arXiv subjects
Publications and source records attributed to K. Mahesh Krishna.
We introduce the notion of noncommutative equiangular lines and derive noncommutative versions of fundamental van Lint-Seidel relative and Gerzon universal bounds.
We introduce the notion of p-adic spherical codes (in particular, p-adic kissing number problem). We show that the one-line proof for a variant of the Delsarte-Goethals-Seidel-Kabatianskii-Levenshtein upper bound for spherical codes, obtained by Pfender \textit{[J. Combin. Theory Ser. A, 2007]}, extends to p-adic Hilbert spaces.
Let $\mathcal{E}$ be a Hilbert C*-module over a unital C*-algebra $\mathcal{A}$. Let $A: \mathcal{D}(A) \subseteq \mathcal{E} \to \mathcal{E}$ and $B: \mathcal{D}(B)\subseteq \mathcal{E}\to \mathcal{E}$ be possibly unbounded self-adjoint morphisms. Then for all $x \in \mathcal{D}(AB)\cap \mathcal{D}(BA)$ with $\langle x, x \rangle =1$, we show that \begin{align*} (1) \quad \quad \quad Δ_x(B)^2d_x(A)^2+Δ_x(A)^2d_x(B)^2\geq \frac{(\langle \{A,B\}x, x \rangle -\{\langle Ax, x \rangle,\langle Bx, x \rangle\})^2-(\langle [A,B]x, x \rangle +[\langle Ax, x \rangle,\langle Bx, x \rangle])^2}{2} \end{align*} and \begin{align*} (2) \quad \quad \quad \quad Δ_x(A)Δ_x(B)\geq \frac{\sqrt{\|(\langle \{A,B\}x, x \rangle -\{\langle Ax, x \rangle,\langle Bx, x \rangle\})^2-(\langle [A,B]x, x \rangle +[\langle Ax, x \rangle,\langle Bx, x \rangle])^2\|}}{2}, \end{align*} where $Δ_x(A):= \|Ax-\langle Ax, x \rangle x \|$, $d_x(A):= \sqrt{\langle Ax, Ax \rangle -\langle Ax, x \rangle^2}$, $[A,B] := AB-BA$, $\{A,B\}:= AB+BA$, $\{\langle Ax, x \rangle,\langle Bx, x \rangle\}:= \langle Ax, x \rangle\langle Bx, x \rangle +\langle Bx, x \rangle\langle Ax, x \rangle$, $[\langle Ax, x \rangle,\langle Bx, x \rangle]:= \langle Ax, x \rangle\langle Bx, x \rangle -\langle Bx, x \rangle\langle Ax, x \rangle$. We call Inequalities (1) and (2) as noncommutative Heisenberg-Robertson-Schrodinger uncertainty principles. They reduce to the Heisenberg-Robertson-Schrodinger uncertainty principle (derived by Schrodinger in 1930) whenever $\mathcal{A}=\mathbb{C}$.
Let $\mathcal{X}$ be a 3-product space. Let $A: \mathcal{D}(A)\subseteq \mathcal{X}\to \mathcal{X}$, $B: \mathcal{D}(B)\subseteq \mathcal{X}\to \mathcal{X}$ and $C: \mathcal{D}(C)\subseteq \mathcal{X}\to \mathcal{X}$ be possibly unbounded 3-self-adjoint operators. Then for all \begin{align*} x \in \mathcal{D}(ABC)\cap\mathcal{D}(ACB) \cap \mathcal{D}(BAC)\cap\mathcal{D}(BCA) \cap \mathcal{D}(CAB)\cap\mathcal{D}(CBA) \end{align*} with $\langle x, x, x \rangle =1$, we show that \begin{align*} (1)\quad \quad Δ_x(3, A) Δ_x(3, B) Δ_x(3, C)\geq |\langle (ABC-a BC-b AC-c AB)x, x, x\rangle +2abc|, \end{align*} where \begin{align*} Δ_x(3, A):= \|Ax-\langle Ax, x, x \rangle x \|, \quad a:= \langle Ax, x, x \rangle, \quad b := \langle Bx, x, x \rangle, \quad c := \langle Cx, x, x \rangle. \end{align*} We call Inequality (1) as 3-Heisenberg-Robertson-Schrodinger uncertainty principle. Classical Heisenberg-Robertson-Schrodinger uncertainty principle (by Schrodinger in 1930) considers two operators whereas Inequality (1) considers three operators.
Motivated from Deutsch entropic uncertainty principle and several product uncertainty principles, we derive an uncertainty principle for the product of entropies using functions.
