being given part of an optimal circuit does not obviously assist in the design of the rest of the circuit. Geodesic analysis thus offers a potentially powerful approach to the problem of proving quantum circuit lower bounds. In this paper we construct several Finsler metrics whose minimal length geodesics provide lower bounds on quantum circuit size. For each Finsler metric we give a procedure to compute the corresponding geodesic equation. We also construct a large class of solutions to the geodesic equation, which we call \emph{Pauli geodesics}, since they arise from isometries generated by the Pauli group. For any unitary U diagonal in the computational basis, we show that: (a) provided the minimal length geodesic is unique, it must be a Pauli geodesic; (b) finding the length of the minimal Pauli geodesic passing from I to U is equivalent to solving an exponential size instance of the closest vector in a lattice problem (CVP); and (c) all but a doubly exponentially small fraction of such unitaries have minimal Pauli geodesics of exponential length.

Pulak Ranjan Giri

Matteo A. C. Rossi, Tommaso Giani, Matteo G. A. Paris

Quantum gravity theories predict a minimal length at the order of magnitude of the Planck length, under which the concepts of space and time lose every physical meaning. In quantum mechanics, the insurgence of such minimal length can be described by introducing a modified position-momentum commutator, which in turn yields a generalized uncertainty principle, where the uncertainty on the position measurement has a lower bound. The value of the minimal length is not predicted by theories and must be evaluated experimentally. In this paper, we address the quantum bound to estimability of the minimal uncertainty length by performing measurements on a harmonic oscillator, which is analytically solvable in the deformed algebra of the Hilbert subspace.