arXiv subjects

Tiju Cherian John

Publications and source records attributed to Tiju Cherian John.

12 recordsLinked to original sources

arXiv2025

Quantum-enhanced quickest change detection of transmission loss

Augmenting a train of bright phase-modulated laser-light pulses of a coherent communications system with infinitesimally small quantum photons per pulse -- entangled across several time bins -- prepared by splitting squeezed light in a temporal-mode interferometer can dramatically enhance a homodyne receiver's ability to detect a sudden change in the channel loss, by up to a factor that is the inverse of the pre-change loss, without affecting the communications rate. We discuss the quantum limit of quickest change detection, and the problem of joint communications and change detection that our study opens up.

quant-ph↗

arXiv2025

What is a Gaussian channel, and when is it physically implementable using a multiport interferometer?

Quantum Gaussian channels are fundamental models for communication and information processing in continuous-variable quantum systems. This work addresses both foundational aspects and physical implementation pathways for these channels. Firstly, we provide a rigorous, unified framework by formally proving the equivalence of three principal definitions of quantum Gaussian channels prevalent in the literature, consolidating theoretical understanding. Secondly, we investigate the physical realization of these channels using multiport interferometers, a key platform in quantum optics. The central research contribution is a precise characterization of the channel parameters that correspond to Gaussian channels physically implementable via linear optical multiport interferometers. This characterization bridges the abstract mathematical description with concrete physical architectures. Along the way, we also resolve some questions posed by Parthasarathy (Indian J. Pure Appl. Math. 46, (2015)).

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↗

arXiv2025

Fundamental Limits Of Quickest Change-point Detection With Continuous-Variable Quantum States

We generalize the quantum CUSUM (QUSUM) algorithm for quickest change-point detection, analyzed in finite dimensions by Fanizza, Hirche, and Calsamiglia (Phys. Rev. Lett. 131, 020602, 2023), to infinite-dimensional quantum systems. Our analysis relies on a novel generalization of a result by Hayashi (Hayashi, J. Phys. A: Math. Gen. 34, 3413, 2001) concerning the asymptotics of quantum relative entropy, which we establish for the infinite-dimensional setting. This enables us to prove that the QUSUM strategy retains its asymptotic optimality, characterized by the relationship between the expected detection delay and the average false alarm time for any pair of states with finite relative entropy. Consequently, our findings apply broadly, including continuous-variable systems (e.g., Gaussian states), facilitating the development of optimal change-point detection schemes in quantum optics and other physical platforms, and rendering experimental verification feasible.

quant-ph↗

arXiv2024

An Order Relation between Eigenvalues and Symplectic Eigenvalues of a Class of Infinite-Dimensional Operators

In this article, we obtain some results in the direction of ``infinite dimensional symplectic spectral theory". We prove an inequality between the eigenvalues and symplectic eigenvalues of a special class of infinite dimensional operators. Let $T$ be an operator such that $T - αI$ is compact for some $α> 0$. Denote by $\{{λ_j^R}^\downarrow(T)\}$, the set of eigenvalues of $T$ lying strictly to the right side of $α$ arranged in the decreasing order and let $\{{λ_j^L}^\uparrow(T)\}$ denote the set of eigenvalues of $T$ lying strictly to the left side of $α$ arranged in the increasing order. Furthermore, let $\{{d_j^R}^\downarrow(T)\}$ denote the symplectic eigenvalues of $T$ lying strictly to the right of $α$ arranged in decreasing order and $\{{d_j^L}^\uparrow(T)\}$ denote the set of symplectic eigenvalues of $T$ lying strictly to the left of $α$ arranged in increasing order, respectively (such an arrangement is possible as it will be shown that the only possible accumulation point for the symplectic eigenvalues is $α$). Then we show that $${d_j^R}^\downarrow(T) \leq {λ_j^R}^\downarrow(T), \quad j = 1,2, \cdots, s_r$$ and $${λ_j^L}^\uparrow(T) \leq {d_j^L}^\uparrow(T), \quad j = 1,2, \cdots, s_l,$$ where $s_r$ and $s_l$ denote the number of symplectic eigenvalues of $T$ strictly to the right and left of $α$, respectively. This generalizes a finite dimensional result obtained by Bhatia and Jain (J. Math. Phys. 56, 112201 (2015)). The class of Gaussian Covariance Operators (GCO) and positive Absolutely Norm attaining Operators ($(\mathcal{A}\mathcal{N})_+$ operators) appear as special cases of the set of operators we consider.

math.SP↗

arXiv2024

Petz-Rényi Relative Entropy of Thermal States and their Displacements

In this article, we obtain the precise range of the values of the parameter $α$ such that Petz-Rényi $α$-relative entropy $D_α(ρ||σ)$ of two displaced thermal states is finite. More precisely, we prove that, given two displaced thermal states $ρ$ and $σ$ with inverse temperature parameters $r_1, r_2,\dots, r_n$ and $s_1,s_2, \dots, s_n$, respectively, we have \[ D_α(ρ||σ)<\infty \Leftrightarrow α< \min \left\{ \frac{s_j}{s_j-r_j}: j \in \{ 1, \ldots , n \} \text{ such that } r_j<s_j \right\}, \] where we adopt the convention that the minimum of an empty set is equal to infinity. Along the way, we prove a special case of a conjecture of Seshdreesan, Lami and Wilde (J. Math. Phys. 59, 072204 (2018)).

