arXiv subjects

Deguang Han

Publications and source records attributed to Deguang Han.

At least 19 recordsLinked to original sources

arXiv2026

Frame phase retrievability and state distinguishability of quantum channels

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.

quant-ph↗

arXiv2026

Enhancing Phase Retrievability of Quantum Channels via Interferometric Coupling

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 $M_i(\theta)=A_i+e^{i\theta}B_i$. 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-ph↗

arXiv2025

Designing optimal dual frames for $\ell^p-$average error optimization

In this paper, we investigates the problem of optimal dual frame selection for signal reconstruction in the presence of erasures. Unlike traditional approaches relying on left inverses, we evaluate performance through the norms of error operators, using the Frobenius norm, spectral radius, and numerical radius as measures. Our central focus is the characterization of dual frames that minimize the $\ell^p-$average under these error operator measurements over all possible erasure patterns. We provide conditions under which the canonical dual frame is uniquely optimal and extend our results to multiple erasures. In the Frobenius norm case, we offer a complete characterization for any number of erasures in uniform tight frames. The paper also examines interconnections between optimality criteria across different norm measures and gives sufficient conditions ensuring uniqueness of the optimal dual.

math.FA↗

arXiv2025

Optimal K dual frames and pairs in the presence of erasures

This paper explores the structure of optimal K-dual frames for a given K-frame and optimal K-dual pairs, within the context of erasures which occur during the transmission of frame coefficients. We address two distinct erasure scenarios and examine their impact on the reconstruction process. The optimality criteria are defined in terms of minimizing the spectral radius and the operator norm of the associated error operators. Through this approach, we provide a comprehensive framework for understanding and mitigating the effects of erasures in frame theory, contributing to enhanced robustness in data transmission and recovery.

math.FA↗

arXiv2025

Dynamical Frames and Hyperinvariant Subspaces

The theory of dynamical frames evolved from practical problems in dynamical sampling where the initial state of a vector needs to be recovered from the space-time samples of evolutions of the vector. This leads to the investigation of structured frames obtained from the orbits of evolution operators. One of the basic problems in dynamical frame theory is to determine the semigroup representations, which we will call central frame representations, whose frame generators are unique (up to equivalence). Recently, Christensen, Hasannasab, and Philipp proved that all frame representations of the semigroup $\Bbb{Z}_{+}$ have this property. Their proof of this result relies on the characterization of the structure of shift-invariant subspaces in $H^2(\mathbb{D})$ due to Beurling. In this paper we settle the general uniqueness problem by presenting a characterization of central frame representations for any semigroup in terms of the co-hyperinvariant subspaces of the left regular representation of the semigroup. This result is not only consistent with the known result of Han-Larson in 2000 for group representation frames, but also proves that all the frame generators of a semigroup generated by any $k$-tuple $(A_1, ... A_k)$ of commuting bounded linear operators on a separable Hilbert space $H$ are equivalent, a case where the structure of shift-invariant subspaces, or submodules, of the Hardy Space on polydisks $H^{2}(\Bbb{D}^k)$ is still not completely characterized.

math.FA↗

arXiv2024

Fidelity preserving and decoherence for mixed unitary quantum channels

Distinguishable and non-distinguishable quantum states are fundamental resources in quantum mechanics and quantum technologies. Interactions with the environment often induce decoherence, impacting both the distinguishability and non-distinguishability between quantum states. In this paper, we investigate mixed unitary quantum channels and the conditions under which fidelity, a measure of quantum state closeness, is preserved. More precisely, for quantum channels in the form $\Phi(\rho) = \sum_{i=1}^N p_i U_i \rho U_i^*$, we analyze their effect on quantum state $|\varphi\rangle$ through the associated purification $|\Psi\rangle$, explore the structure of such quantum channels that preserve either distinguishable or non-distinguishable states and then discuss the challenges of maintaining fidelity, particularly under the influence of phase damping.

