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K. R. Parthasarathy

Publications and source records attributed to K. R. Parthasarathy.

At least 19 recordsLinked to original sources
arXiv2022

Twisted convolution quantum information channels, one-parameter semigroups and their generators

Using the tool of quantum characteristic functions of n-mode states in the boson Fock space Γ(C_n) we construct a semigroup of quantum information channels. This leads to a special class of one-parameter semigroups of such channels. These semigroups are concrete but their generators have unbounded operator coefficients. These one-parameter semigroups are also quantum dynamical semigroups and the form of the generators involve additional features which do not appear in the standard GKSL form. A heuristic discussion of the form of these generators is included. In the wake of this analysis many open problems arise naturally.

quant-ph↗
arXiv2021

Optimal quantum tomography with constrained elementary measurements arising from unitary bases

The purpose of this paper is to introduce techniques of obtaining optimal ways to determine a d-level quantum state or distinguish such states. It entails designing constrained elementary measurements extracted from maximal abelian subsets of a unitary basis U for the operator algebra B(H) of a Hilbert space H of finite dimension d > 3 or, after choosing an orthonormal basis for H, for the *-algebra Md of complex matrices of order d > 3. Illustrations are given for the techniques. It is shown that the Schwinger basis U of unitary operators can give for d, a product of primes p and a, the ideal number d^2 of rank one projectors that have a few quantum mechanical overlaps (or, for that matter, a few angles between the corresponding unit vectors). We also give a combination of the tensor product and constrained elementary measurement techniques to deal with all d. A comparison is drawn for different forms of unitary bases for the Hilbert space and also for different Hilbert space factors of the tensor product. In the process we also study the equivalence relation on unitary bases defined by R. F. Werner [J. Phys. A: Math. Gen. 34 (2001) 7081], connect it to local operations on maximally entangled vectors bases, find an invariant for equivalence classes in terms of certain commuting systems, called fan representations, and, relate it to mutually unbiased bases and Hadamard matrices. Illustrations are given in the context of latin squares and projective representations as well.

quant-ph↗
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↗
arXiv2018

From CCR to Levy Processes: An Excursion in Quantum Probability

This is an expositary article telling a short story made from the leaves of quantum probability with the following ingredients: (i) A special projective, unitary, irreducible and factorizable representation of the euclidean group of a Hilbert space known as the Weyl representation. \item The infinitesimal version of the Weyl representation includes the Heisenberg canonical commutation relations (CCR) of quantum theory. It also yields the three fundamental operator fields known as the creation, conservation and annihilation fields. (ii) The three fundamental fields, with the inclusion of time, lead to quantum stochastic integration and a calculus with an Ito's formula for products of differentials. (iii) Appropriate linear combinations of the fundamental operator processes yield all the L{é}vy processes of classical probability theory along with the bonus of Ito's formula for products of their differentials.

quant-ph↗
arXiv2017

Asymptotic spectral stability of the Gisin-Percival state diffusion

Starting from the Gisin-Percival state diffusion equation for the pure state trajectory of a composite bipartite quantum system and exploiting the purification of a mixed state via its Schmidt decomposition, we write the diffusion equation for the quantum trajectory of the mixed state of a subsystem $S$ of the bipartite system, when the initial state in $S$ is mixed. Denoting the diffused state of the system $S$ at time $t$ by $ρ_t(\mathbf{B})$ for each $t\geq 0$, where $\mathbf{B}$ is the underlying complex $n$-dimensional vector-valued Brownian motion process and using It{ô} calculus, along with an induction procedure, we arrive at the stochastic differential of the scalar-valued moment process ${\rm Tr}[ρ_t^m( \mathbf{B})], \,\,\, m=2,3,\ldots$ in terms of $d\,\mathbf{B}$ and $d\,t$. This shows that each of the processes $\{{\rm Tr}[ρ_t^m( \mathbf{B})], t\geq 0\}$ admits a Doob-Meyer decomposition as the sum of a martingale $M^{(m)}_t(\mathbf{B})$ and a non-negative increasing process $S^{(m)}_t(\mathbf{B})$. This ensures the existence of $\underset{t\rightarrow\infty}{\lim}\, {\rm Tr}[ρ_t^m( \mathbf{B})]$ almost surely with respect to the Wiener probability measure $μ$ of the Brownian motion $\mathbf{B}$, for each $m=2,\, 3,\, \ldots$. In particular, when $S$ is a finite level system, the spectrum and therefore the entropy of $ρ_t (\mathbf{B})$ converge almost surely to a limit as $t\rightarrow \infty$. In the Appendix, by employing probabilistic means, we prove a technical result which implies the almost sure convergence of the spectrum for countably infinite level systems.

