Enhancing Phase Retrievability of Quantum Channels via Interferometric Coupling

Deguang Han; Kai Liu; Omar Nour

arxiv: 2604.24363 · v1 · submitted 2026-04-27 · 🪐 quant-ph

Enhancing Phase Retrievability of Quantum Channels via Interferometric Coupling

Kai Liu , Deguang Han , Omar Nour This is my paper

Pith reviewed 2026-05-08 03:59 UTC · model grok-4.3

classification 🪐 quant-ph

keywords phase retrievabilityquantum channelscomplementary channelsinterferometric couplingoperator-valued framespure-state informational completenessChoi rank

0 comments

The pith

A quantum channel is phase retrievable if and only if its complementary channel is pure-state informationally complete, and interferometric coupling can enhance this property through cross terms.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that phase retrievability of a quantum channel holds exactly when the complementary channel can distinguish all pure states via its measurements. This equivalence yields concrete criteria such as bounds on the dimension of the complementary operator system and results for entanglement-breaking and twirling channels. The authors then construct an interferometric coupling of two arm channels using port operators that introduce interference cross terms, enlarging the complementary operator system and thereby enabling phase retrieval for channels that individually do not possess it.

Core claim

A quantum channel Φ is phase retrievable if and only if its complementary channel Φ^c is pure-state informationally complete. This structural fact implies dimension criteria for the complementary operator system, Choi-rank type bounds, and specific statements for entanglement breaking channels and twirling channels. An interferometric coupling realized by port operators M_i(θ) = A_i + e^{iθ} B_i recombines two arm channels coherently; the resulting cross terms enlarge the complementary operator system relative to classical mixing, and the effect is quantified by injectivity indices for completely positive maps, with explicit examples showing improved phase retrieval even when the separatearm

What carries the argument

The interferometric coupling through port operators M_i(θ)=A_i + e^{iθ}B_i, which generates interference cross terms that enlarge the complementary operator system and thereby improve phase retrievability.

Load-bearing premise

The interferometric coupling with port operators M_i(θ)=A_i + e^{iθ}B_i can be physically realized coherently without introducing additional decoherence or loss that would destroy the cross terms needed to enlarge the complementary operator system.

What would settle it

A calculation or experiment on a concrete pair of arm channels A_i and B_i in which the interferometric output fails to allow unique reconstruction of pure states up to phase, despite the predicted mathematical enlargement of the complementary operator system.

Figures

Figures reproduced from arXiv: 2604.24363 by Deguang Han, Kai Liu, Omar Nour.

**Figure 2.1.** Figure 2.1: A two-path quantum interferometer. The lower arm carries the process ΦA, the upper arm carries the process ΦB, and a relative phase e iθ is inserted in the upper arm before recombination. 5 view at source ↗

**Figure 5.2.** Figure 5.2: Heat maps of I(Ψθ) (left) and Iav(Ψθ) (right) for the interferometric coupling of qubit amplitude damping and Z-dephasing. For the channels considered in the next two examples, each arm is individually not phase retrievable. We show that, after interferometric coupling, the resulting port map becomes phase retrievable for a large range of parameter values, while also exhibiting relatively low compression. 23 view at source ↗

**Figure 5.** Figure 5: shows that, although both arms are individually not phase retrievable, the in view at source ↗

**Figure 5.3.** Figure 5.3: shows that, although both arms are individually not phase retrievable, the interferometric port map has a large region where the injectivity index is positive view at source ↗

**Figure 5.4.** Figure 5.4: Heat maps of I(Ψθ,α) (left) and Iav(Ψθ,α) (right) for the interferometric coupling of the rotated trine family and the trine family. These examples support a common conclusion. The phase retrieval behavior of the port map depends strongly on the interferometric phase, and coherent coupling can substantially improve the behavior of the resulting completely positive map. 6. Summary and outlook In this pap… view at source ↗

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper's core is a clean iff between phase retrievability and the complementary channel being pure-state informationally complete, plus an interferometric mixing construction that can enlarge the operator system under ideal coherence.