quant-ph↗

arXiv2023

Zero Error Correctibility and Phase Retrievability for Twirling Channels

A twirling channel is a quantum channel induced by a continuous unitary representation $\pi = \sum_{i}^{\oplus} m_i\pi_i$, where $\pi_i$ are inequivalent irreducible representations. Motivated by a recent work \cite{Twirling} on minimal mixed unitary rank of $\Phi_{\pi}$, we explore the connections of the independence number, zero error capacity, quantum codes, orthogonality index and phase retrievability of the quantum channel $\Phi_{\pi}$ with the irreducible representation multiplicities $m_i$, the irreducible representation dimensions $\dim H_{\pi_i}$. In particular we show that the independence number of $\Phi_{\pi}$ is the sum of the multiplicities, the orthogonal index of $\Phi_{\pi}$ is exactly the sum of those representation dimensions, and the zero-error capacity is equal to $\log (\sum_{i=1}^{d}m_i)$. We also present a lower bound for the phase retrievability in terms of the minimal length of phase retrievable frames for $C^n$.

quant-ph↗

arXiv2023

Phase retrievability of frames and quantum channels

A phase retrievable quantum channel refers to a quantum channel $\Phi: B(H_A)\to B(H_B)$ such that there is a positive operator valued measure (POVM) $\{F_{j}\}$ in $B(H_{B})$ and $\{\Phi^*(F_j)\}$ is a phase retrievable operator valued frame. In this paper we examine the phase retrievable quantum channels in terms of their Kraus representations. For quantum channels $\Phi$ of Choi's rank-$2$, we obtain a necessary and sufficient condition under which it is phase retrievable. For the general case, we present several necessary and/or sufficient conditions. In particular, a necessary and sufficient condition is obtained in terms of the relevant matrix-valued joint spectrum of the Kraus operators. Additionally, we also examine, by examples, the problem of constructing quantum channels such that there exists a minimal number of rank-one observables $\{F_{j}\}$ such that $\{\Phi^*(F_j)\}$ does phase retrieval for $H_A$. Conversely, for a given set of rank-one observables $\{F_{j}\}_{j=1}^{N}$, we present a sufficient condition under which, for every $1\leq r\leq N$ given, a phase retrievable quantum channel $\Phi$ of Choi's rank-$r$ can be explicitly constructed.

quant-ph↗

arXiv2023

Single-shot phase retrieval: a holography-driven problem in Sobolev space

The phase-shifting digital holography (PSDH) is a widely used approach for recovering signals by their interference (with reference waves) intensity measurements. Such measurements are traditionally from multiple shots (corresponding to multiple reference waves). However, the imaging of dynamic signals requires a single-shot PSDH approach, namely, such an approach depends only on the intensity measurements from the interference with a single reference wave. In this paper, based on the uniform admissibility of plane (or spherical) reference wave and the interference intensity-based approximation to quasi-interference intensity, the nonnegative refinable function is applied to establish the single-shot PSDH in Sobolev space. Our approach is conducted by the intensity measurements from the interference of the signal with a single reference wave. The main results imply that the approximation version from such a single-shot approach converges exponentially to the signal as the level increases. Moreover, like the transport of intensity equation (TIE), our results can be interpreted from the perspective of intensity difference.

math.FA↗

arXiv2022

Determination of compactly supported functions in shift-invariant space by single-angle Radon samples