quant-ph↗

arXiv2023

Quantum $f$-divergences via Nussbaum-Szkoła Distributions and Applications to $f$-divergence Inequalities

The main result in this article shows that the quantum $f$-divergence of two states is equal to the classical $f$-divergence of the corresponding Nussbaum-Szkoła distributions. This provides a general framework for studying certain properties of quantum entropic quantities using the corresponding classical entities. The usefulness of the main result is illustrated by obtaining several quantum $f$-divergence inequalities from their classical counterparts. All results presented here are valid in both finite and infinite dimensions and hence can be applied to continuous variable systems as well. A comprehensive review of the instances in the literature where Nussbaum-Szkoła distributions are used, is also provided in this article.

quant-ph↗

arXiv2023

Relative Entropy via Distribution of Observables

We obtain formulas for Petz-Rényi and Umegaki relative entropy from the idea of distribution of a positive selfadjoint operator. Classical results on Rényi and Kullback-Leibler divergences are applied to obtain new results and new proofs for some known results about Petz-Rényi and Umegaki relative entropy. Most important among these, is a necessary and sufficient condition for the finiteness of the Petz-Rényi $α$-relative entropy. All of the results presented here are valid in both finite and infinite dimensions. In particular, these results are valid for states in Fock spaces and thus are applicable to continuous variable quantum information theory.

quant-ph↗

arXiv2023

Thermal States on Mittag-Leffler Fock Space of the Slitted Plane

Number states and thermal states form an important class of physical states in quantum theory. A mathematical framework for studying these states is that of a Fock space over an appropriate Hilbert space. Several generalizations of the usual Bosonic Fock space have appeared recently due to their importance in many areas of mathematics and other scientific domains. One of the most prominent generalization of Fock spaces is the Mittag-Leffler (ML) Fock space of the slitted plane. Natural generalizations of the basic operators of quantum theory can be obtained on ML Fock spaces. Following the construction of the creation and annihilation operators in the Mittag-Leffler Fock space of the slitted plane by Rosenfeld, Russo, and Dixon, (J. Math. Anal. Appl. 463, 2, 2018). We construct and study the number states and thermal states on the ML Fock space of the slitted plane. Thermal states on usual Fock space form an important subclass of the so called quantum gaussian states, an analogous theory of more general quantum states (like squeezed states and Bell states) on ML Fock spaces is an area open for further exploration.

math.CV↗

arXiv2021

A Common Parametrization for Finite Mode Gaussian States, their Symmetries and associated Contractions with some Applications

Let $Γ(\mathcal{H})$ be the boson Fock space over a finite dimensional Hilbert space $\mathcal{H}$. It is shown that every gaussian symmetry admits a Klauder-Bargmann integral representation in terms of coherent states. Furthermore, gaussian symmetries, gaussian states and second quantization contractions, all of these operators belong to a weakly closed, selfadjoint semigroup $\mathcal{E}_2(\mathcal{H})$ of bounded operators in $Γ(\mathcal{H})$. This yields, a new parametrization of gaussian states, which is a very fruitful alternative to the customary parametrization by position-momentum mean vectors and covariance matrices. This leads to a rich harvest of corollaries: (i) every gaussian state $ρ$ admits a factorization $ρ= Z_{1}^{\dagger}Z_{1}$, where $Z_{1}$ is an element of $\mathcal{E}_2(\mathcal{H})$ and has the form $Z_{1} = \sqrt{c}Γ(\sqrtΛ)\exp{\sum_{r=1}^{n} λ_ra_r+\sum_{r,s=1}^{n} α_{rs}a_{r}a_{s}}$ on the dense linear manifold generated by all exponential vectors, $Λ$ being a positive operator in $\mathcal{H}$, $a_{r}, 1\leq r \leq n$ are the annihilation operators corresponding to the $n$ different modes in $Γ(\mathcal{H})$, $λ_r\in \mathbb{C}$ and $[α_{rs}]$ is a symmetric matrix in $M_n(\mathbb{C})$; (ii) an explicit particle basis expansion of an arbitrary mean zero pure gaussian state vector along with a density matrix formula for a general gaussian state in terms of its $\mathcal{E}_2(\mathcal{H})$-parameters; (iii) an easy test for the entanglement of pure gaussian states and a class of examples of pure $n$-mode gaussian states which are completely entangled; (iv) tomography of an unknown gaussian state in $Γ(\mathbb{C}^n)$ by the estimation of its $\mathcal{E}_2(\mathbb{C}^n)$-parameters using $O(n^2)$ measurements with a finite number of outcomes.

quant-ph↗

arXiv2019

Infinite Mode Quantum Gaussian States

Quantum Gaussian states on Bosonic Fock spaces are quantum versions of Gaussian distributions. In this paper, we explore infinite mode quantum Gaussian states. We extend many of the results of Parthasarathy in \cite{Par10} and \cite{Par13} to the infinite mode case, which includes various characterizations, convexity and symmetry properties.

quant-ph↗

arXiv2019

Real Normal Operators and Williamson's Normal Form

A simple proof is provided to show that any bounded normal operator on a real Hilbert space is orthogonally equivalent to its transpose(adjoint). A structure theorem for invertible skew-symmetric operators, which is analogous to the finite dimensional situation is also proved using elementary techniques. The second result is used to establish the main theorem of this article, which is a generalization of Williamson's normal form for bounded positive operators on infinite dimensional separable Hilbert spaces. This has applications in the study of infinite mode Gaussian states.

math.SP↗