read the letter

The main new piece is the structural result: a channel is phase retrievable exactly when its complement is pure-state informationally complete. This is derived straight from the definitions and immediately gives dimension criteria, Choi-rank bounds, and statements for entanglement-breaking and twirling channels. That part looks tight and useful on its own terms. The second contribution is the interferometric coupling via port operators M_i(θ) = A_i + e^{iθ} B_i. Coherent recombination adds cross terms that can strictly enlarge the complementary operator system, turning some non-retrievable channels into retrievable ones. The frame-theoretic language and injectivity indices give a way to quantify the enlargement, and the Section 5 examples show concrete gains when the arms are taken separately. This ties quantum information, operator frames, and an optical construction together in a direct way. The thinking is clear and the claims stay within what the definitions support. The soft spot is physical realizability. The enhancement requires the cross terms to survive intact, but any real interferometer brings finite visibility, path fluctuations, or loss that damps the off-diagonal blocks. Once those blocks fall below the injectivity threshold the effective operator system shrinks again and the claimed improvement disappears. The calculations assume perfect coherence, so they do not yet bound how much decoherence can be tolerated. This is for people working on quantum channel properties, phase retrieval, or interferometric designs in quantum optics. The characterization is self-contained enough to stand up to checking, and the construction raises concrete questions worth exploring even if the ideal-model results need robustness follow-up. I would send it to referees.

Referee Report

2 major / 3 minor

Summary. The manuscript establishes that a quantum channel is phase retrievable if and only if its complementary channel is pure-state informationally complete. This equivalence yields consequences including dimension criteria for the complementary operator system, Choi-rank bounds, and specific results for entanglement-breaking and twirling channels. The paper then proposes an interferometric coupling of two arm channels via port operators M_i(θ) = A_i + e^{iθ} B_i, which generates interference cross terms that enlarge the complementary operator system and thereby enhance phase retrievability. The construction is framed in terms of coherent coupling of operator-valued frames, with new injectivity indices introduced to quantify the effect. Section 5 examples illustrate significant improvement even when the individual arm channels are not phase retrievable.

Significance. The structural equivalence is a clean, definition-based result that links phase retrievability directly to informational completeness of the complementary map and should prove useful for classifying channels and designing measurements. The interferometric scheme offers a physically motivated method to improve retrievability via coherence, with the frame-theoretic perspective and injectivity indices providing additional analytical tools. If the enhancement construction holds under realistic conditions, the work has clear implications for quantum tomography and state reconstruction protocols.

major comments (2)

[Interferometric coupling construction and Section 5] The enhancement claim rests on the interferometric construction (M_i(θ) = A_i + e^{iθ} B_i) producing cross terms that strictly enlarge the complementary operator system. Section 5 examples are computed assuming perfect coherence, but the manuscript provides no analysis of how finite visibility, path-length fluctuations, or detector inefficiency would damp the off-diagonal blocks and potentially eliminate the phase-retrievability gain once the effective dimension falls below the injectivity threshold. This assumption is load-bearing for the central practical claim.
[Consequences following the structural characterization] The Choi-rank type bounds and dimension criteria for the complementary operator system are listed as consequences of the main equivalence, but the derivations are not tied to explicit equations or operator-system rank calculations. Without these steps, it is difficult to verify whether the bounds are tight or merely restatements of the equivalence.

minor comments (3)

The notation for the port operators and the resulting complementary map after recombination would benefit from an explicit block-matrix expansion showing the cross terms.
A diagram of the interferometer setup (with the two arm channels and the recombination ports) would clarify the physical realization of the coherent coupling.
The definition and properties of the newly introduced injectivity indices should be collected in a dedicated subsection or proposition for easier reference.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and insightful comments on our manuscript. We address each major comment below and have incorporated revisions to strengthen the presentation and clarify the results.

read point-by-point responses

Referee: [Interferometric coupling construction and Section 5] The enhancement claim rests on the interferometric construction (M_i(θ) = A_i + e^{iθ} B_i) producing cross terms that strictly enlarge the complementary operator system. Section 5 examples are computed assuming perfect coherence, but the manuscript provides no analysis of how finite visibility, path-length fluctuations, or detector inefficiency would damp the off-diagonal blocks and potentially eliminate the phase-retrievability gain once the effective dimension falls below the injectivity threshold. This assumption is load-bearing for the central practical claim.

