Physical Review X最新文献

{"title":"Exciton Self-Trapping in Twisted Hexagonal Boron Nitride homostructures","authors":"Sébastien Roux, Christophe Arnold, Etienne Carré, Alexandre Plaud, Lei Ren, Frédéric Fossard, Nicolas Horezan, Eli Janzen, James H. Edgar, Camille Maestre, Bérangère Toury, Catherine Journet, Vincent Garnier, Philippe Steyer, Takashi Taniguchi, Kenji Watanabe, Cédric Robert, Xavier Marie, François Ducastelle, Annick Loiseau, Julien Barjon","doi":"10.1103/physrevx.15.021067","DOIUrl":"https://doi.org/10.1103/physrevx.15.021067","url":null,"abstract":"One of the main interests of 2D materials is their ability to be assembled with many degrees of freedom for tuning and manipulating excitonic properties. There is a need to understand how the structure of the interfaces between atomic layers influences exciton properties. Here we use cathodoluminescence and time-resolved cathodoluminescence experiments to study how excitons interact with the interface between two twisted hexagonal boron nitride (h</a:mi></a:mrow></a:math>-BN) crystals with various angles. An efficient capture of free excitons by the interface is demonstrated, which leads to a population of long-lived and interface-localized (2D) excitons. Temperature-dependent experiments indicate that for high twist angles, these excitons localized at the interface further undergo a self-trapping. It consists in a distortion of the lattice around the exciton on which the exciton traps itself. Our results suggest that this exciton-interface interaction causes the broad 4-eV optical emission of highly twisted <c:math xmlns:c=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><c:mrow><c:mi>h</c:mi></c:mrow></c:math>-BN–<e:math xmlns:e=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><e:mrow><e:mi>h</e:mi></e:mrow></e:math>-BN structures. Exciton self-trapping is finally discussed as a common feature of <g:math xmlns:g=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><g:mi>s</g:mi><g:msup><g:mi>p</g:mi><g:mn>2</g:mn></g:msup></g:math> hybridized boron nitride polytypes and nanostructures due to the ionic nature of the B—N bond and the small size of their excitons. <jats:supplementary-material> <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement> <jats:copyright-year>2025</jats:copyright-year> </jats:permissions> </jats:supplementary-material>","PeriodicalId":20161,"journal":{"name":"Physical Review X","volume":"33 1","pages":""},"PeriodicalIF":12.5,"publicationDate":"2025-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144154195","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"High-Fidelity Electron Spin Gates for Scaling Diamond Quantum Registers","authors":"T. Joas, F. Ferlemann, R. Sailer, P. J. Vetter, J. Zhang, R. S. Said, T. Teraji, S. Onoda, T. Calarco, G. Genov, M. M. Müller, F. Jelezko","doi":"10.1103/physrevx.15.021069","DOIUrl":"https://doi.org/10.1103/physrevx.15.021069","url":null,"abstract":"Diamond is a promising platform for quantum information processing as it can host highly coherent qubits that could allow for the construction of large quantum registers. A prerequisite for such devices is a coherent interaction between nitrogen-vacancy (NV) electron spins enabling scalable entanglement. Entanglement between dipolar-coupled NV spin pairs has been demonstrated but with a limited fidelity, and its error sources have not been characterized. Here, we design and implement a robust two-qubit gate between NV electron spins in diamond and quantify the influence of multiple error sources on the gate performance. Experimentally, we demonstrate a record gate fidelity of F</a:mi></a:mrow>2</a:mn>q</a:mi></a:mrow></a:msub>=</a:mo>(</a:mo>96.0</a:mn>±</a:mo>2.5</a:mn>)</a:mo>%</a:mi></a:mrow></a:math> under ambient conditions. Our identification of the dominant errors paves the way towards NV-NV gates beyond the error correction threshold. <jats:supplementary-material> <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement> <jats:copyright-year>2025</jats:copyright-year> </jats:permissions> </jats:supplementary-material>","PeriodicalId":20161,"journal":{"name":"Physical Review X","volume":"19 