arXiv subjects

Mizanur Rahaman

Publications and source records attributed to Mizanur Rahaman.

At least 19 recordsLinked to original sources

arXiv2026

Generalized Stein's lemma and asymptotic equipartition property for subalgebra entropies

The quantum Stein's lemma is a fundamental result of quantum hypothesis testing in the context of distinguishing two quantum states. The ``generalized quantum Stein's lemma'' asserts that this result is true in a general framework where one of the states is replaced by convex sets of quantum states. Formulated in 2008, the generalized Stein's lemma is one of the most influential results in quantum information theory that links resource convertibility to quantum hypothesis testing. However, in 2023 a logical gap was found in the original proof of the generalized Stein's lemma and since then there has been an enormous effort in resolving this issue. In this work, we show that the assertion of the generalized Stein's lemma is true for the setting where the second hypothesis is the state space of any finite-dimensional subalgebra. This is obtained through a strong asymptotic equipartition property for smooth subalgebra entropies that applies to any fixed smoothing parameter. In fact, we obtain a stronger second-order analysis for the hypothesis-testing relative entropy in this setting. As an application in resource theory, we show that the relative entropy of a subalgebra is the asymptotic dilution cost under suitable operations. This provides a possible route to establishing a connection between different quantum resources based on subalgebras. After finishing this article, we learned that there are two independent works (by Hayashi-Yamasaki, and Lami) that finally resolve the generalized Stein's lemma in its full generality. However, we provide an alternative proof in a special case using different operator-algebraic techniques that may be of independent interest.

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arXiv2026

Amenable traces and the joint numerical radius

We provide necessary and sufficient characterizations of the existence of an amenable trace on a C$^*$-algebra in terms of the joint free numerical radius of tuples of unitaries, isometries, and partial isometries in the algebra. We apply these results to obtain new obstructions to various lifting properties.

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arXiv2026

Fast convergence of Dynamic Capacities of GNS-Symmetric Quantum Channels

We consider a quantum system described by a quantum channel $Φ$ that is applied at every time step and study the time evolution of its information capacities. When $Φ$ is a GNS-symmetric channel (this includes Pauli channels, for example), we give explicit exponential convergence bounds for the classical and quantum capacities. These bounds are in terms of entropic properties of $Φ$. We further illustrate how these results help quantify the performance of active versus passive error-correction setups.

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arXiv2026

Separable neighbourhood of identity in C$^{\ast}$-algebras

We study the structure of separable elements in bipartite C$^{\ast}$-algebras, focusing on the existence and size of a separable neighbourhood around the identity element. While this phenomenon is well understood in the finite-dimensional setting, its extension to general C$^{\ast}$-algebras presents additional challenges. We show that the problem of determining such a neighbourhood can be reduced to estimating the completely bounded norm of contractive positive maps. This approach allows us to characterize the size of such neighbourhoods in terms of structural properties of the algebra, notably its rank. As a consequence, we also resolve a recent conjecture of Musat and Rørdam.

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arXiv2026

Jensen's inequality for partial traces in von Neumann algebras

Motivated by a recent result on finite-dimensional Hilbert spaces, we prove a Jensen's inequality for partial traces in semifinite von Neumann algebras. We also prove a similar inequality in the framework of general (non-tracial) von Neumann algebras.

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arXiv2025

Emulation Capacity between Idempotent Channels

We study the optimal rates of emulation (also called interconversion) between quantum channels. When the source and the target channels are idempotent, we give a single-letter expression for the zero-error emulation capacity in terms of structural properties of the range of the two channels. This expression shows that channel emulation is not reversible for general idempotent channels. Furthermore, we establish a strong converse rate that matches with the zero-error emulation capacity when the source or the target channel is either an identity or a completely dephasing channel.

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arXiv2025

Zero-error communication under discrete-time Markovian dynamics

Consider an open quantum system with (discrete-time) Markovian dynamics. Our task is to store information in the system in such a way that it can be retrieved perfectly, even after the system is left to evolve for an arbitrarily long time. We show that this is impossible for classical (resp. quantum) information precisely when the dynamics is mixing (resp. asymptotically entanglement breaking). Furthermore, we provide tight universal upper bounds on the minimum time after which any such dynamics 'scrambles' the encoded information beyond the point of perfect retrieval. On the other hand, for dynamics that are not of this kind, we show that information must be encoded inside the peripheral space associated with the dynamics in order for it to be perfectly recoverable at any time in the future. This allows us to derive explicit formulas for the maximum amount of information that can be protected from noise in terms of the structure of the peripheral space of the dynamics.

