arXiv subjects

Hemant K. Mishra

Publications and source records attributed to Hemant K. Mishra.

At least 19 recordsLinked to original sources

arXiv2026

Derivatives of symplectic spectral functions

A symplectic spectral function of a $2n \times 2n$ real positive definite matrix $A$ is a function of its symplectic eigenvalues $0< d_1(A) \leq \cdots \leq d_n(A)$, which is given by a composition $f \circ d$ of some symmetric function $f$ on the set of $n$-vectors with positive entries and the symplectic eigenvalue vector map $d(A)=(d_1(A),\ldots, d_n(A))$. In this work, we rigorously study various types of differentiability and Clarke generalized gradient of symplectic spectral functions, and also provide several applications of our findings. We show that the symplectic spectral function $f \circ d$ is Fréchet differentiable at $A$ if and only if $f$ is Fréchet differentiable at $d(A)$, and we compute the derivative expression explicitly. We also show that $f \circ d$ is strictly Fréchet differentiable (respectively, continuously Gâteaux differentiable) at $A$ if and only if $f$ is strictly Fréchet differentiable (respectively, continuously Gâteaux differentiable) at $d(A)$. We determine the Clarke generalized gradient of a symplectic spectral function $f \circ d$ at $A$, which is given in terms of the Clarke generalized gradient of $f$ at $d(A)$. As an application of our work, we show that the purity, von Neumann entropy, and Rényi entropy of a bosonic faithful Gaussian state are Fréchet differentiable functions of the covariance matrix of the Gaussian state. We also provide explicit expressions of their Fréchet derivatives.

math.FA↗

arXiv2026

Lidskii theorem for directional derivatives of symplectic eigenvalues

In this paper, we present a refinement of the symplectic Lidskii theorem for directional derivatives of symplectic eigenvalues, which is inspired by the work of Sendov [Electron. J. Linear Algebra 41(2025), 338-341]. We show that for a $2n \times 2n$ real positive definite matrix $A$ and symmetric matrices $H, K$ \begin{align*} d'(A;H+K)-d'(A;H) \prec_{π_A} d'(A;K). \end{align*} Here $\prec_{π_A}$ represents block-majorization corresponding to the partition $π_A$ of $\{1,\ldots, n\}$ given by the equality blocks of symplectic eigenvalues of $A$. We know that Hermitianization of a symmetric matrix occurs in the directional derivative expression of symplectic eigenvalues. We establish a componentwise inequality comparing the symplectic eigenvalues of a positive definite matrix with the ordinary eigenvalues of its associated Hermitianization, and also characterize the equality case. By using the aforementioned inequality, we then show that the symplectic Lidskii theorem follows from the directional derivative Lidskii theorem.

math.FA↗

arXiv2026

On generalization of Williamson's theorem to real symmetric matrices

Williamson's theorem states that if $A$ is a $2n \times 2n$ real symmetric positive definite matrix then there exists a $2n \times 2n$ real symplectic matrix $M$ such that $M^T A M=D \oplus D$, where $D$ is an $n \times n$ diagonal matrix with positive diagonal entries known as the symplectic eigenvalues of $A$. The theorem is known to be generalized to $2n \times 2n$ real symmetric positive semidefinite matrices whose kernels are symplectic subspaces of $\mathbb{R}^{2n}$, in which case, some of the diagonal entries of $D$ are allowed to be zero. In this paper, we further generalize Williamson's theorem to $2n \times 2n$ real symmetric matrices by allowing the diagonal elements of $D$ to be any real numbers, and thus extending the notion of symplectic eigenvalues to real symmetric matrices. Also, we provide an explicit description of symplectic eigenvalues, construct symplectic matrices achieving Williamson's theorem type decomposition, and establish perturbation bounds on symplectic eigenvalues for a class of $2n \times 2n$ real symmetric matrices denoted by $\operatorname{EigSpSm}(2n)$. Our perturbation bounds on symplectic eigenvalues for $\operatorname{EigSpSm}(2n)$ generalize known perturbation bounds on symplectic eigenvalues of positive definite matrices given by Bhatia and Jain \textit{[J. Math. Phys. 56, 112201 (2015)]}.

math.FA↗

arXiv2026

Barycentric bounds on the error exponents of quantum hypothesis exclusion

Quantum state exclusion is an operational task with application to ontological interpretations of quantum states. In such a task, one is given a system whose state is randomly selected from a finite set, and the goal is to identify a state from the set that is not the true state of the system. An error occurs if and only if the state identified is the true state. In this paper, we study the optimal error probability of quantum state exclusion and its error exponent from an information-theoretic perspective. Our main finding is a single-letter upper bound on the error exponent of state exclusion given by the multivariate log-Euclidean Chernoff divergence, and we prove that this improves upon the best previously known upper bound. We also extend our analysis to quantum channel exclusion, and we establish a single-letter and efficiently computable upper bound on its error exponent, admitting the use of adaptive strategies. We derive both upper bounds, for state and channel exclusion, based on one-shot analysis and formulate them as a type of multivariate divergence measure called a barycentric Chernoff divergence. Moreover, our result on channel exclusion has implications in two important special cases. First, when there are two hypotheses, our result provides the first known efficiently computable upper bound on the error exponent of symmetric binary channel discrimination. Second, when all channels are classical, we show that our upper bound is achievable by a parallel strategy, thus solving the exact error exponent of classical channel exclusion.

