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Information Theory

55,580 papers in this slice of arXiv.

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2608.12428
3 days ago

MindMemOS: A Portable and Self-Evolving Memory Operating Layer for AI Agents

Kaichao Liang, Yuqi Cui, Hao Kong +13

Memory is a core component of AI agents, enabling them to accumulate experience, maintain personalization, and adapt over long-term interactions. However, existing memory systems often remain fixed after development, limiting their ability to adapt their memory models, organization strategies, and procedural knowledge through continued use. We present MindMemOS, a portable and self-evolving memory operating layer that organizes open-world information using a unified entity property timestructure. MindMemOS supports scenario-adaptive memory modeling, higher-order pattern discovery, autonomous memory refinement, and continuous skill evolution. Its MindMemEvolve algorithm employs validation-driven evolutionary search to optimize memory schemas for target scenarios, whiledreaming consolidates accumulated memories by merging redundant records and resolving conflicts. In addition, implicit corrective feedback serves as a human-in-the-loop signal for identifying and revising potentially inaccurate or misaligned memories. Its MindSkillEvolve algorithm further transforms agent execution trajectories into reusable and progressively refined skills. MindMemOS achieves 94.03% accuracy on LOCOMO and 70.63% on PersonaMem. MindSkillEvolve improves SpreadsheetBench success by 9.2 percentage points over the initial-skill baseline.

Artificial Intelligence
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Information Retrieval
Information Theory
2608.11850
3 days ago

Uniformly Stable Minimal Weyl--Heisenberg Measurements Approaching the SIC Benchmark

Xiuwu Zhu, Yu Wang

Informational completeness (IC) guarantees that an inverse exists, not that it is statistically well conditioned. For minimal rank-one Weyl--Heisenberg (WH) measurements, covariance makes the nonidentity projector-Gram spectrum proportional to the fiducial's ambiguity intensities, with eigenvalues d∣χφ(u)∣2d|χ_φ(u)|^2d∣χφ​(u)∣2, turning stability into an explicit worst-direction design problem; write λλλ for its smallest nonidentity eigenvalue. Haar fiducials are IC almost surely while E[λ−1]=∞\mathbb E[λ^{-1}]=\inftyE[λ−1]=∞, and an explicit geometric family used to establish balanced informationally complete measurements in every dimension has a normalized spectral floor bounded above by an exponentially decaying envelope. We then construct a hierarchy of minimal measurements. A cyclic family with exactly d2d^2d2 outcomes in every integer dimension has floors Θ(d−3)Θ(d^{-3})Θ(d−3) for odd ddd and Θ(d−5)Θ(d^{-5})Θ(d−5) for even ddd; a finite-field family for q=2mq=2^mq=2m obeys the uniform bound λ≥4/9λ\ge4/9λ≥4/9. Our main result treats every prime-power dimension of characteristic p≥5p\ge5p≥5. A balanced one-coordinate perturbation repairs the zero ambiguity axis of a cubic Alltop state, gives an attained floor uniformly bounded below by a positive constant, and confines the entire nonidentity spectrum to [Lq,Uq][L_q,U_q][Lq​,Uq​] with Uq/Lq→1U_q/L_q\to1Uq​/Lq​→1. Its SIC-normalized minimum tends to one, and λ(φq)/Λq⋆→1λ(φ_q)/Λ_q^\star\to1λ(φq​)/Λq⋆​→1 for the global finite-field WH max--min optimum Λq⋆Λ_q^\starΛq⋆​, without assuming SIC existence. The complete spectrum determines the exact finite-sample Hilbert--Schmidt error of canonical linear inversion at I/dI/dI/d, while its lower edge controls local Fisher efficiency and canonical-shadow bounds.

