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arXiv:2608.30006

Frame phase retrievability and state distinguishability of quantum channels

Deguang Han, Kai Liu

quant-phSubmitted Aug 30, 2026• Announced Sep 1, 2026• 0 citations
v1 · Aug 30, 2026v2 · Sep 25, 2026
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Abstract

This survey introduces the role of frame phase retrieval in pure-state identification and information preservation by quantum channels. For unit vectors, the lift $x\mapsto xx^*$ identifies vectors that differ only by a global phase with the same rank-one quantum state, converts frame intensities into linear functionals of the lifted state, and makes the adjoint pullback of output observables an operator-valued measurement on the input system. From this viewpoint, a channel is phase retrievable exactly when it is injective on pure states. We develop this correspondence through Choi-rank-two linear combinations of Kraus operators, higher-rank relative spectra, structural obstructions, and constructions with prescribed Choi rank. We distinguish injective identification from full tomography, perfect one-shot discrimination, zero-error classical communication, and exact quantum correction. Twirling channels then provide a structured setting in which commutants, irreducible dimensions, multiplicities, orbit-frame orthogonality, coding indices, and phase-retrievable subspaces can be read from group representations.

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This survey introduces the role of frame phase retrieval in pure-state identification and information preservation by quantum channels. For unit vectors, the lift x↦xx∗x\mapsto xx^*x↦xx∗ identifies vectors that differ only by a global phase with the same rank-one quantum state, converts frame intensities into linear functionals of the lifted state, and makes the adjoint pullback of output observables an operator-valued measurement on the input system. From this viewpoint, a channel is phase retrievable exactly when it is injective on pure states. We develop this correspondence through Choi-rank-two linear combinations of Kraus operators, higher-rank relative spectra, structural obstructions, and constructions with prescribed Choi rank. We distinguish injective identification from full tomography, perfect one-shot discrimination, zero-error classical communication, and exact quantum correction. Twirling channels then provide a structured setting in which commutants, irreducible dimensions, multiplicities, orbit-frame orthogonality, coding indices, and phase-retrievable subspaces can be read from group representations.
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 Mi(θ)=Ai+eiθBiM_i(θ)=A_i+e^{iθ}B_iMi​(θ)=Ai​+eiθBi​. 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.
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.
A twirling channel is a quantum channel induced by a continuous unitary representation π=∑i⊕miπiπ= \sum_{i}^{\oplus} m_iπ_iπ=∑i⊕​mi​πi​, where πiπ_iπi​ are inequivalent irreducible representations. Motivated by a recent work \cite{Twirling} on minimal mixed unitary rank of ΦπΦ_πΦπ​, we explore the connections of the independence number, zero error capacity, quantum codes, orthogonality index and phase retrievability of the quantum channel ΦπΦ_πΦπ​ with the irreducible representation multiplicities mim_imi​, the irreducible representation dimensions dim⁡Hπi\dim H_{π_i}dimHπi​​. In particular we show that the independence number of ΦπΦ_πΦπ​ is the sum of the multiplicities, the orthogonal index of ΦπΦ_πΦπ​ is exactly the sum of those representation dimensions, and the zero-error capacity is equal to log⁡(∑i=1dmi)\log (\sum_{i=1}^{d}m_i)log(∑i=1d​mi​). We also present a lower bound for the phase retrievability in terms of the minimal length of phase retrievable frames for CnC^nCn.
This thesis investigates the preservation of fundamental quantum properties under noisy quantum processes. Motivated by the challenges of maintaining coherence and information integrity in realistic quantum systems, we study three distinct structural properties: phase retrievability, orbit injectivity, and fidelity preservation. Each property reflects a unique facet of how quantum information can be robustly encoded, distinguished, or compared in the presence of environmental interactions. In the first part of the thesis, we analyze phase retrievable quantum channels which allow recovery of a quantum state up to a global phase. We provide structural characterizations using the Kraus representation and the joint spectrum of the Kraus operators. We then extend this study to orbit-injective channels, which distinguish quantum states up to more general group actions. This generalization captures symmetry-induced ambiguities and deepens the understanding of information distinguishability in structured quantum systems. The second part focuses on fidelity, a widely used measure of quantum similarity. We explore conditions under which mixed unitary quantum channels preserve fidelity between quantum states. These investigations reveal how fidelity preservation can constrain the structure of a quantum channel, providing insight into error tolerance and stability in quantum communication. Together, these results contribute to a unified perspective on how different aspects of quantum information-state reconstruction, distinguishability, and similarity-can be maintained under noise, offering both theoretical insight and tools for practical quantum technologies.
Knill-Laflamme coefficients characterize exact quantum error detection through scalar compressions of error operators to the code space. For prescribed Pauli observables and a fixed code dimension, attainable coefficient vectors form a joint higher-rank numerical range, whose Euclidean norm image is the signature spectrum. We study how operator representation, code dimension, and detection constraints govern this geometry. Our main theorem identifies representation multiplicity as sufficient to lift active expectation data to exact scalar compressions: a lift exists whenever each occupied block has multiplicity at least the code dimension times the rank of the corresponding active state. When multiplicity suffices to realize all active states, the coefficient range is the full active expectation body and is therefore convex and connected, with a closed interval as its signature spectrum. This framework unifies commuting Pauli families, where sufficient common-eigenspace degeneracy yields polytopes, and subsystem stabilizer codes, where protected logical subsystems supply multiplicity spaces for noncommuting gauge observables. Explicit examples show how increasing the code dimension can make a range shrink, collapse, or become empty, while increasing multiplicity can turn a Bloch sphere into a filled ball. In the subsystem setting, parent-Hamiltonian constructions realize exact detecting codes as degenerate ground spaces, selecting distinguished vectors and continuous paths within these ranges. A final three-qubit example shows additional noncommuting detection constraints reduce a tetrahedral range to its barycenter and four vertices, producing a disconnected signature spectrum. The results provide a structural framework for Knill-Laflamme coefficient geometry and motivate sharper criteria for connected coefficient ranges and interval signature spectra under structured error models.
Degradable quantum channels are among the only channels whose quantum and private classical capacities are known. As such, determining the structure of these channels is a pressing open question in quantum information theory. We give a comprehensive review of what is currently known about the structure of degradable quantum channels, including a number of new results as well as alternate proofs of some known results. In the case of qubits, we provide a complete characterization of all degradable channels with two dimensional output, give a new proof that a qubit channel with two Kraus operators is either degradable or anti-degradable, and present a complete description of anti-degradable unital qubit channels with a new proof. For higher output dimensions we explore the relationship between the output and environment dimensions (dB and dE, respectively) of degradable channels. For several broad classes of channels we show that they can be modeled with an environment that is “small” in the sense of ΦC. Such channels include all those with qubit or qutrit output, those that map some pure state to an output with full rank, and all those which can be represented using simultaneously diagonal Kraus operators, even in a non-orthogonal basis. Perhaps surprisingly, we also present examples of degradable channels with “large” environments, in the sense that the minimal dimension dE>dB. Indeed, one can have dE>14dB2. These examples can also be used to give a negative answer to the question of whether additivity of the coherent information is helpful for establishing additivity for the Holevo capacity of a pair of channels. In the case of channels with diagonal Kraus operators, we describe the subclasses that are complements of entanglement breaking channels. We also obtain a number of results for channels in the convex hull of conjugations with generalized Pauli matrices. However, a number of open questions remain about these channels and the more general case of random unitary channels.