Phase retrievability of a quantum channel asks whether pure states can be reconstructed from suitable measurements. In this paper, we study this problem from three complementary viewpoints: quantum information theory, operator-valued frames, and the physical realization through quantum interferometry. We first show that a quantum channel is phase retrievable if and only if its complementary channel is pure-state informationally complete. This structural characterization leads to several consequences for phase retrievability, including criteria involving the dimension of the complementary operator system, Choi-rank type bounds, and specific results for entanglement breaking channels and twirling channels. We then introduce an interferometric coupling in which two arm channels are coherently recombined through port operators $M_i(θ)=A_i+e^{iθ}B_i$ . Unlike classical mixing, this construction produces interference cross terms that can enlarge the complementary operator system and thereby enhance phase retrievability. From the frame theory viewpoint, the interferometer realizes a coherent coupling of operator-valued frames. To quantify this effect, we introduce injectivity indices for completely positive maps. The examples in Section~5 show that coherent interference can significantly improve phase retrieval behavior even when the arm channels are individually not phase retrievable.Phase retrievability of a quantum channel asks whether pure states can be reconstructed from suitable measurements. In this paper, we study this problem from three complementary viewpoints: quantum information theory, operator-valued frames, and the physical realization through quantum interferometry. We first show that a quantum channel is phase retrievable if and only if its complementary channel is pure-state informationally complete. This structural characterization leads to several consequences for phase retrievability, including criteria involving the dimension of the complementary operator system, Choi-rank type bounds, and specific results for entanglement breaking channels and twirling channels. We then introduce an interferometric coupling in which two arm channels are coherently recombined through port operators $M_i(θ)=A_i+e^{iθ}B_i$. Unlike classical mixing, this construction produces interference cross terms that can enlarge the complementary operator system and thereby enhance phase retrievability. From the frame theory viewpoint, the interferometer realizes a coherent coupling of operator-valued frames. To quantify this effect, we introduce injectivity indices for completely positive maps. The examples in Section~5 show that coherent interference can significantly improve phase retrieval behavior even when the arm channels are individually not phase retrievable.

quant-pharXiv:2604.24363Journal of Physics A Mathematical and TheoreticalApr 28, 20260 citations

Phase Retrievability of Super Operators and Measurements

Javier Alejandro Chávez-Domínguez, Jeremiah Ifeanyi Isu

We continue the study of Phase Retrieval in the Quantum Information setting in the style of \cite{liu2023phase}, in a manner which is more general in two primary ways: (a) Instead of only pure states we consider also states (and positive semidefinite operators) of bounded rank, as is done in Quantum Tomography; (b) We consider general super operators instead of only quantum channels. We show that in order to have phase retrieval with respect to positive semidefinite operators of bounded rank, it suffices to discriminate between perfectly distinguishable pairs (i.e. those with orthogonal supports). From this we prove that phase retrievability is always Lipschitz stable with respect to the trace norm, and then we use the Fuchs-van de Graaf inequalities to deduce stability with respect to the Bures--Wasserstein distance (thus generalizing the known stability results for phase retrieval in Frame Theory). For Hermitian-preserving super operators taking values in block-diagonal matrices, we characterize phase retrievability in terms of conditions inspired by the classical complement property from Frame Theory.We continue the study of Phase Retrieval in the Quantum Information setting in the style of \cite{liu2023phase}, in a manner which is more general in two primary ways: (a) Instead of only pure states we consider also states (and positive semidefinite operators) of bounded rank, as is done in Quantum Tomography; (b) We consider general super operators instead of only quantum channels. We show that in order to have phase retrieval with respect to positive semidefinite operators of bounded rank, it suffices to discriminate between perfectly distinguishable pairs (i.e. those with orthogonal supports). From this we prove that phase retrievability is always Lipschitz stable with respect to the trace norm, and then we use the Fuchs-van de Graaf inequalities to deduce stability with respect to the Bures--Wasserstein distance (thus generalizing the known stability results for phase retrieval in Frame Theory). For Hermitian-preserving super operators taking values in block-diagonal matrices, we characterize phase retrievability in terms of conditions inspired by the classical complement property from Frame Theory.

quant-pharXiv:2607.26474Jul 30, 20260 citations

Experimental Approaches to the Phase Problem of Optical Coherence Theory.

David Köhler

Jan 1, 19730 citations

Some Theorems Concerning the Phase Problem of Coherence Theory

Demosthenes Dialetis

The spectral density of a fluctuating light beam may be determined from the knowledge of both the modulus and the phase of the complex degree of self-coherence γ(τ) of the beam. The phase itself may be determined from the modulus and from the location of the zeros of the analytic continuation of γ(τ) in the lower half of the complex τ plane. In the present paper results of an investigation are presented which show that the determination of the zeros is equivalent to the solution of a certain inhomogeneous eigenvalue problem of the Sturm-Liouville type on a semi-infinite frequency range. This eigenvalue problem is found to be equivalent to a certain stability problem in mechanics. Although no general technique for the solution of this type of an eigenvalue problem appears to be known, the new formulation may be used to determine spectral profiles for which the associated degree of self-coherence has zeros at prescribed points in the complex τ plane. Some illustrative examples are given.The spectral density of a fluctuating light beam may be determined from the knowledge of both the modulus and the phase of the complex degree of self-coherence γ(τ) of the beam. The phase itself may be determined from the modulus and from the location of the zeros of the analytic continuation of γ(τ) in the lower half of the complex τ plane. In the present paper results of an investigation are presented which show that the determination of the zeros is equivalent to the solution of a certain inhomogeneous eigenvalue problem of the Sturm-Liouville type on a semi-infinite frequency range. This eigenvalue problem is found to be equivalent to a certain stability problem in mechanics. Although no general technique for the solution of this type of an eigenvalue problem appears to be known, the new formulation may be used to determine spectral profiles for which the associated degree of self-coherence has zeros at prescribed points in the complex τ plane. Some illustrative examples are given.

Journal of Mathematical PhysicsAug 1, 19679 citations

M_i(θ)=A_i+e^{iθ}B_i

. Unlike classical mixing, this construction produces interference cross terms that can enlarge the complementary operator system and thereby enhance phase retrievability. From the frame theory viewpoint, the interferometer realizes a coherent coupling of operator-valued frames. To quantify this effect, we introduce injectivity indices for completely positive maps. The examples in Section~5 show that coherent interference can significantly improve phase retrieval behavior even when the arm channels are individually not phase retrievable.