While traditionally the computerized tomography of a function $f\in L^{2}(\mathbb{R}^{2})$ depends on the samples of its Radon transform at multiple angles, the real-time imaging sometimes requires the reconstruction of $f$ by the samples of its Radon transform $\mathcal{R}_{\emph{\textbf{p}}}f$ at a single angle $\theta$, where $\emph{\textbf{p}}=(\cos\theta, \sin\theta)$ is the direction vector. This naturally leads to the question of identifying those functions that can be determined by their Radon samples at a single angle $\theta$. The shift-invariant space $V(\varphi, \mathbb{Z}^2)$ generated by $\varphi$ is a type of function space that has been widely considered in many fields including wavelet analysis and signal processing. In this paper we examine the single-angle reconstruction problem for compactly supported functions $f\in V(\varphi, \mathbb{Z}^2)$. The central issue for the problem is to identify the eligible $\emph{\textbf{p}}$ and sampling set $X_{\emph{\textbf{p}}}\subseteq \mathbb{R}$ such that $f$ can be determined by its single-angle Radon (w.r.t $\emph{\textbf{p}}$) samples at $X_{\emph{\textbf{p}}}$. For the general generator $\varphi$, we address the eligible $\emph{\textbf{p}}$ for the two cases: (1) $\varphi$ being nonvanishing ($\int_{\mathbb{R}^{2}}\varphi(\emph{\textbf{x}})d\emph{\textbf{x}}\neq0$) and (2) being vanishing ($\int_{\mathbb{R}^2}\varphi(\emph{\textbf{x}})d\emph{\textbf{x}}=0$). We prove that eligible $X_{\emph{\textbf{p}}}$ exists for general $\varphi$. In particular, $X_{\emph{\textbf{p}}}$ can be explicitly constructed if $\varphi\in C^{1}(\mathbb{R}^{2})$. The single-angle problem corresponding to the case that $\varphi$ being positive definite is addressed such that $X_{\emph{\textbf{p}}}$ can be constructed easily.

math.FA↗

arXiv2022

Phase retrieval of analytic signals from short-time Fourier transform measurements

Analytic signals constitute a class of signals that are widely applied in time-frequency analysis such as extracting instantaneous frequency (IF) or phase derivative in the characterization of ultrashort laser pulse. The purpose of this paper is to investigate the phase retrieval (PR) problem for analytic signals in $\mathbb{C}^{N}$ by short-time Fourier transform (STFT) measurements since they enjoy some very nice structures. Since generic analytic signals are generally not sparse in the time domain, the existing PR results for sparse (in time domain) signals do not apply to analytic signals. We will use bandlimited windows that usually have the full support length $N$ which allows us to get much better resolutions on low frequencies. More precisely, by exploiting the structure of the STFT for analytic signals, we prove that the STFT based phase retrieval (STFT-PR for short) of generic analytic signals can be achieved by their $(3\lfloor\frac{N}{2}\rfloor+1)$ measurements. Since the generic analytic signals are $(\lfloor \frac{N}{2}\rfloor+1)$-sparse in the Fourier domain, such a number of measurements is lower than $4N+\hbox{O}(1)$ and $\hbox{O}(k^{3})$ which are required in the literature for STFT-PR of all signals and of $k^{2}$-sparse (in the Fourier domain) signals in $\mathbb{C}^{N^{2}}$, respectively. Moreover, we also prove that if the length $N$ is even and the windows are also analytic, then the number of measurements can be reduced to $(\frac{3 N}{2}-1)$. As an application of this we get that the instantaneous frequency (IF) of a generic analytic signal can be exactly recovered from the STFT measurements.

cs.IT↗

arXiv2021

Dilations for operator-valued quantum measures

This paper concerns the dilations of Banach space operator-valued quantum measures. While the recently developed general dilation theory can lead to a projection (idempotent) valued dilation for any quantum measure over the projection lattice for a von Neumann algebra that dose not contain type $I_{2}$ direct summand, such a dilation does not necessarily guarantee the preservation of countable additivity of the quantum measure. So it remain an open question whether every countably additive $B(X)$-valued quantum measure can be dilated to a countably additive projection-valued measure.The main purpose of this paper is to prove that such a dilation can be constructed if one of the following two conditions is satisfied: (i) the underling Banach space $X = \ell_{p}$ $(1\leq p < 2$) or it has Schur property, (ii) the quantum measure has bounded $p$-variation for some $ 1\leq p < \infty $. All of these were achieved by establishing a non-commutative version of a minimal dilation theory on the so-called elementary dilation space equipping with an appropriate dilation norm. In particular, the newly introduced $p$-variation norm on the elementary dilation space allows us to prove that every operator-valued quantum measure with bounded $p$-variation has a projection-valued quantum measure dilation that preserves the boundedness of the $p$-variation.