1","pages":""},"PeriodicalIF":12.5,"publicationDate":"2025-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144154303","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"“Morphogenetic Action” Principle for 3D Shape Formation by the Growth of Thin Sheets","authors":"Dillon J. Cislo, Anastasios Pavlopoulos, Boris I. Shraiman","doi":"10.1103/physrevx.15.021068","DOIUrl":"https://doi.org/10.1103/physrevx.15.021068","url":null,"abstract":"How does growth encode form in developing organisms? Many different spatiotemporal growth profiles may sculpt tissues into the same target 3D shapes, but only specific growth patterns are observed in animal and plant development. In particular, growth profiles may differ in their degree of spatial variation and growth anisotropy; however, the criteria that distinguish observed patterns of growth from other possible alternatives are not understood. Here we exploit the mathematical formalism of quasiconformal transformations to formulate the problem of “growth pattern selection” quantitatively in the context of 3D shape formation by growing 2D epithelial sheets. We propose that nature settles on growth patterns that are the “simplest” in a certain way. Specifically, we demonstrate that growth pattern selection can be formulated as an optimization problem and solved for the trajectories that minimize spatiotemporal variation in areal growth rates and deformation anisotropy. The result is a complete prediction for the growth of the surface, including not only a set of intermediate shapes, but also a prediction for cell displacement along those surfaces in the process of growth. Optimization of growth trajectories for both idealized surfaces and those observed in nature show that relative growth rates can be uniformized at the cost of introducing anisotropy. Minimizing the variation of programmed growth rates can therefore be viewed as a generic mechanism for growth pattern selection and may help us to understand the prevalence of anisotropy in developmental programs. <jats:supplementary-material> <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement> <jats:copyright-year>2025</jats:copyright-year> </jats:permissions> </jats:supplementary-material>","PeriodicalId":20161,"journal":{"name":"Physical Review X","volume":"3 1","pages":""},"PeriodicalIF":12.5,"publicationDate":"2025-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144154197","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"High-Rate Measurement-Device-Independent Quantum Communication without Optical Reference Light","authors":"Shan-Feng Shao, Lai Zhou, Jinping Lin, Mariella Minder, Chengfang Ge, Yuan-Mei Xie, Ao Shen, Zhengyu Yan, Hua-Lei Yin, Zhiliang Yuan","doi":"10.1103/physrevx.15.021066","DOIUrl":"https://doi.org/10.1103/physrevx.15.021066","url":null,"abstract":"In the realm of long-distance quantum communication, asynchronous measurement-device-independent quantum key distribution (AMDI-QKD) stands out for its experimental simplicity and high key rate generation. Despite these advantages, there exists a challenge in finding a balance between simplifying the laser system further and achieving high key rates. To address this challenge, we have devised a postmeasurement compensation scheme to accurately estimate the mutual frequency offset between two compact lasers using just the announced quantum-signal detection results, thereby obviating the need for optical reference light. As a result, we demonstrate an AMDI-QKD system operating at 2.5 GHz and achieving secure key rates (SKRs) of 537 and 101</a:mn></a:mtext></a:mtext>kbit</a:mi>/</a:mo>s</a:mi></a:mrow></a:math> at distances of 100 and 201 km, respectively, showcasing a significant key rate improvement compared to similar setups. By leveraging ultrastable lasers, we achieve the highest SKRs with measurement-device-independent security within the 100–400-km range. Over 100 km, we reach a remarkable key rate of <d:math xmlns:d=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><d:mrow><d:mn>1.03</d:mn><d:mtext> </d:mtext><d:mtext> </d:mtext><d:mi>Mbit</d:mi><d:mo>/</d:mo><d:mi mathvariant=\"normal\">s</d:mi></d:mrow></d:math>, which could enable real-time one-time-pad video encryption. These findings render