Authors: We agree that the examples in Section 5 assume ideal coherence, as is standard for establishing the theoretical enhancement mechanism. The interferometric construction is intended to demonstrate how coherent coupling via port operators can enlarge the complementary operator system through cross terms, thereby improving phase retrievability beyond what is possible with individual channels. We acknowledge that real-world imperfections such as finite visibility would reduce the magnitude of the off-diagonal interference terms, potentially limiting the gain. In the revised manuscript, we will add a paragraph in Section 5 discussing this limitation qualitatively, noting that the enhancement scales with the visibility factor and that the injectivity indices provide a tool to assess the required coherence threshold. A full quantitative analysis under specific noise models (e.g., path fluctuations) lies beyond the scope of the current theoretical work but could be pursued in follow-up studies; the core structural result on operator-system enlargement holds in the ideal case. revision: partial
Referee: [Consequences following the structural characterization] The Choi-rank type bounds and dimension criteria for the complementary operator system are listed as consequences of the main equivalence, but the derivations are not tied to explicit equations or operator-system rank calculations. Without these steps, it is difficult to verify whether the bounds are tight or merely restatements of the equivalence.

Authors: We appreciate this observation. The dimension criteria and Choi-rank bounds follow directly from the equivalence between phase retrievability and pure-state informational completeness of the complementary channel, combined with standard facts about the rank of the Choi operator and the dimension of the associated operator system. In the revised version, we will expand the relevant subsection to include explicit derivations: starting from the definition of the complementary operator system, we will show the dimension lower bound via the informational completeness condition, and derive the Choi-rank inequality by relating the kernel of the complementary map to the support of the Choi matrix. These steps will confirm that the bounds are tight, as demonstrated by the entanglement-breaking and twirling channel examples already present in the paper. revision: yes

Circularity Check

0 steps flagged

No circularity: structural iff result and interferometric construction are derived from definitions without reduction to inputs or self-citation chains

full rationale

The central characterization (phase-retrievable channel iff complementary channel is pure-state IC) is presented as following directly from the definitions of phase retrievability and informational completeness, with no equations or steps that equate the output to a fitted parameter or renamed input by construction. The interferometric port operators M_i(θ) = A_i + e^{iθ}B_i and resulting cross terms are introduced as an explicit new construction to enlarge the complementary operator system; this is not a prediction fitted to data but a proposed physical model whose consequences are then analyzed. No load-bearing uniqueness theorem, ansatz, or self-citation is invoked to force the result. The derivation chain remains self-contained against external benchmarks of operator theory and frame theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on standard definitions from quantum information theory and operator-valued frame theory; the interferometric construction introduces a new physical model whose assumptions are not detailed in the abstract.

axioms (2)

standard math Standard axioms of quantum mechanics and completely positive trace-preserving maps
Invoked implicitly when defining quantum channels and their complements.
domain assumption Existence of a physical realization for coherent recombination via port operators without extra decoherence
Required for the interferometric enhancement to work as described.

invented entities (1)

Interferometric coupling via port operators M_i(θ) = A_i + e^{iθ} B_i no independent evidence
purpose: To produce interference cross terms that enlarge the complementary operator system and enhance phase retrievability
New construction introduced in the paper; no independent evidence provided in abstract.

pith-pipeline@v0.9.0 · 5507 in / 1455 out tokens · 58760 ms · 2026-05-08T03:59:13.835232+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

37 extracted references · 4 canonical work pages · 1 internal anchor

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C. Branciard, A. Cl´ ement, M. Mhalla, and S. Perdrix, Coherent control and distinguishability of quan- tum channels via PBS-diagrams, Leibniz International Proceedings in Informatics (LIPIcs) 202 (2021) 22:1–22:20

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A. Vanrietvelde and G. Chiribella, Universal control of quantum processes using sector-preserving chan- nels, Quantum Information and Computation. 21 (2021) 1320–1352

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P. Balazs, J.-P. Antoine, and A. Grybos, Weighted and controlled frames: Mutual relationship and first numerical properties, International Journal of Wavelets, Multiresolution and Information Processing. 8 (2010) 109–132

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J. Wechs, H. Dourdent, A. A. Abbott, and C. Branciard, Quantum circuits with classical versus quantum control of causal order, PRX Quantum. 2 (2021) 030335