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arXiv2025

Exactness and LLP Results via Operator System Methods

In this paper, we employ operator system techniques to investigate structural properties of C*-algebras. In particular, we provide more direct proofs of results concerning exactness and the local lifting property (LLP) of group C$^*$-algebras that avoid relying on the traditional heavy machinery of C$^*$-algebra theory. Briefly, these methods allow us to deduce that any C*-algebra containing $n$ unitaries whose $\textit{ joint numerical radius}$, in the sense defined by \cite{FKP}, is strictly less than $n$, must fail certain of these properties.

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arXiv2025

Asymptotic Equipartition Theorems in von Neumann algebras

The Asymptotic Equipartition Property (AEP) in information theory establishes that independent and identically distributed (i.i.d.) states behave in a way that is similar to uniform states. In particular, with appropriate smoothing, for such states both the min and the max relative entropy asymptotically coincide with the relative entropy. In this paper, we generalize several such equipartition properties to states on general von Neumann algebras. First, we show that the smooth max relative entropy of i.i.d. states on a von Neumann algebra has an asymptotic rate given by the quantum relative entropy. In fact, our AEP not only applies to states, but also to quantum channels with appropriate restrictions. In addition, going beyond the i.i.d. assumption, we show that for states that are produced by a sequential process of quantum channels, the smooth max relative entropy can be upper bounded by the sum of appropriate channel relative entropies. Our main technical contributions are to extend to the context of general von Neumann algebras a chain rule for quantum channels, as well as an additivity result for the channel relative entropy with a replacer channel.

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arXiv2025

Capacities of quantum Markovian noise for large times

Given a quantum Markovian noise model, we study the maximum dimension of a classical or quantum system that can be stored for arbitrarily large time. We show that, unlike the fixed time setting, in the limit of infinite time, the classical and quantum capacities are characterized by efficiently computable properties of the peripheral spectrum of the quantum channel. In addition, the capacities are additive under tensor product, which implies in the language of Shannon theory that the one-shot and the asymptotic i.i.d. capacities are the same. We also provide an improved algorithm for computing the structure of the peripheral subspace of a quantum channel, which might be of independent interest.

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arXiv2024

Completely Bounded Norms of $k$-positive Maps

Given an operator system $\mathcal{S}$, we define the parameters $r_k(\mathcal{S})$ (resp. $d_k(\mathcal{S})$) defined as the maximal value of the completely bounded norm of a unital $k$-positive map from an arbitrary operator system into $\mathcal{S}$ (resp. from $\mathcal{S}$ into an arbitrary operator system). In the case of the matrix algebras $M_n$, for $1 \leq k \leq n$, we compute the exact value $r_k(M_n) = \frac{2n-k}{k}$ and show upper and lower bounds on the parameters $d_k(M_n)$. Moreover, when $\mathcal{S}$ is a finite-dimensional operator system, adapting recent results of Passer and the 4th author, we show that the sequence $(r_k( \mathcal{S}))$ tends to $1$ if and only if $\mathcal{S}$ is exact and that the sequence $(d_k(\mathcal{S}))$ tends to $1$ if and only if $\mathcal{S}$ has the lifting property.

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arXiv2023

Operator Algebra Generalization of a Theorem of Watrous and Mixed Unitary Quantum Channels

We establish an operator algebra generalization of Watrous' theorem \cite{watrous2009} on mixing unital quantum channels (completely positive trace-preserving maps) with the completely depolarizing channel, wherein the more general objects of focus become (finite-dimensional) von Neumann algebras, the unique trace preserving conditional expectation onto the algebra, the group of unitary operators in the commutant of the algebra, and the fixed point algebra of the channel. As an application, we obtain a result on the asymptotic theory of quantum channels, showing that all unital channels are eventually mixed unitary. We also discuss the special case of the diagonal algebra in detail, and draw connections to the theory of correlation matrices and Schur product maps.