AMDI-QKD as a promising contender for the establishment of high performance and cost-effective large-scale intercity quantum networks. <jats:supplementary-material> <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement> <jats:copyright-year>2025</jats:copyright-year> </jats:permissions> </jats:supplementary-material>","PeriodicalId":20161,"journal":{"name":"Physical Review X","volume":"6 1","pages":""},"PeriodicalIF":12.5,"publicationDate":"2025-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144122485","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Fast and Parallelizable Logical Computation with Homological Product Codes","authors":"Qian Xu, Hengyun Zhou, Guo Zheng, Dolev Bluvstein, J. Pablo Bonilla Ataides, Mikhail D. Lukin, Liang Jiang","doi":"10.1103/physrevx.15.021065","DOIUrl":"https://doi.org/10.1103/physrevx.15.021065","url":null,"abstract":"Quantum error correction is necessary to perform large-scale quantum computation but requires extremely large overheads in both space and time. High-rate quantum low-density-parity-check (qLDPC) codes promise a route to reduce qubit numbers, but performing computation while maintaining low space cost has required serialization of operations and extra time costs. In this work, we design fast and parallelizable logical gates for qLDPC codes and demonstrate their utility for key algorithmic subroutines such as the quantum adder. Our gate gadgets utilize transversal logical s between a data qLDPC code and a suitably constructed ancilla code to perform parallel Pauli product measurements (PPMs) on the data logical qubits. For hypergraph product codes, we show that the ancilla can be constructed by simply modifying the base classical codes of the data code, achieving parallel PPMs on a subgrid of the logical qubits with a lower space-time cost than existing schemes for an important class of circuits. Generalizations to 3D and 4D homological product codes further feature fast PPMs in constant depth. While prior work on qLDPC codes has focused on individual logical gates, we initiate the study of fault-tolerant compilation with our expanded set of native qLDPC code operations, constructing algorithmic primitives for preparing k</a:mi></a:math>-qubit Greenberger-Horne-Zeilinger states and distilling or teleporting <c:math xmlns:c=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><c:mi>k</c:mi></c:math> magic states with <e:math xmlns:e=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><e:mi>O</e:mi><e:mo stretchy=\"false\">(</e:mo><e:mn>1</e:mn><e:mo stretchy=\"false\">)</e:mo></e:math> space overhead in <i:math xmlns:i=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><i:mi>O</i:mi><i:mo stretchy=\"false\">(</i:mo><i:mn>1</i:mn><i:mo stretchy=\"false\">)</i:mo></i:math> and <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><m:mi>O</m:mi><m:mo stretchy=\"false\">(</m:mo><m:msqrt><m:mi>k</m:mi></m:msqrt><m:mi>log</m:mi><m:mi>k</m:mi><m:mo stretchy=\"false\">)</m:mo></m:math> logical cycles, respectively. We further generalize this to key algorithmic subroutines, demonstrating the efficient implementation of quantum adders using parallel operations. Our constructions are naturally compatible with reconfigurable architectures such as neutral atom arrays, paving the way to large-scale quantum computation with low space and time overheads. <jats:supplementary-material> <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement> <jats:copyright-year>2025</jats:copyright-year> </jats:permissions> </jats:supplementary-material>","PeriodicalId":20161,"journal":{"name":"Physical Review X","volume":"20 1","pages":""},"PeriodicalIF":12.5,"publicationDate":"2025-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144122402","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Hyperdisordered Cell Packing on a Growing Surface","authors":"R. J. H. Ross, Giovanni D. Masucci, Chun Yen Lin, Teresa L. Iglesias, Sam Reiter, Simone Pigolotti","doi":"10.1103/physrevx.15.021064","DOIUrl":"https://doi.org/10.1103/physrevx.15.021064","url":null,"abstract":"While the physics of disordered packing in nongrowing systems is well understood, unexplored phenomena can emerge when packing takes