Quantum PhysicsInformation Theory
2608.11782
3 days ago

Proof-Valid Caching under Premise Erasures: Local Structural Limits and Shared-Workload Gains

Jianfeng Xu

We study reliable query recovery under independent premise erasures in semantically transparent caching systems, where every cached object must be a logical consequence of the premise base. Recovery succeeds only when the query remains derivable from surviving premises and the cache. Under a deterministic canonical-witness regime, we prove a query-local projection theorem and an exact residual-leaf law: recovery fails exactly when an erased base leaf retains a cache-free path to the query. Single-query design becomes weighted partial path interception. For shared workloads, we introduce semantic modules and derive exact reliability laws under joint and maximal-error criteria. The shared-module cache is exactly optimal under exact module routing and homogeneous costs, whereas optimal selection in general derivation DAGs is NP-complete at depth two. Against a coded benchmark recovering workload-relevant leaf payloads, MDS parity caching is optimal up to one packet. Leaf-only transparency incurs a first-order overhead inversely proportional to the erasure rate; shared modules multiply that inverse-erasure-rate scaling by the module-to-leaf cost ratio divided by the number of protected leaves. A Datalog instance and Monte Carlo checks illustrate the theory. For derivation-structured content, the results provide exact stochastic-erasure counterparts of function-correcting storage and an exact distributional quantification of maximal recoverability.

Information Theory
2608.11757
3 days ago

Complete characterization of the differential spectrum of a Niho type power function

Nian Li, Xi Xie, Rui Xu +2

Power functions with Niho exponents have attracted considerable attention due to their important applications in sequence design, coding theory, and cryptography. This paper investigates the differential properties of Niho type power functions of the form F(x)=xs(2m−1)+1F(x)=x^{s(2^m-1)+1}F(x)=xs(2m−1)+1 over F22m\mathbb{F}_{2^{2m}}F22m​ with 2≤s≤2m2\leq s\leq 2^m2≤s≤2m. We first establish a general characterization of the differential spectrum of F(x)F(x)F(x) having at most three nonzero values via its Walsh spectrum. Focusing subsequently on the case s=(2k+1)−1(mod2m+1)s=(2^k+1)^{-1} \pmod{2^m+1}s=(2k+1)−1(mod2m+1) where gcd⁡(k,m)=e\gcd(k,m)=egcd(k,m)=e, we employ a refined analysis of the number of solutions to certain equations over finite fields. Specifically, it is proved that F(x)F(x)F(x) is locally differentially 2e2^e2e-uniform when gcd⁡(2k−1,2m+1)=2e+1\gcd(2^k-1,2^m+1)=2^e+1gcd(2k−1,2m+1)=2e+1 and locally differentially (22e−2e)(2^{2e}-2^e)(22e−2e)-uniform when gcd⁡(2k−1,2m+1)=1\gcd(2^k-1,2^m+1)=1gcd(2k−1,2m+1)=1, and their differential spectra are completely determined. These results completely characterize the differential properties of this family and yield new infinite families of locally differentially 444-uniform power functions.

Information Theory
2608.11754
3 days ago

Characterizing Nonlinearities in IM-DD Links via the Best Linear Approximation: Distortion Analysis and Modulation Optimization

Sebastian Fraga Fernández, Leonardo Minelli, Fernando de Bernardinis +3

We use the best linear approximation (BLA) to characterize nonlinearities in IM-DD links. The orthogonal nonlinear distortion is shown to be non-Gaussian. The BLA is used to approximate the optimum modulation depth for different equalizer structures.

Signal ProcessingInformation Theory
2608.11666
3 days ago

Sparse Rotatable Arrays (SRA): Unifying Array Aperture and Antenna Directivity for Wireless Communications

Ailing Zheng, Qingqing Wu, Xiyuan Liu +1

Sparse arrays reduce the hardware cost of extremely large-scale multiple-input multiple-output systems while preserving a large effective aperture, but may suffer from severe sidelobe and grating-lobe leakage. Rotatable antennas (RAs) provide directional leakage suppression, whereas frequent boresight adaptation to instantaneous user locations incurs considerable control overhead. In practice, long-term user distributions are often non-uniform, with compact hotspots coexisting with scattered users, which enables low-complexity RA configuration based on group-level geometry. Motivated by this observation, we propose a group-aware sparse RA architecture in which users are organized into service groups and the activated RAs are partitioned into group-specific sparse subarrays. We maximize the minimum signal-to-interference-plus-noise ratio (SINR) among all users by jointly optimizing the sparse-aperture allocation, RA orientations, and transmit beamforming. We show that RA-induced inter-group leakage suppression and sparse-aperture-induced intra-group orthogonalization jointly yield an approximately decoupled group-wise beamforming structure, providing physical guidance for group-wise antenna-number allocation and non-periodic sparse-position initialization. Guided by these insights, we develop a structured low-complexity two-layer algorithm that embeds a closed-form projected group-center RA orientation rule into the sparse-aperture allocation and beamforming design. The algorithm combines analysis-guided initialization, sampled multi-start antenna-allocation search, and bisection-based second-order cone programming for beamforming. Numerical results show that the proposed design closely approaches the fully shared and orientation-optimized benchmarks while substantially outperforming the compact and omnidirectional sparse-array schemes.