quant-ph↗
arXiv2017

From quantum stochastic differential equations to Gisin-Percival state diffusion

Starting from the quantum stochastic differential equations of Hudson and Parthasarathy (Comm. Math. Phys. 93, 301 (1984)) and exploiting the Wiener-Ito-Segal isomorphism between the Boson Fock reservoir space $Γ(L^2(\mathbb{R}_+)\otimes (\mathbb{C}^{n}\oplus \mathbb{C}^{n}))$ and the Hilbert space $L^2(μ)$, where $μ$ is the Wiener probability measure of a complex $n$-dimensional vector-valued standard Brownian motion $\{\mathbf{B}(t), t\geq 0\}$, we derive a non-linear stochastic Schrodinger equation describing a classical diffusion of states of a quantum system, driven by the Brownian motion $\mathbf{B}$. Changing this Brownian motion by an appropriate Girsanov transformation, we arrive at the Gisin-Percival state diffusion equation (J. Phys. A, 167, 315 (1992)). This approach also yields an explicit solution of the Gisin-Percival equation, in terms of the Hudson-Parthasarathy unitary process and a radomized Weyl displacement process. Irreversible dynamics of system density operators described by the well-known Gorini-Kossakowski-Sudarshan-Lindblad master equation is unraveled by coarse-graining over the Gisin-Percival quantum state trajectories.

quant-ph↗
arXiv2016

On the Kolmogorov--Wiener--Masani spectrum of a multi-mode weakly stationary quantum process

We introduce the notion of a $k$-mode weakly stationary quantum process $\varrho$ based on the canonical Schrödinger pairs of position and momentum observables in copies of $L^2(\mathbb{R}^k)$, indexed by an additive abelian group $D$ of countable cardinality. Such observables admit an autocovariance map $\widetilde{K}$ from $D$ into the space of real $2k \times 2k$ matrices. The map $\widetilde{K}$ on the discrete group $D$ admits a spectral representation as the Fourier transform of a $2k \times 2k$ complex Hermitain matrix-valued totally finite measure $Φ$ on the compact character group $\widehat{D}$, called the Kolmogorov-Wiener-Masani (KWM) spectrum of the process $\varrho$. Necessary and sufficient conditions on a $2k \times 2k$ complex Hermitian matrix-valued measure $Φ$ on $\widehat{D}$ to be the KWM spectrum of a process $\varrho$ are obtained. This enables the construction of examples. Our theorem reveals the dramatic influence of the uncertainty relations among the position and momentum observables on the KWM spectrum of the process $\varrho$. In particular, KWM spectrum cannot admit a gap of positive Haar measure in $\widehat{D}$. The relationship between the number of photons in a particular mode at any site of the process and its KWM spectrum needs further investigation.

quant-ph↗
arXiv2016

On the equivalence of separability and extendability of quantum states

Motivated by the notions of $k$-extendability and complete extendability of the state of a finite level quantum system as described by Doherty et al (Phys. Rev. A, 69:022308), we introduce parallel definitions in the context of Gaussian states and using only properties of their covariance matrices derive necessary and sufficient conditions for their complete extendability. It turns out that the complete extendability property is equivalent to the separability property of a bipartite Gaussian state. Following the proof of quantum de Finetti theorem as outlined in Hudson and Moody (Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 33(4):343--351), we show that separability is equivalent to complete extendability for a state in a bipartite Hilbert space where at least one of which is of dimension greater than 2. This, in particular, extends the result of Fannes, Lewis, and Verbeure (Lett. Math. Phys. 15(3): 255--260) to the case of an infinite dimensional Hilbert space whose C* algebra of all bounded operators is not separable.