place in growing domains. We study the arrangements of pigment cells (chromatophores) on squid skin as a biological example of a packed system on an expanding surface. We find that relative density fluctuations in cell numbers grow with spatial scale. We term this behavior “hyperdisordered,” in contrast with hyperuniform behavior in which relative fluctuations tend to zero at large scales. We find that hyperdisordered scaling, akin to that of a critical system, is quantitatively reproduced by a model in which hard disks are randomly inserted in a homogeneously growing surface. In addition, we find that chromatophores increase in size during animal development but maintain a stationary size distribution. The physical mechanisms described in our work may apply to a broad class of growing dense systems. <jats:supplementary-material> <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement> <jats:copyright-year>2025</jats:copyright-year> </jats:permissions> </jats:supplementary-material>","PeriodicalId":20161,"journal":{"name":"Physical Review X","volume":"57 1","pages":""},"PeriodicalIF":12.5,"publicationDate":"2025-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144122403","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Topological Phases with Average Symmetries: The Decohered, the Disordered, and the Intrinsic","authors":"Ruochen Ma, Jian-Hao Zhang, Zhen Bi, Meng Cheng, Chong Wang","doi":"10.1103/physrevx.15.021062","DOIUrl":"https://doi.org/10.1103/physrevx.15.021062","url":null,"abstract":"Global symmetries greatly enrich the landscape of topological quantum phases, playing an essential role from topological insulators to fractional quantum Hall effect. Topological phases in mixed quantum states, originating from decoherence in open quantum systems or disorders in imperfect crystalline solids, have recently garnered significant interest. Unlike pure states, mixed quantum states can exhibit average symmetries—symmetries that keep the total ensemble invariant but not on each individual state. In this work, we present a systematic classification and characterization of average symmetry-protected topological (ASPT) phases applicable to generic symmetry groups, encompassing both average and exact symmetries, for bosonic and fermionic systems. Moreover, we formulate the theory of average symmetry-enriched topological (ASET) orders in disordered bosonic systems. Our systematic approach helps clarify nuanced issues in previous literature and uncovers compelling new physics. Notably, we discover that (1) the definition and classification of ASPT phases in decohered and disordered systems exhibit subtle differences, (2) despite these differences, ASPT phases in both settings can be classified and characterized under a unified framework of defect decoration and spectral sequence, (3) this systematic classification uncovers a plethora of ASPT phases that are intrinsically mixed, implying they can exclusively manifest in decohered or disordered systems where part of the symmetry is average, and (4) similarly for ASET, we find intrinsically disordered phases exhibiting exotic anyon behaviors—the ground states of such phases necessarily contain localized anyons, with gapless (yet still localized) excitation blue spectra. <jats:supplementary-material> <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement> <jats:copyright-year>2025</jats:copyright-year> </jats:permissions> </jats:supplementary-material>","PeriodicalId":20161,"journal":{"name":"Physical Review X","volume":"136 1","pages":""},"PeriodicalIF":12.5,"publicationDate":"2025-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144113490","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Minimal Fractional Topological Insulator in Half-Filled Conjugate Moiré Chern Bands","authors":"Chao-Ming Jian, Meng Cheng, Cenke Xu","doi":"10.1103/physrevx.15.021063","DOIUrl":"https://doi.org/10.1103/physrevx.15.021063","url":null,"abstract":"We propose a “minimal” fractional topological insulator (mFTI), motivated by the recent experimental report on the fractional quantum spin-Hall effect in a transition metal dichalcogenide moiré system. The observed effect suggests the possibility of