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Q. Dong, S. Nakayama, A. Soeda, and M. Murao, Controlled quantum operations and combs, and their applications to universal controllization of divisible unitary operations, arXiv:1911.01645 (2019)

work page arXiv 1911
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S. Luo, Y. Sun, Quantifying the irreversibility of channels, Theoretical and Mathematical Physics. 218 (2024) 426–451

2024
[35]

Y. Li, S. Luo, Y. Sun, S. Wang, Quantifying information loss of quantum channels, Physical Review A. 112 (2025) 062440

2025
[36]

Y. Guo, S. Luo, L. Zhang, Quantifying channel incompatibility via Jordan negativity, Physics Letters A. 543 (2025) 130444

2025
[37]

Sun and N

Y. Sun and N. Li, Reversibility-unitality-disturbance tradeoff in quantum channels, Physical Review A. 109 (2024) 012407. 27 Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA Email address:kai.liu@ucf.edu Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA Email address:deguang.han@ucf.edu Depart...

2024

[1] [1]

Balan, P

R. Balan, P. G. Casazza and D. Edidin, On signal reconstruction without phase, Applied and Compu- tational Harmonic Analysis. 20 (2006) 345-356

2006

[2] [2]

Heinosaari, L

T. Heinosaari, L. Mazzarella, and M. M. Wolf, Quantum tomography under prior information, Com- munications in Mathematical Physics. 318 (2013) 355-374

2013

[3] [3]

Zambrano, L

L. Zambrano, L. Pereira, and A. Delgado, Minimal orthonormal bases for pure quantum state estima- tion, Quantum. 8 (2024) 1244

2024

[4] [4]

Huang, R

H. Huang, R. Kueng, and J. Preskill, Predicting many properties of a quantum system from very few measurements, Nature Physics. 16 (2020) 1050-1057

2020

[5] [5]

Grier, H

D. Grier, H. Pashayan, and L. Schaeffer, Sample-optimal classical shadows for pure states, Quantum. 8 (2024) 1373

2024

[6] [6]

Z. Qin, J. M. Lukens, B. T. Kirby, and Z. Zhu, Enhancing quantum state reconstruction with structured classical shadows, npj Quantum Information. 11 (2025) 147

2025

[7] [7]

Liu and D

K. Liu and D. Han, Phase retrievability of frames and quantum channels, Linear Algebra and its Applications. 725 (2025) 378-400

2025

[8] [8]

Han and K

D. Han and K. Liu, Zero-error correctibility and phase retrievability for twirling channels, Reports on Mathematical Physics. 93 (2024) 87-102

2024

[9] [9]

A. A. Mele and L. Bittel, Optimal learning of quantum channels in diamond distance, arXiv:2512.10214 (2025)

work page arXiv 2025

[10] [10]

K. Chen, N. Yu, and Z. Zhang, Quantum channel tomography and estimation by local test, arXiv:2512.13614 (2025)

work page arXiv 2025

[11] [11]

Reconstructing the unitary part of a noisy quantum channel

A. Romer, D. M. Reich, and C. P. Koch, Reconstructing the unitary part of a noisy quantum channel, arXiv:2507.17648 (2025)

work page internal anchor Pith review Pith/arXiv arXiv 2025

[12] [12]

Jayakumar, S

A. Jayakumar, S. Chessa, C. Coffrin, A. Y. Lokhov, M. Vuffray and S. Misra, Universal framework for simultaneous tomography of quantum states and SPAM noise, Quantum. 8 (2024) 1426

2024

[13] [13]

Girard, D

M. Girard, D. Leung, J. Levick, C.-K. Li, V. I. Paulsen, Y. T. Poon, and J. Watrous, On the mixed- unitary rank of quantum channels, Communications in Mathematical Physics. 394 (2022) 919-951

2022

[14] [14]

Y. Li, S. Luo, and Y. Sun, Structure of reversible quantum channels, Annalen der Physik. 538 (2026) 2500473. 26

2026

[15] [15]

Kaftal, D

V. Kaftal, D. R. Larson, and S. Zhang, Operator-valued frames, Transactions of the American Mathe- matical Society. 361 (2009) 6349-6385

2009

[16] [16]

D. Han, P. Li, B. Meng, and W. Tang, Operator valued frames and structured quantum channels, Science China Mathematics. 54 (2011) 2361-2372

