arXiv subjects

Marius Junge

Publications and source records attributed to Marius Junge.

At least 55 records · Page 3Linked to original sources

arXiv2019

Stability of logarithmic Sobolev inequalities under a noncommutative change of measure

We generalize Holley-Stroock's perturbation argument from commutative to quantum Markov semigroups. As a consequence, results on (complete) modified logarithmic Sobolev inequalities and logarithmic Sobolev inequalities for self-adjoint quantum Markov process can be used to prove estimates on the exponential convergence in relative entropy of quantum Markov systems which preserve a fixed state. This leads to estimates for the decay to equilibrium for coupled systems and to estimates for mixed state preparation times using Lindblad operators. Our techniques also apply to discrete time settings, where we show that the strong data processing inequality constant of a quantum channel can be controlled by that of a corresponding unital channel.

quant-ph↗

arXiv2019

Relative entropy for von Neumann subalgebras

We revisit the connection between index and relative entropy for an inclusion of finite von Neumann algebras. We observe that the Pimsner-Popa index connects to sandwiched Renyi $p$-relative entropy for all $1/2\le p\le \infty$, including Umegaki's relative entropy at $p=1$. Based on that, we introduce a new notation of relative entropy to a subalgebra which generalizes subfactors index. This relative entropy has application in estimating decoherence time of quantum Markov semigroups.

math.OA↗

arXiv2019

Quantum Euclidean Spaces with Noncommutative Derivatives

Quantum Euclidean spaces, as Moyal deformations of Euclidean spaces, are the model examples of noncompact noncommutative manifold. In this paper, we study the quantum Euclidean space equipped with partial derivatives satisfying canonical commutation relation (CCR). This gives an example of semi-finite spectral triple with non-flat geometric structure. We develop an abstract symbol calculus for the pseudo-differential operators with noncommuting derivatives. We also obtain a simplified local index formula (even case) that is similar to the commutative setting.

math.OA↗

arXiv2019

Algebraic Calderón-Zygmund theory

Calderón-Zygmund theory has been traditionally developed on metric measure spaces satisfying additional regularity properties. In the lack of good metrics, we introduce a new approach for general measure spaces which admit a Markov semigroup satisfying purely algebraic assumptions. We shall construct an abstract form of "Markov metric" governing the Markov process and the naturally associated BMO spaces, which interpolate with the Lp-scale and admit endpoint inequalities for Calderón-Zygmund operators. Motivated by noncommutative harmonic analysis, this approach gives the first form of Calderón-Zygmund theory for arbitrary von Neumann algebras, but is also valid in classical settings like Riemannian manifolds with nonnegative Ricci curvature or doubling/nondoubling spaces. Other less standard commutative scenarios like fractals or abstract probability spaces are also included. Among our applications in the noncommutative setting, we improve recent results for quantum Euclidean spaces and group von Neumann algebras, respectively linked to noncommutative geometry and geometric group theory.

math.FA↗

arXiv2019

Unifying Entanglement with Uncertainty via Symmetries of Observable Algebras

Strong subadditivity goes beyond the tensored subsystem and commuting operator models. As previously noted by Petz and later by Araki and Moriya, two subalgebras of observables satisfy a generalized SSA-like inequality if they form a commuting square. We explore the interpretation and consequences in finite dimensions, connecting various entropic uncertainty relations for mutually unbiased bases with the positivity of a generalized conditional mutual information (CMI), and with inequalities on relative entropies of coherence and asymmetry. We obtain a bipartite resource theory of operations under which the two subalgebras are respectively invariant and covariant, with CMI as a monotone, and generalized non-classical monotones based on squashed entanglement and entanglement of formation. Free transformations support conversion between entanglement and uncertainty-based configurations, as "EPR <-> 2UCR." Our theory quantifies the common non-classicality in entanglement and uncertainty, implying a strong conceptual link between these fundamentally quantum phenomena.

quant-ph↗

arXiv2018

On the spectral gap of random quantum channels

In this work, we prove a lower bound on the difference between the first and second singular values of quantum channels induced by random isometries, that is tight in the scaling of the number of Kraus operators. This allows us to give an upper bound on the difference between the first and second largest (in modulus) eigenvalues of random channels with same large input and output dimensions for finite number of Kraus operators $k\geq 169$. Moreover, we show that these random quantum channels are quantum expanders, answering a question posed by Hastings. As an application, we show that ground states of infinite 1D spin chains, which are well-approximated by matrix product states, fulfill a principle of maximum entropy.