Breakthrough Sparsity Theorem, derived independently by Donoho and Elad \textit{[Proc. Natl. Acad. Sci. USA, 2003]}, Gribonval and Nielsen \textit{[IEEE Trans. Inform. Theory, 2003]} and Fuchs \textit{[IEEE Trans. Inform. Theory, 2004]} says that unique sparse solution to NP-Hard $\ell_0$-minimization problem can be obtained using unique solution of P-Type $\ell_1$-minimization problem. In this paper, we derive noncommutative version of their result using frames for Hilbert C*-modules.
Let $\{τ_n\}_{n=1}^\infty$ and $\{ω_m\}_{m=1}^\infty$ be two modular Parseval frames for a Hilbert C*-module $\mathcal{E}$. Then for every $x \in \mathcal{E}\setminus\{0\}$, we show that \begin{align} (1) \quad \quad \quad \quad \|θ_τx \|_0 \|θ_ωx \|_0 \geq \frac{1}{\sup_{n, m \in \mathbb{N}} \|\langle τ_n, ω_m\rangle \|^2}. \end{align} We call Inequality (1) as \textbf{Noncommutative Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle}. Inequality (1) is the noncommutative analogue of breakthrough Ricaud-Torrésani uncertainty principle \textit{[IEEE Trans. Inform. Theory, 2013]}. In particular, Inequality (1) extends Elad-Bruckstein uncertainty principle \textit{[IEEE Trans. Inform. Theory, 2002]} and Donoho-Stark uncertainty principle \textit{[SIAM J. Appl. Math., 1989]}.
Khosravi, Drnovšek and Moslehian [\textit{Filomat, 2012}] derived Buzano inequality for Hilbert C*-modules. Using this inequality we derive Deutsch entropic uncertainty principle for Hilbert C*-modules over commutative unital C*-algebras.
In 2002, Krishna and Parthasarathy [\textit{Sankhyā Ser. A}] derived discrete quantum version of Maassen-Uffink [\textit{Phys. Rev. Lett., 1988}] entropic uncertainty principle. In this paper, using the notion of continuous operator-valued frames, we derive an entropic uncertainty principle for arbitrary family of operators indexed by measure spaces having finite measure. We give an application to the special case of compact groups.
Let $(\{f_n\}_{n=1}^\infty, \{τ_n\}_{n=1}^\infty)$ and $(\{g_n\}_{n=1}^\infty, \{ω_n\}_{n=1}^\infty)$ be unbounded continuous p-Schauder frames ($0<p<1$) for a disc Banach space $\mathcal{X}$. Then for every $x \in ( \mathcal{D}(θ_f) \cap\mathcal{D}(θ_g))\setminus\{0\}$, we show that \begin{align}\label{UB} (1) \quad \quad \quad \quad \|θ_f x\|_0\|θ_g x\|_0 \geq \frac{1}{\left(\displaystyle\sup_{n,m \in \mathbb{N} }|f_n(ω_m)|\right)^p\left(\displaystyle\sup_{n, m \in \mathbb{N}}|g_m(τ_n)|\right)^p}, \end{align} where \begin{align*} & θ_f: \mathcal{D}(θ_f) \ni x \mapsto θ_fx := \{f_n(x)\}_{n=1}^\infty\in \ell^p(\mathbb{N}), \quad θ_g: \mathcal{D}(θ_g) \ni x \mapsto θ_gx := \{g_n(x)\}_{n=1}^\infty\in \ell^p(\mathbb{N}). \end{align*} Inequality (1) is unexpectedly different from both bounded uncertainty principle arXiv:2308.00312v1 and unbounded uncertainty principle arXiv:2312.00366v1 for Banach spaces.
We show that one of the two important uncertainty principles derived by Maccone and Pati \textit{[Phys. Rev. Lett., 2014]} can be derived for arbitrary maps defined on subsets of $\mathcal{L}^p$ spaces for $1< p<\infty$. Our main tool is the Clarkson inequalities. We also derive a nonlinear uncertainty principle for weak parallelogram spaces and Type-p Banach spaces.
