Weak density of states

Rainer Dick

Rashid Ahmad

Abstract We investigate weak values in a many-body quantum system by introducing the weak-valued single-particle reduced density matrix as a probe of conditional local coherence. As a concrete example, we consider a three-particle W-state evolving under a discrete-time quantum walk and evaluate the weak-valued reduced density matrix at a selected lattice site. We find that the weak-valued particle density vanishes identically, while the off-diagonal coherence remains finite. The resulting weak-valued density matrix is therefore traceless but exhibits nonzero spin coherence. This separation arises from quantum interference among the paths contributing to a pre-and post-selected ensemble and provides an explicit many-body realization of a Cheshire-cat–like weak-value structure. The effect is obtained analytically within the standard Two-State Vector Formalism and confirmed numerically to persist under continuous deformations of the quantum coin. The robustness of the nonvanishing coherence contrasts with the exact cancellation of the weak-valued density, revealing a distinct interference pattern in the weak-valued reduced state. Our results demonstrate that weak-valued coherence and weak-valued density can exhibit qualitatively different spatial distributions in multi-particle quantum walks. This phenomenon reveals a distinctive weak-value structure in many-body quantum systems, where coherence and density exhibit markedly different conditional spatial distributions. By introducing the weak-valued reduced density matrix as a diagnostic tool, our work provides a new perspective on the interplay between coherence, interference, and post-selection in quantum dynamics, opening a route toward the systematic study of conditional coherence phenomena in multi-particle quantum walks and other complex quantum systems.

Rainer Dick

Edward R. Pike, Sarben Sarkar

Using the theory of generalized imprimitivities for the description of weakly localized single-photon states, the spatial dependence of the photodetection of such states is calculated and an ${x}^{\mathrm{\ensuremath{-}}7}$ falloff is found. A general technique for determining the falloff of densities associated with weakly localized states is given and in particular the same spatial dependence is found for the falloff of energy density in agreement with the previous work of Amrein.

Y. Ben-Aryeh

Explicit separable density matrices, for mixed two qubits states, are derived by the use of Hilbert Schmidt decompositions and Peres Horodecki criterion. A strongly separable two qubits mixed state is defined by multiplications of two density matrices given with pure states while weakly separable two qubits state is defined by multiplications of two density matrices which includes non-pure states. We find the sufficient and necessary condition for separability of two-qubits density matrices and show that under this condition the two-qubit density matrices are strongly separable.

Rainer Dick