2011

[17] [17]

L. E. Fischer, T. Dao, I. Tavernelli, and F. Tacchino, Dual frame optimization for informationally complete quantum measurements, Physical Review A 109 (2024) 062415

2024

[18] [18]

Han and T

D. Han and T. Juste, Phase-retrievable operator-valued frames and representations of quantum channels, Linear Algebra and its Applications. 579 (2019) 148-168

2019

[19] [19]

D. K. L. Oi, Interference of quantum channels, Physical Review Letters. 91 (2003) 067902

2003

[20] [20]

D. K. L. Oi and J. ˚Aberg, Fidelity and coherence measures from interference, Physical Review Letters. 97 (2006) 220404

2006

[21] [21]

Tan and P

S.-H. Tan and P. P. Rohde, The resurgence of the linear optics quantum interferometer: Recent advances and applications, Reviews in Physics 4 (2019) 100030

2019

[22] [22]

A. A. Abbott, J. Wechs, D. Horsman, M. Mhalla, and C. Branciard, Communication through coherent control of quantum channels, Quantum 4 (2020) 333

2020

[23] [23]

Branciard, A

C. Branciard, A. Cl´ ement, M. Mhalla, and S. Perdrix, Coherent control and distinguishability of quan- tum channels via PBS-diagrams, Leibniz International Proceedings in Informatics (LIPIcs) 202 (2021) 22:1–22:20

2021

[24] [24]

M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information, Cambridge Univer- sity Press, Cambridge, 2000

2000

[25] [25]

J. R. Magnus and H. Neudecker, Matrix Differential Calculus with Applications in Statistics and Econo- metrics, John Wiley & Sons, 1999

1999

[26] [26]

˚Aberg, Operations and single-particle interferometry, Physical Review A

J. ˚Aberg, Operations and single-particle interferometry, Physical Review A. 70 (2004) 012103

2004

[27] [27]

Vanrietvelde and G

A. Vanrietvelde and G. Chiribella, Universal control of quantum processes using sector-preserving chan- nels, Quantum Information and Computation. 21 (2021) 1320–1352

2021

[28] [28]

Sun, G-frames and g-Riesz bases, Journal of Mathematical Analysis and Applications

W. Sun, G-frames and g-Riesz bases, Journal of Mathematical Analysis and Applications. 322 (2006) 437–452

2006

[29] [29]

Balazs, J.-P

P. Balazs, J.-P. Antoine, and A. Grybos, Weighted and controlled frames: Mutual relationship and first numerical properties, International Journal of Wavelets, Multiresolution and Information Processing. 8 (2010) 109–132

2010

[30] [30]

D. T. Stoeva and P. Balazs, A survey on the unconditional convergence and the invertibility of frame multipliers with implementation, in: Sampling: Theory and Applications, 169–192, 2020

2020

[31] [31]

X. Yuan, H. Zhou, M. Gu, and X. Ma, Unification of nonclassicality measures in interferometry, Physical Review A. 97 (2018) 012331

2018

[32] [32]

Wechs, H

J. Wechs, H. Dourdent, A. A. Abbott, and C. Branciard, Quantum circuits with classical versus quantum control of causal order, PRX Quantum. 2 (2021) 030335

2021

[33] [33]

Q. Dong, S. Nakayama, A. Soeda, and M. Murao, Controlled quantum operations and combs, and their applications to universal controllization of divisible unitary operations, arXiv:1911.01645 (2019)

work page arXiv 1911

[34] [34]

S. Luo, Y. Sun, Quantifying the irreversibility of channels, Theoretical and Mathematical Physics. 218 (2024) 426–451

2024

[35] [35]

Y. Li, S. Luo, Y. Sun, S. Wang, Quantifying information loss of quantum channels, Physical Review A. 112 (2025) 062440

2025

[36] [36]

Y. Guo, S. Luo, L. Zhang, Quantifying channel incompatibility via Jordan negativity, Physics Letters A. 543 (2025) 130444

2025

[37] [37]

Sun and N

Y. Sun and N. Li, Reversibility-unitality-disturbance tradeoff in quantum channels, Physical Review A. 109 (2024) 012407. 27 Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA Email address:kai.liu@ucf.edu Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA Email address:deguang.han@ucf.edu Depart...

2024