Information Theory
2608.12414
3 days ago

New optimal linear codes over ₄

Hopein Christofen Tang, Djoko Suprijanto

In this work, we present novel approaches for constructing linear codes over ₄ from the known ones. We succeeded in obtaining new linear codes, many of which are optimal. In particular, we found all optimal codes for k1=2, k2=0k_1=2,~k_2=0k1​=2, k2​=0 and many optimal codes for k1=3, k2=0.k_1=3,~k_2=0.k1​=3, k2​=0.

Information TheoryCombinatorics
2608.11599
3 days ago

An AoI-oriented Time-Frequency Distributed Access Mechanism in Wireless Sensor Networks with Spectrum Division

Jingwei Liu, Fang Liu, Wing Shing Wong +2

The increasing adoption of spectrum-division techniques enables concurrent uplink transmissions over multiple orthogonal resources, yet low-overhead access design with effective information freshness remains insufficiently studied for large-scale randomly activated sensor networks. In this paper, we apply the age of information (AoI) to measure information freshness and propose an AoI-efficient deterministic time-frequency distributed access (D-TFDA) mechanism. D-TFDA combines centralized configuration and distributed operation through a periodic token-based time-frequency structure, which provides sensors with collision-free and predictable transmission opportunities without considerable run-time overhead. We develop an analytical framework to characterize the long-term average AoI (AAoI) by exploiting the periodicity of the token assignment pattern and modeling the steady local state of each sensor with a one-dimensional discrete-time Markov chain (DTMC). We further reveal structural properties of the token assignment pattern and identify AoI-equivalent token clusters, which substantially reduce the search space of the AAoI-optimal token allocation problem. Based on this structure, we formulate the reduced problem as a linear programming (LP) problem and develop an AAoI-optimal search algorithm, together with an auction-inspired heuristic algorithm of lower complexity. Simulation results validate the proposed AAoI analysis, demonstrate the effectiveness of the token allocation algorithms, and show that D-TFDA achieves substantially lower AAoI than optimized random access baselines by avoiding collisions and exploiting heterogeneous sensor--resource transmission reliability.

Information Theory
2608.11551
4 days ago

Semantic Error Control Coding with Foundation Models for Future Communications

Chentao Yue, Gaoyang Pang, Branka Vucetic +1

Classical channel decoding typically treats all information sequences as equally likely and relies primarily on the channel observations and code structure, without exploiting statistical or semantic structure in the source data. Although source compression is designed to remove redundancy, practical source coding can leave substantial residual structure that conventional channel decoders do not exploit. Modern multimodal data sources, including text, speech, and images, exhibit rich statistical and semantic dependencies that foundation models can learn and exploit to improve channel decoding. This article introduces semantic error control coding (SECC), which seamlessly integrates the semantic structure of the source into encoding and decoding through a foundation model. The semantic source prior, represented by the model's a priori probability of the source content, directs code redundancy toward semantically important content at the encoder, and improves reliability estimation, candidate search, and error detection/correction at the decoder. The channel code keeps its algebraic structure, and its constraints ensure that the semantic suggestions from the foundation model comply with this structure. We describe the SECC framework, classify its design methods into three approaches, and demonstrate representative schemes on text sources. The demonstrated schemes show several decibels of coding gain over conventional decoding on AWGN channels, and reach error rates below the normal approximation bound. Finally, we identify several open challenges.

Information Theory
2608.11160
4 days ago

Quantum Codes with Arbitrary Z-Rotation logical Gates and Applications to Fault-Tolerant Code Switching