quant-ph↗
arXiv2015

Exchangeable, stationary and entangled chains of Gaussian states

We explore conditions on the covariance matrices of a consistent chain of mean zero finite mode Gaussian states in order that the chain may be exchangeable or stationary. For an exchangeable chain our conditions are necessary and sufficient. Every stationary Gaussian chain admits an asymptotic entropy rate. Whereas an exchangeable chain admits a simple expression for its entropy rate, in our examples of stationary chains the same admits an integral formula based on the asymptotic eigenvalue distribution for Toeplitz matrices. An example of a stationary entangled Gaussian chain is given.

quant-ph↗
arXiv2015

From particle counting to Gaussian tomography

All the $n(2n+3)$ mean and covariance parameters of an $n$-mode Gaussian states are expressed in terms of the expectation values of the same number of conjugates of the total number observable. This permits a complete tomography of the state. The same is applied to outputs of a Gaussian channel corresponding to selected coherent states to perform the complete tomography of the channel. This leads to some interesting problems concerning the distribution of the number operator and also tomographic complexity.

quant-ph↗
arXiv2014

Quantum Stochastic Calculus and Quantum Gaussian Processes

In this lecture we present a brief outline of boson Fock space stochastic calculus based on the creation, conservation and annihilation operators of free field theory, as given in the 1984 paper of Hudson and Parthasarathy. We show how a part of this architecture yields Gaussian fields stationary under a group action. Then we introduce the notion of semigroups of quasifree completely positive maps on the algebra of all bounded operators in the boson Fock space $Γ(\mathbb{C}^n)$ over $\mathbb{C}^n.$ These semigroups are not strongly continuous but their preduals map Gaussian states to Gaussian states. They were first introduced and their generators were shown to be of the Lindblad type by Vanheuverzwijn. They were recently investigated in the context of quantum information theory by Heinosaari, Holevo and Wolf. Here we present the exact noisy Schrödinger equation which dilates such a semigroup to a quantum Gaussian Markov process.

math-ph↗
arXiv2014

Symplectic Dilations, Gaussian States and Gaussian Channels

By elementary matrix algebra we show that every real $2n \times 2n$ matrix admits a dilation to an element of the real symplectic group $Sp (2(n+m))$ for some nonnegative integer $m.$ Our methods do not yield the minimum value of $m,$ for which such a dilation is possible. After listing some of the main properties of Gaussian states in $L^2 (\mathbb{R}^n),$ we analyse the implications of symplectic dilations in the study of quantum Gaussian channels which lead to some interesting open problems, particularly, in the context of the work of Heinosaari, Holevo and Wolf \cite{3}.