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arXiv2023

Positive maps and entanglement in real Hilbert spaces

The theory of positive maps plays a central role in operator algebras and functional analysis, and has countless applications in quantum information science. The theory was originally developed for operators acting on complex Hilbert spaces, and little is known about its variant on real Hilbert spaces. In this article we study positive maps acting on a full matrix algebra over the reals, pointing out a number of fundamental differences with the complex case and discussing their implications in quantum information. We provide a necessary and sufficient condition for a real map to admit a positive complexification, and connect the existence of positive maps with non-positive complexification with the existence of mixed states that are entangled in real Hilbert space quantum mechanics, but separable in the complex version, providing explicit examples both for the maps and for the states. Finally, we discuss entanglement breaking and PPT maps, and we show that a straightforward real version of the PPT-squared conjecture is false even in dimension 2. Nevertheless, we show that the original PPT-squared conjecture implies a different conjecture for real maps, in which the PPT property is replaced by a stronger property of invariance under partial transposition (IPT). When the IPT property is assumed, we prove an asymptotic version of the conjecture.

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arXiv2022

Entanglement Breaking Rank via Complementary Channels and Multiplicative Domains

Quantum entanglement can be studied through the theory of completely positive maps in a number of ways, including by making use of the Choi-Jamilkowski isomorphism, which identifies separable states with entanglement breaking quantum channels, and optimal ensemble length with entanglement breaking rank. The multiplicative domain is an important operator structure in the theory of completely positive maps. We introduce a new technique to determine if a channel is entanglement breaking and to evaluate entanglement breaking rank, based on an analysis of multiplicative domains determined by complementary quantum channels. We give a full description of the class of entanglement breaking channels that have a projection as their Choi matrix, and we show the entanglement breaking and Choi ranks of such channels are equal.

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arXiv2022

Counterexamples to the extendibility of positive unital norm-one maps

Arveson's extension theorem guarantees that every completely positive map defined on an operator system can be extended to a completely positive map defined on the whole C*-algebra containing it. An analogous statement where complete positivity is replaced by positivity is known to be false. A natural question is whether extendibility could still hold for positive maps satisfying stronger conditions, such as being unital and norm 1. Here we provide three counterexamples showing that positive norm-one unital maps defined on an operator subsystem of a matrix algebra cannot be extended to a positive map on the full matrix algebra. The first counterexample is an unextendible positive unital map with unit norm, the second counterexample is an unextendible positive unital isometry on a real operator space, and the third counterexample is an unextendible positive unital isometry on a complex operator space.

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arXiv2021

An extension of Bravyi-Smolin's construction for UMEBs

We extend Bravyi and Smolin's construction for obtaining unextendible maximally entangled bases (UMEBs) from equiangular lines. We show that equiangular real projections of rank more than 1 also exhibit examples of UMEBs. These projections arise in the context of optimal subspace packing in Grassmannian spaces. This generalization yields new examples of UMEBs in infinitely many dimensions of the underlying system. Consequently, we find a set of orthogonal unitary bases for symmetric subspaces of matrices in odd dimensions. This finding validates a recent conjecture about the mixed unitary rank of the symmetric Werner-Holevo channel in infinitely many dimensions.

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arXiv2021

Positively Factorizable Maps

We initiate a study of linear maps on $M_n(\mathbb{C})$ that have the property that they factor through a tracial von Neumann algebra $(\mathcal{A,τ})$ via operators $Z\in M_n(\mathcal{A})$ whose entries consist of positive elements from the von-Neumann algebra. These maps often arise in the context of non-local games, especially in the synchronous case. We establish a connection with the convex sets in $\mathbb{R}^n$ containing self-dual cones and the existence of these maps. The Choi matrix of a map of this kind which factors through an abelian von-Neumann algebra turns out to be a completely positive (CP) matrix. We fully characterize positively factorizable maps whose Choi rank is 2. We also provide some applications of this analysis in finding doubly nonnegative matrices which are not CPSD. A special class of these examples is found from the concept of Unextendible Product Bases in quantum information theory.

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arXiv2021

Entanglement Breaking Channels, Stochastic Matrices, and Primitivity

We consider the important class of quantum operations (completely positive trace-preserving maps) called entanglement breaking channels. We show how every such channel induces stochastic matrix representations that have the same non-zero spectrum as the channel. We then use this to investigate when entanglement breaking channels are primitive, and prove this depends on primitivity of the matrix representations. This in turn leads to tight bounds on the primitivity index of entanglement breaking channels in terms of the primitivity index of the associated stochastic matrices. We also present examples and discuss open problems generated by the work.

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