Reza Dastbasteh, Ruben M. Otxoa, Pedro M. Crespo +1

A technique for realizing a universal set of fault-tolerant quantum operations is the code switching method, which leverages two quantum codes with complementary sets of transversal gates. To date, the application of this technique has been largely limited to families of color codes supporting a logical TTT gate. No analogous code switching protocols exist for many other prominent families, such as rotated surface codes, or for finer ZZZ-rotation gates. In this work, we first utilize the doubling technique as a unified framework to construct a class of quantum color codes encoding a single logical qubit with an arbitrarily large minimum distance, enabling the transversal realization of arbitrary small logical ZZZ-rotation gates. We investigate the structural properties of this code family, demonstrating that they improve upon the parameters of state-of-the-art triorthogonal codes, achieve lower qubit overhead compared to certain known color codes, and admit single-shot decoding of ZZZ-syndromes via meta-checks. Furthermore, we show that this framework extends beyond color codes; specifically, it enables the generation of rrr-orthogonal quantum codes, r≥2r \ge 2r≥2, that inherit the local geometry of rotated surface codes. We then provide an overhead optimization protocol alongside several candidate codes tailored for realizing logical ZZZ-rotation gates within rotated surface codes. Finally, we extend the fault-tolerant code switching protocol based on transversal CNOT gates to incorporate fault-tolerant realization of ZZZ-rotation gates at any level of the Clifford hierarchy for geometries compatible with rotated surface codes. We present the first demonstration of fault-tolerant magic state preparation by means of code switching within a distance-three rotated surface code using a total footprint of only 45 physical qubits, and evaluate its performance through a simulation.

Quantum PhysicsInformation Theory
2608.11151
4 days ago

Breaking the Quadratic Barrier for von Neumann Entropy Estimation

Minbo Gao, Qisheng Wang

We study the sample complexity of estimating the von Neumann entropy of an unknown ddd-dimensional quantum state. All previously known estimators require Ω(d2)Ω(d^2)Ω(d2) samples, and plug-in estimators are known to face a quadratic barrier. We give the first subquadratic-sample estimator: for additive error ε\varepsilonε, our estimator uses O ⁣(d2log⁡2(log⁡(d))log⁡(1/ε)ε2log⁡2(d)+log⁡2(d/ε)ε2)O\!\left(\frac{d^2 \log^2(\log(d)) \log(1/\varepsilon)}{\varepsilon^2 \log^2(d)} + \frac{\log^2(d/\varepsilon)}{\varepsilon^2}\right)O(ε2log2(d)d2log2(log(d))log(1/ε)​+ε2log2(d/ε)​) samples. In particular, for constant ε\varepsilonε, the complexity is Oε(d2log⁡2(log⁡(d))/log⁡2(d))=o(d2)O_\varepsilon(d^2\log^2(\log(d))/\log^2(d))=o(d^2)Oε​(d2log2(log(d))/log2(d))=o(d2). Our analysis introduces a new pinching inequality that bounds the entropy loss under a space direct-sum decomposition, together with a bias-corrected estimator for large eigenvalues and a new bounded-coefficient polynomial estimator for small eigenvalues.

Quantum PhysicsInformation Theory
2608.11056
4 days ago

Twisted Conjugacy and the Classification of Induced Centrosymmetric Alternant Codes

Ousmane Ndiaye, Massamba Sow

This paper presents a classification of induced centrosymmetric alternant codes through the study of the automorphism structures inherited from Generalized Reed--Solomon (GRS) codes. We introduce the twisted conjugation action naturally associated with projective semilinear transformations and establish its correspondence with ordinary conjugacy in the projective semilinear group. This correspondence enables the application of Shintani's theorem to classify the γpjγ_{p^j}γpj​-similarity classes of involutions. As a consequence, we obtain necessary and sufficient conditions for an alternant code to admit a centrosymmetric structure induced by a projective semilinear automorphism. The resulting classification unifies the different families of induced centrosymmetric alternant codes within a common automorphism-based framework.

Information TheoryCombinatorics
2608.11031
4 days ago

A Bayesian Proof of the Bernoulli Theorem

Jingbo Liu, Ilias Zadik

We give a new proof of the Bernoulli theorem, conjectured by Talagrand and proved in the seminal work of Bednorz and Latała. Our approach is based on information-theoretic ideas: lower bounds on the supremum of a Bernoulli process are translated to the fundamental limits of Bayesian estimation in a Cauchy additive channel. This leads to a new information-theoretic functional that characterizes Bernoulli-process suprema and plays a role analogous to Fernique's majorizing-measure functional for Gaussian processes. The same viewpoint yields a distributional strengthening: for any prescribed law of the index, we characterize the largest expected value attainable over all couplings of that index with the Bernoulli process. This extends to Bernoulli processes a phenomenon previously understood for Gaussian processes through the work of Fernique and Talagrand.