math.FA↗

arXiv2021

FROG-measurement based phase retrieval for analytic signals

While frequency-resolved optical gating (FROG) is widely used in characterizing the ultrafast pulse in optics, analytic signals are often considered in time-frequency analysis and signal processing, especially when extracting instantaneous features of events. In this paper we examine the phase retrieval (PR) problem of analytic signals in $\Bbb{C}^N$ by their FROG measurements. After establishing the ambiguity of the FROG-PR of analytic signals, we found that the FROG-PR of analytic signals of even lengths is different from that of analytic signals of odd lengths, and it is also different from the case of $B$-bandlimited signals with $B \leq N/2$. The existing approach to bandlimited signals can be applied to analytic signals of odd lengths, but it does not apply to the even length case. With the help of two relaxed FROG-PR problems and a translation technique, we develop an approach to FROG-PR for the analytic signals of even lengths, and prove that in this case the generic analytic signals can be uniquely (up to the ambiguity) determined by their $(3N/2+1)$ FROG measurements.

eess.SP↗

arXiv2020

Gabor single-frame and multi-frame multipliers in any given dimension

Functional Gabor single-frame or multi-frame generator multipliers are the matrices of function entries that preserve Parseval Gabor single-frame or multi-frame generators. An interesting and natural question is how to characterize all such multipliers. This question has been answered for several special cases including the case of single-frame generators in two dimensions and the case of multi-frame generators in one-dimension. In this paper we completely characterize multipliers for Gabor single-frame and multi-frame generators with respect to separable time-frequency lattices in any given dimension. Our approach is general and applies to the previously known cases as well.

math.FA↗

arXiv2020

Nonuniform sampling and approximation in Sobolev space from the perturbation of framelet system

The Sobolev space $H^{\varsigma}(\mathbb{R}^{d})$, where $\varsigma > d/2$, is an important function space that has many applications in various areas of research. Attributed to the inertia of a measurement instrument, it is desirable in sampling theory to recover a function by its nonuniform sampling. In the present paper, based on dual framelet systems for the Sobolev space pair $(H^{s}(\mathbb{R}^{d}), H^{-s}(\mathbb{R}^{d}))$, where $d/2<s<\varsigma$, we investigate the problem of constructing the approximations to all the functions in $H^{\varsigma}(\mathbb{R}^{d})$ by nonuniform sampling. We first establish the convergence rate of the framelet series in $(H^{s}(\mathbb{R}^{d}), H^{-s}(\mathbb{R}^{d}))$, and then construct the framelet approximation operator that acts on the entire space $H^{\varsigma}(\mathbb{R}^{d})$. We examine the stability property for the framelet approximation operator with respect to the perturbations of shift parameters, and obtain an estimate bound for the perturbation error. Our result shows that under the condition $d/2<s<\varsigma$, the approximation operator is robust to shift perturbations. Motivated by some recent work on nonuniform sampling and approximation in Sobolev space (e.g., [20]), we don't require the perturbation sequence to be in $\ell^{\alpha}(\mathbb{Z}^{d})$. Our results allow us to establish the approximation for every function in $H^{\varsigma}(\mathbb{R}^{d})$ by nonuniform sampling. In particular, the approximation error is robust to the jittering of the samples.

