Energy states

Joseph Akeyo Omolo

Factorization in normal order form provides positive energy solutions, while the alternative anti-normal order form provides negative energy solutions of the time-independent Schroedinger equation for a linear harmonic oscillator. The positive and negative energy state spaces are related by quantum conjugation effected through sign-reversal of Planck's quantization constant . � Photons occupying negative energy states have the same displacement xand Hamiltonian H, but opposite momentum p ˆ - compared to photons occupying the corresponding positive energy partner states. Emission of positive energy photons from a positive energy quantum state is equivalent to absorption of negative energy photons into a negative energy quantum state, leading to state lowering, while absorption of positive energy photons into a positive energy quantum state is equivalent to emission of negative energy photons from a negative energy quantum state, leading to state raising. The complete positive- negative energy spectrum of the quantized oscillator may then be

André F. Verbeure

Stephan Weis, Максим Евгеньевич Широков

We show that for any energy observable every extreme point of the set of quantum states with bounded energy is a pure state. This allows us to write every state with bounded energy in terms of a continuous convex combination of pure states of bounded energy. Furthermore, we prove that any quantum state with finite energy can be represented as a continuous convex combination of pure states with the same energy. We discuss examples from quantum information theory.

Joan L. Robinson

Daochen Wang

Quantum algorithms for estimating the ground state energy of a quantum system often operate by preparing a classically accessible quantum state and then applying quantum phase estimation. Whether this approach yields quantum advantage hinges on the state's energy spectrum, that is, the sequence of the state's overlaps with the energy eigenstates of the system Hamiltonian. We show that the energy spectrum of any entanglement-compressed quantum state must have large support if most energy eigenstates are highly entangled, an assumption supported by the eigenstate thermalization hypothesis. Furthermore, we show that if the compressed quantum state minimizes expected energy, then its energy spectrum decays with the inverse-squared energy eigenvalues under a convex relaxation of the compression constraint. This explains the main empirical finding of Silvester et al. [Phys. Rev. Lett. 134, 126503 (2025).PRLTAO0031-900710.1103/PhysRevLett.134.126503] that the energy spectra of matrix product states do not decay exponentially.