quant-ph↗

arXiv2026

Simultaneous symplectic spectral decomposition of positive semidefinite matrices

We establish necessary and sufficient conditions on simultaneous symplectic spectral decomposition of a family of $2n \times 2n$ real positive semidefinite matrices with symplectic kernels. We also provide a precise algebraic condition on a $2n \times 2n$ real positive semidefinite matrix with symplectic kernel for orthosymplectic spectral diagonalization, which generalizes a known result for positive definite matrices.

math-ph↗

arXiv2026

Majorization between symplectic spectra of positive semidefinite matrices

Given $2n \times 2n$ real symmetric positive semidefinite matrix $A$ with symplectic kernel, there exists a real $2n \times 2n$ \emph{symplectic matrix} $M$ such that $M^TAM= D \oplus D$, where $D$ is an $n \times n$ non-negative diagonal matrix which is unique up to permutation of its diagonal entries. The diagonal entries of $D$ are called the \emph{symplectic eigenvalues} or symplectic spectrum of $A$. In this work, we investigate some majorization and weak supermajorization relations between the symplectic spectra of two positive semidefinite matrices. More explicitly, suppose $A$ and $B$ are $2n \times 2n$ real symmetric positive semidefinite matrices with symplectic kernels. We show that if the symplectic spectrum of $A$ is majorized by the symplectic spectrum of $B$, then $A$ lies in the convex hull of the symplectic orbit of $B$. We also establish that only a weak converse of this statement holds; i.e., if $A$ lies in the convex hull of the symplectic orbit of $B$ then the symplectic spectrum of $A$ is \emph{weakly supermajorized} by the symplectic spectrum of $B$. Several consequences of our results are also presented. Our methods make use of well-known connections between the theory of majorization, doubly stochastic, doubly superstochastic, and symplectic matrices.

math.FA↗

arXiv2025

Converse bounds for quantum hypothesis exclusion: A divergence-radius approach

Hypothesis exclusion is an information-theoretic task in which an experimenter aims at ruling out a false hypothesis from a finite set of known candidates, and an error occurs if and only if the hypothesis being ruled out is the ground truth. For the tasks of quantum state exclusion and quantum channel exclusion -- where hypotheses are represented by quantum states and quantum channels, respectively -- efficiently computable upper bounds on the asymptotic error exponents were established in a recent work of the current authors [Ji et al., arXiv:2407.13728 (2024)], where the derivation was based on nonasymptotic analysis. In this companion paper of our previous work, we provide alternative proofs for the same upper bounds on the asymptotic error exponents of quantum state and channel exclusion, but using a conceptually different approach from the one adopted in the previous work. Specifically, we apply strong converse results for asymmetric binary hypothesis testing to distinguishing an arbitrary ``dummy'' hypothesis from each of the concerned candidates. This leads to the desired upper bounds in terms of divergence radii via a geometrically inspired argument.

quant-ph↗

arXiv2025

Multivariate Fidelities

The main contribution of our paper is to introduce a number of multivariate quantum fidelities and show that they satisfy several desirable properties that are natural extensions of those of the Uhlmann and Holevo fidelities. We propose three variants that reduce to the average pairwise fidelity for commuting states: average pairwise $z$-fidelities, the multivariate semi-definite programming (SDP) fidelity, and a multivariate fidelity inspired by an existing secrecy measure. The second one is obtained by extending the SDP formulation of the Uhlmann fidelity to more than two states. All three of these variants satisfy the following properties: (i) reduction to multivariate classical fidelities for commuting states, (ii) the data-processing inequality, (iii) invariance under permutations of the states, (iv) its values are in the interval $[0,1]$; they are faithful, that is, their values are equal to one if and only if all the states are equal, and they satisfy orthogonality, that is their values are equal to zero if and only if the states are mutually orthogonal to each other, (v) direct-sum property, (vi) joint concavity, and (vii) uniform continuity bounds under certain conditions. Furthermore, we establish inequalities relating these different variants, indeed clarifying that all these definitions coincide with the average pairwise fidelity for commuting states. Lastly, we introduce another multivariate fidelity called multivariate log-Euclidean fidelity, which is a quantum generalization of the Matusita multivariate fidelity. We also show that it satisfies most of the desirable properties listed above, it is a function of a multivariate log-Euclidean divergence, and has an operational interpretation in terms of quantum hypothesis testing with an arbitrarily varying null hypothesis.