ProbabilityInformation TheoryStatistics Theory
2608.11003
4 days ago

Information Bottleneck under Perfect Privacy

Junle Zhong, Mohamad Assaad, Sreejith Sreekumar

In this work, we study the information bottleneck under perfect privacy, with particular emphasis on the active-rate regime, where the representation-rate constraint is binding and directly limits the achievable utility. The goal is to construct a representation that preserves utility-relevant information while remaining statistically independent of a sensitive variable. This exact independence requirement introduces an additional constraint beyond the classical rate-relevance tradeoff and must be explicitly incorporated into the optimization. To this end, we develop an alternating direction method of multipliers (ADMM)-based method tailored to the resulting problem structure. Under suitable regularity conditions, we establish global convergence of the generated sequence, characterize its convergence rate through the Kurdyka-Lojasiewicz exponent, and extend the analysis to inexact block updates.

Information TheoryMachine Learning
2608.11000
4 days ago

The entanglement-assisted transmission capacity is a strong converse bound for identification

Satvik Singh

Classical identification via a noisy channel is a communication task in which the receiver is not required to reconstruct the full transmitted message, but only to decide whether it coincides with a message of interest. This relaxation allows the number of identifiable messages to grow doubly exponentially with the blocklength. For quantum channels, the resulting (doubly exponential) identification capacity CIDC_{\mathrm{ID}}CID​ can strictly exceed the ordinary (exponential) transmission capacity CCC. In this paper, we prove that the entanglement-assisted transmission capacity CEC_ECE​ is a strong converse bound for this task: CID≤CEC_{\mathrm{ID}}\leq C_ECID​≤CE​. For sufficiently low-noise channels, this bound can also be achieved via the Hayden-Winter (quantum) identification + fingerprinting codes. This yields an exact characterization CID=CEC_{\mathrm{ID}}=C_ECID​=CE​ of identification capacity for such channels. However, for general channels, we prove that this upper bound can be strict. We exhibit an explicit family of transpose-depolarizing channels for which CID<CEC_{\mathrm{ID}}<C_ECID​<CE​. As a consequence, we also obtain the first example of strict superadditivity of the identification capacity CIDC_{\mathrm{ID}}CID​.

Quantum PhysicsInformation TheoryMetric Geometry
2608.10967
4 days ago

Search-to-Decision Reductions for the Linear and General Code Equivalence Problems

Abhinaba Mazumder

In this paper, we present efficient search-to-decision reductions for the Linear Code Equivalence (LCE) and Generalised Code Equivalence (GCE) problems. Our methodology is inspired by the recent search-to-decision reduction for Permutation Code Equivalence. We demonstrate how to recover the permutation component of the equivalence using a decision oracle, and subsequently show one way of recovering the diagonal and field automorphism components in deterministic polynomial time by leveraging the elegant Engel-Schneider algorithm for diagonal equivalence.

Information Theory
2608.10922
4 days ago

Bounds for Pure Disjoint (r,δ)(r,δ)(r,δ)-Quantum Locally Recoverable Codes

Evagoras Stylianou, Holger Boche

We study pure disjoint (r,δ)(r,δ)(r,δ)-quantum locally recoverable codes (qLRCs) without assuming a stabilizer structure. We formulate local Knill--Laflamme conditions for recovery from up to δ−1δ-1δ−1 erasures within a recovery block, and introduce blockwise Shor--Laflamme and unitary weight enumerators that capture how error weight is distributed across recovery sets. We establish several properties of these enumerators and use them to derive a Singleton-like bound that strengthens the known bound for disjoint (r,δ)(r,δ)(r,δ)-qLRCs under a purity assumption, as well as a linear-programming upper bound on the code dimension. These results provide a non-stabilizer, weight-enumerator-based approach to the study of pure disjoint (r,δ)(r,δ)(r,δ)-qLRCs.