math.FA↗

arXiv2018

A duality principle for groups II: Multi-frames meet super-frames

The duality principle for group representations developed in \cite{DHL-JFA, HL_BLM} exhibits a fact that the well-known duality principle in Gabor analysis is not an isolated incident but a more general phenomenon residing in the context of group representation theory. There are two other well-known fundamental properties in Gabor analysis: The Wexler-Raz biorthogonality and the Fundamental Identity of Gabor analysis. In this paper we will show that these fundamental properties remain to be true for general projective unitary group representations. The main purpose of this paper is present a more general duality theorem which shows that that muti-frame generators meet super-frame generators through a dual commutant pairs. In particular, for the Gabor representations $\pi_{\Lambda}$ and $\pi_{\Lambda^{o}}$ with respect to a pair of dual time-frequency lattices $\Lambda$ and $\Lambda^{o}$ in $\R^{d}\times \R^{d}$ we have that $\{\pi_{\Lambda}(m, n)g_{1} \oplus ... \oplus \pi_{\Lambda}(m, n)g_{k}\}_{m, n \in \Z^{d}}$ is a frame for $L^{2}(\R^{d})\oplus ... \oplus L^{2}(\R^{d})$ if and only if $\cup_{i=1}^{k}\{\pi_{\Lambda^{o}}(m, n)g_{i}\}_{m, n\in\Z^{d}}$ is a Riesz sequence, and $\cup_{i=1}^{k}\{\pi_{\Lambda}(m, n)g_{i}\}_{m, n\in\Z^{d}}$ is a frame for $L^{2}(\R^{d})$ if and only if $\{\pi_{\Lambda^{o}}(m, n)g_{1} \oplus ... \oplus \pi_{\Lambda^{o}}(m, n)g_{k}\}_{m, n \in \Z^{d}}$ is a Riesz sequence. This appears to be new even in the context of Gabor analysis.

math.FA↗

arXiv2018

Two-step PR-scheme for recovering signals in detectable union of cones by magnitude measurements

Motivated by the research on sampling problems for a union of subspaces (UoS), we investigate in this paper the phase-retrieval problem for the signals that are residing in a union of (finitely generated) cones (UoC for short) in $\mathbb{R}^{n}$. We propose a two-step PR-scheme: $\hbox{PR}=\hbox{detection}+\hbox{recovery}$. We first establish a sufficient and necessary condition for the detectability of a UoC, and then design a detection algorithm that allows us to determine the cone where the target signal is residing. The phase-retrieval will be then performed within the detected cone, which can be achieved by using at most $\Gamma$-number of measurements and with very low complexity, where $\Gamma (\leq n)$ is the maximum of the ranks of the generators for the UoC. Numerical experiments are provided to demonstrate the efficiency of our approach, and to exhibit comparisons with some existing phase-retrieval methods.

cs.IT↗

arXiv2017

Framelet perturbation and application to nouniform sampling approximation for Sobolev space

The Sobolev space $H^{s}(\mathbb{R}^{d})$, where $s > d/2$, is an important function space that has many applications in various areas of research. Attributed to the inertia of a measuring instrument, it is desirable in sampling theory to reconstruct a function by its nonuniform samples. In the present paper, we investigate the problem of constructing the approximation to all the functions in $H^{s}(\mathbb{R}^{d})$ with nonuniform samples by utilizing dual framelet systems for the Sobolev space pair $(H^{s}(\mathbb{R}^{d}), H^{-s}(\mathbb{R}^{d}))$. We first establish the convergence rates of the framelet series in $(H^{s}(\mathbb{R}^{d}), H^{-s}(\mathbb{R}^{d}))$, and then construct the framelet approximation operator holding for the entire space $H^{s}(\mathbb{R}^{d})$. Using the approximation operator, any function in $H^{s}(\mathbb{R}^{d})$ can be approximated at the exponential rate with respect to the scale level. We examine the stability property for the perturbations of the framelet approximation operator with respect to shift parameters, and obtain an estimate bound for the perturbation error. Our result shows that under the condition $s > d/2$, the approximation operator is robust to the shift perturbation. These results are used to establish the nonuniform sampling approximation for every function in $H^{s}(\mathbb{R}^{d})$. In particular, the new nonuniform sampling approximation error is robust to the jittering of the samples.

math.FA↗