quant-ph↗

arXiv2025

Near-optimal performance of square-root measurement for general score functions and quantum ensembles

The Barnum-Knill theorem states that the optimal success probability in the multiple state discrimination task is not more than the square root of the success probability when the pretty good or square-root measurement is used for this task. An assumption of the theorem is that the underlying ensemble consists of finitely many quantum states over a finite-dimensional quantum system. Motivated in part by the fact that the success probability is not a relevant metric for continuous ensembles, in this paper we provide a generalization of the notion of pretty good measurement and the Barnum-Knill theorem for general quantum ensembles, including those described by a continuous parameter space and an infinite-dimensional Hilbert space. To achieve this, we also design a general metric of performance for quantum measurements that generalizes the success probability, namely, the expected gain of the measurement with respect to a positive score function. A notable consequence of the main result is that, in a Bayesian estimation task, the mean square error of the generalized pretty good measurement does not exceed twice the optimal mean square error.

quant-ph↗

arXiv2025

A Complete Characterization of Passive Unitary Normalizable (PUN) Gaussian States

We provide a complete characterization of the class of multimode quantum Gaussian states that can be reduced to a tensor product of thermal states using only a passive unitary operator. We call these states \textit{passive unitary normalizable} (PUN) Gaussian states. The characterization of PUN Gaussian states is given in three different ways: $(i)$ in terms of their covariance matrices, $(ii)$ using gauge-invariance (a special class of Glauber--Sudarshan $p$-functions), and $(iii)$ with respect to the recently obtained $(A,Λ)$ parametrization of Gaussian states in [J. Math. Phys. 62, 022102 (2021)]. In terms of the covariance matrix, our characterization states that an $n$-mode quantum Gaussian state is PUN if and only if its $2n\times 2n$ quantum covariance matrix $S$ commutes with the standard symplectic matrix $J$. It is well-known that the so-called gauge-invariant Gaussian states are PUN, but whether the converse is true is not known in the literature to the best of our knowledge. We establish the converse in affirmation. Lastly, in terms of the $(A,Λ)$-parameterization, we show that a Gaussian state with parameters $(A,Λ)$ is PUN if and only if $A=0$.

quant-ph↗

arXiv2024

Equality in some symplectic eigenvalue inequalities

In the last decade, numerous works have investigated several properties of symplectic eigenvalues. Remarkably, the results on symplectic eigenvalues have been found to be analogous to those of eigenvalues of Hermitian matrices with appropriate interpretations. In particular, symplectic analogs of famous eigenvalue inequalities are known today such as Weyl's inequalities, Lidskii's inequalities, and Schur--Horn majorization inequalities. In this paper, we provide necessary and sufficient conditions for equality in the symplectic analogs of the aforementioned inequalities. The equality conditions for the symplectic Weyl's and Lidskii's inequalities turn out to be analogous to the known equality conditions for eigenvalues.

math.FA↗

arXiv2024

Majorization in some symplectic weak supermajorizations

Symplectic eigenvalues are known to satisfy analogs of several classic eigenvalue inequalities. Of these is a set of weak supermajorization relations concerning symplectic eigenvalues that are weaker analogs of some majorization relations corresponding to eigenvalues. The aim of this letter is to establish necessary and sufficient conditions for the saturation of the symplectic weak supermajorization relations by majorization.

math.FA↗

arXiv2024

On the optimal error exponents for classical and quantum antidistinguishability

The concept of antidistinguishability of quantum states has been studied to investigate foundational questions in quantum mechanics. It is also called quantum state elimination, because the goal of such a protocol is to guess which state, among finitely many chosen at random, the system is not prepared in (that is, it can be thought of as the first step in a process of elimination). Antidistinguishability has been used to investigate the reality of quantum states, ruling out $ψ$-epistemic ontological models of quantum mechanics [Pusey et al., Nat. Phys., 8(6):475-478, 2012]. Thus, due to the established importance of antidistinguishability in quantum mechanics, exploring it further is warranted. In this paper, we provide a comprehensive study of the optimal error exponent -- the rate at which the optimal error probability vanishes to zero asymptotically -- for classical and quantum antidistinguishability. We derive an exact expression for the optimal error exponent in the classical case and show that it is given by the multivariate classical Chernoff divergence. Our work thus provides this divergence with a meaningful operational interpretation as the optimal error exponent for antidistinguishing a set of probability measures. For the quantum case, we provide several bounds on the optimal error exponent: a lower bound given by the best pairwise Chernoff divergence of the states, a single-letter semi-definite programming upper bound, and lower and upper bounds in terms of minimal and maximal multivariate quantum Chernoff divergences. It remains an open problem to obtain an explicit expression for the optimal error exponent for quantum antidistinguishability.