a topological state living in a pair of half-filled conjugate Chern bands with Chern numbers C</a:mi>=</a:mo>±</a:mo>1</a:mn></a:math>. We propose the mFTI as a novel candidate topological state in the half-filled conjugate Chern bands. The mFTI is characterized by the following features. (1) It is a fully gapped topological order (TO) with 16 Abelian anyons if the electron is considered trivial (32 including electrons), (2) the minimally charged anyon carries electric charge <c:math xmlns:c=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><c:msup><c:mi>e</c:mi><c:mo>*</c:mo></c:msup><c:mo>=</c:mo><c:mi>e</c:mi><c:mo>/</c:mo><c:mn>2</c:mn></c:math>, together with the fractional quantum spin-Hall conductance, implying the robustness of the mFTI’s gapless edge state whenever time-reversal symmetry and charge conservation are present, and (3) the mFTI is minimal in the sense that it has the smallest total quantum dimension (a metric for the TO’s complexity) within all the TOs that can potentially be realized at the same electron filling and with the same Hall transports; the mFTI is also the minimal TO that respects time-reversal symmetry. (4) The mFTI is the common descendant of multiple valley-decoupled “product TOs” with larger quantum dimensions. It can also be viewed as the result of gauging multiple symmetry-protected topological states. Similar mFTIs are classified and constructed for a pair of <e:math xmlns:e=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><e:mn>1</e:mn><e:mo>/</e:mo><e:mi>q</e:mi></e:math>-filled conjugate Chern bands. We further classify the mFTIs via the stability of the gapless interfaces between them. <jats:supplementary-material> <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement> <jats:copyright-year>2025</jats:copyright-year> </jats:permissions> </jats:supplementary-material>","PeriodicalId":20161,"journal":{"name":"Physical Review X","volume":"77 1","pages":""},"PeriodicalIF":12.5,"publicationDate":"2025-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144113492","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Polariton Chern Bands in 2D Photonic Crystals beyond Dirac Cones","authors":"Xin Xie, Kai Sun, Hui Deng","doi":"10.1103/physrevx.15.021061","DOIUrl":"https://doi.org/10.1103/physrevx.15.021061","url":null,"abstract":"Polaritons, formed by strong light-matter interactions, open new avenues for studying topological phases, where the spatial and time symmetries can be controlled via the light and matter components, respectively. However, most research on topological polaritons has been confined to hexagonal photonic lattices featuring Dirac cones at large wave numbers. This restricts key topological properties and device performance, including sub-meV gap sizes that hinder further experimental investigations and future applications of polariton Chern insulator systems. In this study, we move beyond the traditional Dirac cone framework and introduce two alternative band structures in photonic crystals (PhCs) as promising platforms for realizing polariton Chern bands: bands with symmetry-protected bound states in the continuum and bands with symmetry-protected degeneracies at the Γ</a:mi></a:math> points. These band structures are prevalent in various PhC lattices and have features crucial for experimental studies. We show examples of higher Chern number bands, more uniform Berry curvature distributions, and experimentally feasible systems capable of achieving topological gap greater than 10 meV. Our findings show the broad applicability of polariton Chern bands in 2D PhCs and provide design principles for enhancing the functionality and performance of topological photonic devices, opening up exciting possibilities for better understanding and using topological physics. <jats:supplementary-material> <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement> <jats:copyright-year>2025</jats:copyright-year> </jats:permissions> </jats:supplementary-material>","PeriodicalId":20161,"journal":{"name":"Physical Review X","volume":"79 1","pages":""},"PeriodicalIF":12.5,"publicationDate":"2025-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144104264","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Locally