Information TheoryQuantum Physics
2608.10912
4 days ago

CSS Quantum LRCs with Intersecting Recovery Sets: Constructions and Bounds

Evagoras Stylianou, Vinayak Ramkumar, Holger Boche +1

In this work, we study (r,t,x)(r,t,x)(r,t,x) quantum locally recoverable codes (qLRCs) with locality rrr, ttt recovery sets per qudit, and intersection parameter xxx. We first show that, assuming the underlying classical codes have dual minimum distance at least two, a CSS code is an (r,t,x)(r,t,x)(r,t,x)-qLRC if and only if the underlying classical codes are (r,t,x)(r,t,x)(r,t,x) classical LRCs (cLRCs) with common recovery sets. We then use subset-inclusion matrices to construct families of binary dual-containing (r,t,x)(r,t,x)(r,t,x)-cLRCs, which yield binary (r,t,x)(r,t,x)(r,t,x)-qLRCs via the CSS construction. For CSS (r,t,x)(r,t,x)(r,t,x)-qLRCs, we derive upper bounds on the dimension and rate, minimum-distance bounds in the pure case, and a Singleton-like dimension bound in the exact case. Finally, we show that these families attain high rates and nontrivial minimum distances.

Information TheoryQuantum Physics
2608.10816
4 days ago

Capacity regimes for Boolean function computation via channels

Jingge Zhu, Matthias Frey

Consider a point-to-point communication system in which the transmitter holds a binary message of length mmm and transmits a corresponding codeword of length nnn. The receiver's goal is to recover a Boolean function of that message, where the function is unknown to the transmitter, but chosen from a known class FFF. We are interested in the asymptotic relationship of mmm and nnn: given nnn, how large can mmm be (asymptotically), such that the value of the Boolean function can be recovered reliably? This problem generalizes the identification-via-channels framework introduced by Ahlswede and Dueck. In this paper, we formulate the notion of computation capacity, and derive achievability and converse results for a large class of functions FFF, characterized by the Hamming weight of functions. Different from the classical transmission problem, the performance of the function computation problem is jointly characterized by the computation capacity and the rate function, namely how mmm scales with nnn asymptotically. Our results give a complete characterization of the rate function of the computation problem, and provide upper and lower bounds on the computation capacity, where they differ by a factor of at most 222.

Information Theory
2608.10674
4 days ago

Optimal fidelity estimation when one state is pure via algorithmic Uhlmann transform

Yupan Liu, Qisheng Wang

The Uhlmann fidelity F(ρ0,ρ1)=tr∣ρ0ρ1∣{\rm F}(ρ_0,ρ_1) = {\rm tr}|\sqrt{ρ_0}\sqrt{ρ_1}|F(ρ0​,ρ1​)=tr∣ρ0​​ρ1​​∣ is one of the most fundamental quantities in quantum information theory for quantifying the closeness between two quantum states. Estimating the Uhlmann fidelity to within additive error ε\varepsilonε requires a number of copies of the states, or queries to their state-preparation circuits, that depends at least linearly on the smaller of the ranks of ρ0ρ_0ρ0​ and ρ1ρ_1ρ1​. Consequently, this rank dependence disappears when either state is pure, in which case the query and sample complexities depend only polynomially on 1/ε1/\varepsilon1/ε. However, the known optimal estimator for F(ρ,∣ψ⟩ ⁣⟨ψ∣){\rm F}(ρ,|ψ\rangle\!\langleψ|)F(ρ,∣ψ⟩⟨ψ∣) due to Fang and Wang (ESA 2025) requires prior knowledge of which state is pure. In this work, we remove this mathematically unnecessary prior-knowledge requirement and establish an optimal estimator for F(ρ,∣ψ⟩ ⁣⟨ψ∣){\rm F}(ρ, |ψ\rangle\!\langleψ|)F(ρ,∣ψ⟩⟨ψ∣) under the sole promise that one of the two states is pure, without knowing which one. Our estimator is obtained by specializing the refined algorithmic Uhlmann transform of Utsumi, Nakata, Wang, and Takagi (2025) to the case where one state is pure. In this setting, the Uhlmann fidelity can be recovered as follows: apply a unitary dilation of trA(∣ψ0⟩ ⁣⟨ψ1∣){\rm tr}_{\sf A}(|ψ_0\rangle\!\langleψ_1|)trA​(∣ψ0​⟩⟨ψ1​∣) (or its inverse) to the reference register R\sf RR of the purification ∣ψ1⟩|ψ_1\rangle∣ψ1​⟩ (or ∣ψ0⟩|ψ_0\rangle∣ψ0​⟩) on the registers A\sf AA and R\sf RR, estimate the corresponding square-root amplitude in each case, and take the maximum of the resulting two estimates.

Quantum PhysicsData Structures and AlgorithmsInformation Theory