quant-ph↗

arXiv2024

Orthosymplectic diagonalization in Williamson's theorem

In this paper, we provide an algebraic condition on any $2n \times 2n$ real symmetric positive definite matrix which is necessary and sufficient for the matrix to be diagonalized by an orthosymplectic matrix in the sense of Williamson's theorem.

math.FA↗

arXiv2023

Block perturbation of symplectic matrices in Williamson's theorem

Williamson's theorem states that for any $2n \times 2n$ real positive definite matrix $A$, there exists a $2n \times 2n$ real symplectic matrix $S$ such that $S^TAS=D \oplus D$, where $D$ is an $n\times n$ diagonal matrix with positive diagonal entries which are known as the symplectic eigenvalues of $A$. Let $H$ be any $2n \times 2n$ real symmetric matrix such that the perturbed matrix $A+H$ is also positive definite. In this paper, we show that any symplectic matrix $\tilde{S}$ diagonalizing $A+H$ in Williamson's theorem is of the form $\tilde{S}=S Q+\mathcal{O}(\|H\|)$, where $Q$ is a $2n \times 2n$ real symplectic as well as orthogonal matrix. Moreover, $Q$ is in $\textit{symplectic block diagonal}$ form with the block sizes given by twice the multiplicities of the symplectic eigenvalues of $A$. Consequently, we show that $\tilde{S}$ and $S$ can be chosen so that $\|\tilde{S}-S\|=\mathcal{O}(\|H\|)$. Our results hold even if $A$ has repeated symplectic eigenvalues. This generalizes the stability result of symplectic matrices for non-repeated symplectic eigenvalues given by Idel, Gaona, and Wolf [$\textit{Linear Algebra Appl., 525:45-58, 2017}$].

math.FA↗

arXiv2023

Pretty good measurement for bosonic Gaussian ensembles

The pretty good measurement is a fundamental analytical tool in quantum information theory, giving a method for inferring the classical label that identifies a quantum state chosen probabilistically from an ensemble. Identifying and constructing the pretty good measurement for the class of bosonic Gaussian states is of immediate practical relevance in quantum information processing tasks. Holevo recently showed that the pretty good measurement for a bosonic Gaussian ensemble is a bosonic Gaussian measurement that attains the accessible information of the ensemble (IEEE Trans. Inf. Theory, 66(9):5634-564, 2020). In this paper, we provide an alternate proof of Gaussianity of the pretty good measurement for a Gaussian ensemble of multimode bosonic states, with a focus on establishing an explicit and efficiently computable Gaussian description of the measurement. We also compute an explicit form of the mean square error of the pretty good measurement, which is relevant when using it for parameter estimation. Generalizing the pretty good measurement is a quantum instrument, called the pretty good instrument. We prove that the post-measurement state of the pretty good instrument is a faithful Gaussian state if the input state is a faithful Gaussian state whose covariance matrix satisfies a certain condition. Combined with our previous finding for the pretty good measurement and provided that the same condition holds, it follows that the expected output state is a faithful Gaussian state as well. In this case, we compute an explicit Gaussian description of the post-measurement and expected output states. Our findings imply that the pretty good instrument for bosonic Gaussian ensembles is no longer merely an analytical tool, but that it can also be implemented experimentally in quantum optics laboratories.

quant-ph↗

arXiv2023

Optimal input states for quantifying the performance of continuous-variable unidirectional and bidirectional teleportation

Continuous-variable (CV) teleportation is a foundational protocol in quantum information science. A number of experiments have been designed to simulate ideal teleportation under realistic conditions. In this paper, we detail an analytical approach for determining optimal input states for quantifying the performance of CV unidirectional and bidirectional teleportation. The metric that we consider for quantifying performance is the energy-constrained channel fidelity between ideal teleportation and its experimental implementation, and along with this, our focus is on determining optimal input states for distinguishing the ideal process from the experimental one. We prove that, under certain energy constraints, the optimal input state in unidirectional, as well as bidirectional, teleportation is a finite entangled superposition of twin-Fock states saturating the energy constraint. Moreover, we also prove that, under the same constraints, the optimal states are unique; that is, there is no other optimal finite entangled superposition of twin-Fock states.

quant-ph↗

arXiv2020

First order sensitivity analysis of symplectic eigenvalues

For every $2n \times 2n$ positive definite matrix $A$ there are $n$ positive numbers $d_1(A) \leq \ldots \leq d_n(A)$ associated with $A$ called the symplectic eigenvalues of $A.$ It is known that $d_m$ are continuous functions of $A$ but are not differentiable in general. In this paper, we show that the directional derivative of $d_m$ exists and derive its expression. We also discuss various subdifferential properties of $d_m$ such as Clarke and Michel-Penot subdifferentials.

math.FA↗