Purified Density Operators for Symmetry-Protected Topological Phases in Mixed States","authors":"Yuchen Guo, Jian-Hao Zhang, Hao-Ran Zhang, Shuo Yang, Zhen Bi","doi":"10.1103/physrevx.15.021060","DOIUrl":"https://doi.org/10.1103/physrevx.15.021060","url":null,"abstract":"We propose a tensor network approach known as the locally purified density operator (LPDO) to investigate the classification and characterization of symmetry-protected topological phases in open quantum systems. We extend the concept of injectivity, originally associated with matrix product states and projected entangled pair states, to LPDOs in (&lt;/a:mo&gt;1&lt;/a:mn&gt;+&lt;/a:mo&gt;1&lt;/a:mn&gt;&lt;/a:mrow&gt;)&lt;/a:mo&gt;D&lt;/a:mi&gt;&lt;/a:mrow&gt;&lt;/a:math&gt; and &lt;f:math xmlns:f=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"&gt;&lt;f:mrow&gt;&lt;f:mo stretchy=\"false\"&gt;(&lt;/f:mo&gt;&lt;f:mrow&gt;&lt;f:mn&gt;2&lt;/f:mn&gt;&lt;f:mo&gt;+&lt;/f:mo&gt;&lt;f:mn&gt;1&lt;/f:mn&gt;&lt;/f:mrow&gt;&lt;f:mo stretchy=\"false\"&gt;)&lt;/f:mo&gt;&lt;f:mi mathvariant=\"normal\"&gt;D&lt;/f:mi&gt;&lt;/f:mrow&gt;&lt;/f:math&gt; systems, unveiling two distinct types of injectivity conditions that are inherent for short-range entangled density matrices. Within the LPDO framework, we outline a classification scheme for decohered average symmetry-protected topological (ASPT) phases, consistent with earlier results obtained through spectrum sequence techniques. However, our approach offers an intuitive and explicit construction of ASPT states with the decorated domain-wall picture emerging naturally. We illustrate our framework with ASPT phases protected by a weak global symmetry and strong fermion parity symmetry and then extend it to a general group structure. Moreover, we derive both the classification data and the explicit forms of the obstruction functions using the LPDO formalism, particularly in the case of nontrivial group extension between strong and weak symmetries, where intrinsic ASPT phases may emerge. We demonstrate constructions of fixed-point LPDOs for ASPT phases in both &lt;k:math xmlns:k=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"&gt;&lt;k:mrow&gt;&lt;k:mo stretchy=\"false\"&gt;(&lt;/k:mo&gt;&lt;k:mrow&gt;&lt;k:mn&gt;1&lt;/k:mn&gt;&lt;k:mo&gt;+&lt;/k:mo&gt;&lt;k:mn&gt;1&lt;/k:mn&gt;&lt;/k:mrow&gt;&lt;k:mo stretchy=\"false\"&gt;)&lt;/k:mo&gt;&lt;k:mi mathvariant=\"normal\"&gt;D&lt;/k:mi&gt;&lt;/k:mrow&gt;&lt;/k:math&gt; and &lt;p:math xmlns:p=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"&gt;&lt;p:mrow&gt;&lt;p:mo stretchy=\"false\"&gt;(&lt;/p:mo&gt;&lt;p:mrow&gt;&lt;p:mn&gt;2&lt;/p:mn&gt;&lt;p:mo&gt;+&lt;/p:mo&gt;&lt;p:mn&gt;1&lt;/p:mn&gt;&lt;/p:mrow&gt;&lt;p:mo stretchy=\"false\"&gt;)&lt;/p:mo&gt;&lt;p:mi mathvariant=\"normal\"&gt;D&lt;/p:mi&gt;&lt;/p:mrow&gt;&lt;/p:math&gt; and discuss their physical realization in decohered or disordered systems. In particular, we construct examples of intrinsic ASPT states in &lt;u:math xmlns:u=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"&gt;&lt;u:mrow&gt;&lt;u:mo stretchy=\"false\"&gt;(&lt;/u:mo&gt;&lt;u:mrow&gt;&lt;u:mn&gt;1&lt;/u:mn&gt;&lt;u:mo&gt;+&lt;/u:mo&gt;&lt;u:mn&gt;1&lt;/u:mn&gt;&lt;/u:mrow&gt;&lt;u:mo stretchy=\"false\"&gt;)&lt;/u:mo&gt;&lt;u:mi mathvariant=\"normal\"&gt;D&lt;/u:mi&gt;&lt;/u:mrow&gt;&lt;/u:math&gt; and &lt;z:math xmlns:z=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"&gt;&lt;z:mrow&gt;&lt;z:mo stretchy=\"false\"&gt;(&lt;/z:mo&gt;&lt;z:mrow&gt;&lt;z:mn&gt;2&lt;/z:mn&gt;&lt;z:mo&gt;+&lt;/z:mo&gt;&lt;z:mn&gt;1&lt;/z:mn&gt;&lt;/z:mrow&gt;&lt;z:mo stretchy=\"false\"&gt;)&lt;/z:mo&gt;&lt;z:mi mathvariant=\"normal\"&gt;D&lt;/z:mi&gt;&lt;/z:mrow&gt;&lt;/z:math&gt; using the LPDO formalism. &lt;jats:supplementary-material&gt; &lt;jats:copyright-statement&gt;Published by the American Physical Society&lt;/jats:copyright-statement&gt; &lt;jats:copyright-year&gt;2025&lt;/jats:c","PeriodicalId":20161,"journal":{"name":"Physical Review X","volume":"24 1","pages":""},"PeriodicalIF":12.5,"publicationDate":"2025-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144104266","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}