quant-ph↗
arXiv2011

Two remarks on Normality Preserving Borel Automorphisms of R^n

Let $T$ be a bijective map on $\mathbb{R}^n$ such that both $T$ and $T^{-1}$ are Borel measurable. For any $\btheta \in \mathbb{R}^n$ and any real $n \times n$ positive definite matrix $Σ,$ let $N (\btheta, Σ)$ denote the $n$-variate normal (gaussian) probability measure on $\mathbb{R}^n$ with mean vector $\btheta$ and covariance matrix $Σ.$ Here we prove the following two results: (1) Suppose $N(\btheta_j, I)T^{-1}$ is gaussian for $0 \leq j \leq n$ where $I$ is the identity matrix and $\{\btheta_j - \btheta_0, 1 \leq j \leq n \}$ is a basis for $\mathbb{R}^n.$ Then $T$ is an affine linear transformation; (2) Let $Σ_j = I + ε_j \mathbf{u}_j \mathbf{u}_j^{\prime},$ $1 \leq j \leq n$ where $ε_j > -1$ for every $j$ and ${\mathbf{u}_j, 1 \leq j \leq n}$ is a basis of unit vectors in $\mathbb{R}^n$ with $\mathbf{u}_j^{\prime}$ denoting the transpose of the column vector $\mathbf{u}_j.$ Suppose $N(\mathbf{0}, I)T^{-1}$ and $N (\mathbf{0}, Σ_j)T^{-1},$ $1 \leq j \leq n$ are gaussian. Then $T(\mathbf{x}) = \sum\limits_{\mathbf{s}} 1_{E_{\mathbf{s}}} V \mathbf{s} U \mathbf{x}$ a.e. $\mathbf{x}$ where $\mathbf{s}$ runs over the set of $2^n$ diagonal matrices of order $n$ with diagonal entries $\pm 1,$ $U,\, V$ are $n \times n$ orthogonal matrices and $\{E_{\mathbf{s}}\}$ is a collection of $2^n$ Borel subsets of $\mathbb{R}^n$ such that $\{E_{\mathbf{s}}\}$ and $\{V \mathbf{s} U (E_{\mathbf{s}})\}$ are partitions of $\mathbb{R}^n$ modulo Lebesgue-null sets and for every $j,$ $V \mathbf{s} U Σ_j (V \mathbf{s} U)^{-1}$ is independent of all $\mathbf{s}$ for which the Lebesgue measure of $E_{\mathbf{s}}$ is positive. The converse of this result also holds. \vskip0.1in Our results constitute a sharpening of the results of S. Nabeya and T. Kariya

math.PR↗
arXiv2011

A note on gaussian distributions in R^n

Given any finite set F of (n - 1)-dimensional subspaces of R^n we give examples of nongaussian probability measures in R^n whose marginal distribution in each subspace from F is gaussian. However, if F is an infinite family of such (n - 1)-dimensional subspaces then such a nongaussian probability measure in R^n does not exist.

math.ST↗
arXiv2011

The Symmetry Group of Gaussian States in $L^2 (\mathbb{R}^n)$

This is a continuation of the expository article \cite{krp} with some new remarks. Let $S_n$ denote the set of all Gaussian states in the complex Hilbert space $L^2 (\mathbb{R}^n),$ $K_n$ the convex set of all momentum and position covariance matrices of order $2n$ in Gaussian states and let $\mathcal{G}_n$ be the group of all unitary operators in $L^2 (\mathbb{R}^n)$ conjugations by which leave $S_n$ invariant. Here we prove the following results. $K_n$ is a closed convex set for which a matrix $S$ is an extreme point if and only if $S=\frac{1}{2} L^{T} L$ for some $L$ in the symplectic group $Sp (2n, \mathbb{R}).$ Every element in $K_n$ is of the form $\frac{1}{2} (L^{T} L + M^{T} M)$ for some $L,M$ in $Sp (2n, \mathbb{R}).$ Every Gaussian state in $L^2 (\mathbb{R}^n)$ can be purified to a Gaussian state in $L^2 (\mathbb{R}^{2n}).$ Any element $U$ in the group $\mathcal{G}_n$ is of the form $U = λW ({\bm α}) Γ(L)$ where $λ$ is a complex scalar of modulus unity, ${\bm α} \in \mathbb{C}^n,$ $L \in Sp (2n, \mathbb{R}),$ $W({\bm α})$ is the Weyl operator corresponding to ${\bm α} $ and $Γ(L)$ is a unitary operator which implements the Bogolioubov automorphism of the Lie algebra generated by the canonical momentum and position observables induced by the symplectic linear transformation $L.$

math.PR↗
arXiv2009

On the philosophy of Cramér-Rao-Bhattacharya Inequalities in Quantum Statistics

To any parametric family of states of a finite level quantum system we associate a space of Fisher maps and introduce the natural notions of Cramér-Rao-Bhattacharya tensor and Fisher information form. This leads us to an abstract Cramér-Rao-Bhattacharya lower bound for the covariance matrix of any finite number of unbiased estimators of parameteric functions. A number of illustrative examples is included. Modulo technical assumptions of various kinds our methods can be applied to infinite level quantum systems as well as parametric families of classical probability distributions on Borel spaces.

math.PR↗