quant-ph↗

arXiv2018

Generalized $q$-Gaussian von Neumann algebras with coefficients, I. Relative strong solidity

We define $Γ_q(B,S \otimes H)$, the generalized $q$-gaussian von Neumann algebras associated to a sequence of symmetric independent copies $(π_j,B,A,D)$ and to a subset $1 \in S = S^* \subset A$ and, under certain assumptions, prove their strong solidity relative to $B$. We provide many examples of strongly solid generalized $q$-gaussian von Neumann algebras. We also obtain non-isomorphism and non-embedability results about some of these von Neumann algebras.

math.OA↗

arXiv2018

Universal recovery maps and approximate sufficiency of quantum relative entropy

The data processing inequality states that the quantum relative entropy between two states $ρ$ and $σ$ can never increase by applying the same quantum channel $\mathcal{N}$ to both states. This inequality can be strengthened with a remainder term in the form of a distance between $ρ$ and the closest recovered state $(\mathcal{R} \circ \mathcal{N})(ρ)$, where $\mathcal{R}$ is a recovery map with the property that $σ= (\mathcal{R} \circ \mathcal{N})(σ)$. We show the existence of an explicit recovery map that is universal in the sense that it depends only on $σ$ and the quantum channel $\mathcal{N}$ to be reversed. This result gives an alternate, information-theoretic characterization of the conditions for approximate quantum error correction.

quant-ph↗

arXiv2018

Fisher Information and Logarithmic Sobolev Inequality for Matrix Valued Functions

We prove a version of Talagrand's concentration inequality for subordinated sub-Laplacian on a compact Riemannian manifold using tools from noncommutative geometry. As an application, motivated by quantum information theory, we show that on a finite dimensional matrix algebra the set of self-adjoint generators satisfying a tensor stable modified logarithmic Sobolev inequality is dense.

math.FA↗

arXiv2018

On the symmetrized arithmetic-geometric mean inequality for opertors

We study the symmetrized noncommutative arithmetic geometric mean inequality introduced(AGM) by Recht and Ré $$ \|\frac{(n-d)!}{n!}\sum\limits_{ j_1,...,j_d \mbox{ different} }A_{j_{1}}^*A_{j_{2}}^*...A_{j_{d}}^*A_{j_{d}}...A_{j_{2}}A_{j_{1}} \| \leq C(d,n) \|\frac{1}{n} \sum_{j=1}^n A_j^*A_j\|^d .$$ Complementing the results from Recht and Ré, we find upper bounds for C(d,n) under additional assumptions. Moreover, using free probability, we show that $C(d, n) > 1$, thereby disproving the most optimistic conjecture from Recht and Ré.We also prove a deviation result for the symmetrized-AGM inequality which shows that the symmetric inequality almost holds for many classes of random matrices. Finally we apply our results to the incremental gradient method(IGM).

math.OA↗

arXiv2018

Quantum Teleportation and Super-dense Coding in Operator Algebras

Let $\mathcal{B}_d$ be the unital $C^*$-algebra generated by the elements $u_{jk}, \, 0 \le i, j \le d-1$, satisfying the relations that $[u_{j,k}]$ is a unitary operator, and let $C^*(\mathbb{F}_{d^2})$ be the full group $C^*$-algebra of free group of $d^2$ generators. Based on the idea of teleportation and super-dense coding in quantum information theory, we exhibit the two $*$-isomorphisms $M_d(C^*(\mathbb{F}_{d^2}))\cong \mathcal{B}_d\rtimes \mathbb{Z}_d\rtimes \mathbb{Z}_d$ and $M_d(\mathcal{B}_d)\cong C^*(\mathbb{F}_{d^2})\rtimes \mathbb{Z}_d\rtimes \mathbb{Z}_d$, for certain actions of $\mathbb{Z}_d$. As an application, we show that for any $d,m\ge 2$ with $(d,m)\neq (2,2)$, the matrix-valued generalization of the (tensor product) quantum correlation set of $d$ inputs and $m$ outputs is not closed.

math.OA↗

arXiv2017

Capacity Estimates via comparison with TRO channels

A ternary ring of operators (TRO) in finite dimensions is a diagonal sum of spaces of rectangular matrices. TRO as operator space corresponds to quantum channels that are diagonal sums of partial traces, which we call TRO channels. TRO channels admits simple, single-letter capacity formula. Using operator space and complex interpolation techniques, we give perturbative capacities estimates for a wider class of quantum channels by comparison to TRO channels. Our estimates applies mainly for quantum and private capacity and also strong converse rates. The examples includes random unitary from group representations which in general are non-degradable channels.

quant-ph↗

arXiv2017

Heralded Channel Holevo Superadditivity Bounds from Entanglement Monogamy

We show that for a particular class of quantum channels, which we call heralded channels, monogamy of squashed entanglement limits the superadditivity of Holevo capacity. Heralded channels provide a means to understand the quantum erasure channel composed with an arbitrary other quantum channel, as well as common situations in experimental quantum information that involve frequent loss of qubits or failure of trials. We also show how entanglement monogamy applies to non-classicality in quantum games, and we consider how faithful, monogamous entanglement measures may bound other entanglement-dependent quantities in many-party scenarios.

quant-ph↗

arXiv2017

Harmonic analysis approach to Gromov--Hausdorff convergence for noncommutative tori

We show that the rotation algebras are limit of matrix algebras in a very strong sense of convergence for algebras with additional Lipschitz structure. Our results generalize to higher dimensional noncommutative tori and operator valued coefficients. In contrast to previous results by Rieffel, Li, Kerr, and Latrémolière we use Lipschitz norms induced by the `carré du champ' of certain natural dynamical systems, including the heat semigroup.

math.OA↗

arXiv2017

Classical vs. quantum communication in XOR games

In this work we introduce an intermediate setting between quantum nonlocality and communication complexity problems. More precisely, we study the value of XOR games $G$ when Alice and Bob are allowed to use a limited amount of one-way classical communication $ω_{o.w.-c}(G)$ (resp. one-way quantum communication $ω_{o.w.-c}^*(G)$), where $c$ denotes the number of bits (resp. qubits). The key quantity here is the quotient $ω_{o.w.-c}^*(G)/ω_{o.w.-c}(G)$. We provide a universal way to obtain Bell inequality violations of general Bell functionals from XOR games for which the quotient $ω_{o.w.-c}^*(G)/ω_{o.w.-2c}(G)$ is larger than 1. This allows, in particular, to find (unbounded) Bell inequality violations from communication complexity problems in the same spirit as the recent work by Buhrman et al. (2016). We also provide an example of a XOR game for which the previous quotient is optimal (up to a logarithmic factor) in terms of the amount of information $c$. Interestingly, this game has only polynomially many inputs per player. For the related problem of separating the classical vs quantum communication complexity of a function, the known examples attaining exponential separation require exponentially many inputs per party.

quant-ph↗

arXiv2017

Generalized notions of sparsity and restricted isometry property. Part I: A unified framework

The restricted isometry property (RIP) is an integral tool in the analysis of various inverse problems with sparsity models. Motivated by the applications of compressed sensing and dimensionality reduction of low-rank tensors, we propose generalized notions of sparsity and provide a unified framework for the corresponding RIP, in particular when combined with isotropic group actions. Our results extend an approach by Rudelson and Vershynin to a much broader context including commutative and noncommutative function spaces. Moreover, our Banach space notion of sparsity applies to affine group actions. The generalized approach in particular applies to high order tensor products.

stat.ML↗

arXiv2017

Generalized notions of sparsity and restricted isometry property. Part II: Applications

The restricted isometry property (RIP) is a universal tool for data recovery. We explore the implication of the RIP in the framework of generalized sparsity and group measurements introduced in the Part I paper. It turns out that for a given measurement instrument the number of measurements for RIP can be improved by optimizing over families of Banach spaces. Second, we investigate the preservation of difference of two sparse vectors, which is not trivial in generalized models. Third, we extend the RIP of partial Fourier measurements at optimal scaling of number of measurements with random sign to far more general group structured measurements. Lastly, we also obtain RIP in infinite dimension in the context of Fourier measurement concepts with sparsity naturally replaced by smoothness assumptions.

stat.ML↗

arXiv2017

Singular integrals in quantum Euclidean spaces

In this paper, we establish the core of singular integral theory and pseudodifferential calculus over the archetypal algebras of noncommutative geometry: quantum forms of Euclidean spaces and tori. Our results go beyond Connes' pseudodifferential calculus for rotation algebras, thanks to a new form of Calderón-Zygmund theory over these spaces which crucially incorporates nonconvolution kernels. We deduce $L_p$-boundedness and Sobolev $p$-estimates for regular, exotic and forbidden symbols in the expected ranks. In the $L_2$ level both Calderón-Vaillancourt and Bourdaud theorems for exotic and forbidden symbols are also generalized to the quantum setting. As a basic application of our methods, we prove $L_p$-regularity of solutions for elliptic PDEs.

math.OA↗