Let $\mathcal{X}$ be a Banach space. Let $\{τ_j\}_{j=1}^n, \{ω_k\}_{k=1}^m\subseteq \mathcal{X}$ and $\{f_j\}_{j=1}^n$, $\{g_k\}_{k=1}^m\subseteq \mathcal{X}^*$ satisfy $ |f_j(τ_j)|\geq 1$ for all $ 1\leq j \leq n$, $|g_k(ω_k)|\geq 1 $ for all $1\leq k \leq m$. If $x \in \mathcal{X}\setminus \{0\}$ is such that $x=θ_τθ_f x=θ_ωθ_g x$, then we show that \begin{align}\label{FKDB} (1) \quad\quad\quad\quad \|θ_fx\|_0\|θ_gx\|_0\geq \frac{\bigg[1-(\|θ_fx\|_0-1)\max\limits_{1\leq j,r \leq n,j\neq r}|f_j(τ_r)|\bigg]^+\bigg[1-(\|θ_g x\|_0-1)\max\limits_{1\leq k,s \leq m,k\neq s}|g_k(ω_s)|\bigg]^+}{\left(\displaystyle\max_{1\leq j \leq n, 1\leq k \leq m}|f_j(ω_k)|\right)\left(\displaystyle\max_{1\leq j \leq n, 1\leq k \leq m}|g_k(τ_j)|\right)}. \end{align} We call Inequality (1) as \textbf{Functional Kuppinger-Durisi-Bölcskei Uncertainty Principle}. Inequality (1) improves the uncertainty principle obtained by Kuppinger, Durisi and Bölcskei \textit{[IEEE Trans. Inform. Theory (2012)]} (which improved the Donoho-Stark-Elad-Bruckstein uncertainty principle \textit{[SIAM J. Appl. Math. (1989), IEEE Trans. Inform. Theory (2002)]}). We also derive functional form of the uncertainity principle obtained by Studer, Kuppinger, Pope and Bölcskei \textit{[EEE Trans. Inform. Theory (2012)]}.
Let $(Ω, μ)$, $(Δ, ν)$ be measure spaces and $p=1$ or $p=\infty$. Let $(\{f_α\}_{α\in Ω}, \{τ_α\}_{α\in Ω})$ and $(\{g_β\}_{β\in Δ}, \{ω_β\}_{β\in Δ})$ be unbounded continuous p-Schauder frames for a Banach space $\mathcal{X}$. Then for every $x \in ( \mathcal{D}(θ_f) \cap\mathcal{D}(θ_g))\setminus\{0\}$, we show that \begin{align}\label{UB} (1) \quad \quad \quad \quad μ(\operatorname{supp}(θ_f x))ν(\operatorname{supp}(θ_g x)) \geq \frac{1}{\left(\displaystyle\sup_{α\in Ω, β\in Δ}|f_α(ω_β)|\right)\left(\displaystyle\sup_{α\in Ω, β\in Δ}|g_β(τ_α)|\right)}, \end{align} where \begin{align*} &θ_f:\mathcal{D}(θ_f) \ni x \mapsto θ_fx \in \mathcal{L}^p(Ω, μ); \quad θ_fx: Ω\ni α\mapsto (θ_fx) (α):= f_α(x) \in \mathbb{K},\\ &θ_g: \mathcal{D}(θ_g) \ni x \mapsto θ_gx \in \mathcal{L}^p(Δ, ν); \quad θ_gx: Δ\ni β\mapsto (θ_gx) (β):= g_β(x) \in \mathbb{K}. \end{align*} We call Inequality (1) as \textbf{Unbounded Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle}. Along with recent \textbf{Functional Continuous Uncertainty Principle} [arXiv:2308.00312], Inequality (1) also improves Ricaud-Torrésani uncertainty principle [IEEE Trans. Inform. Theory, 2013]. In particular, it improves Elad-Bruckstein uncertainty principle [IEEE Trans. Inform. Theory, 2002] and Donoho-Stark uncertainty principle [SIAM J. Appl. Math., 1989].
Let $(Ω, μ)$ be a measure space and $\{τ_α\}_{α\in Ω}$ be a normalized continuous Bessel family for a real Hilbert space $\mathcal{H}$. If the diagonal $Δ:= \{(α, α):α\in Ω\}$ is measurable in the measure space $Ω\times Ω$, then we show that \begin{align} (1) \quad\quad\quad\quad \sup _{α, β\in Ω, α\neq β}\langle τ_α, τ_β\rangle \geq \frac{-(μ\timesμ)(Δ)}{(μ\timesμ)((Ω\timesΩ)\setminusΔ)}. \end{align} We call Inequality (1) as continuous Rankin bound. It improves 76 years old result of Rankin [\textit{Ann. of Math., 1947}]. It also answers one of the questions asked by K. M. Krishna in the paper [Continuous Welch bounds with applications, \textit{Commun. Korean Math. Soc., 2023}]. We also derive Banach space version of Inequality (1).
Let $(Ω, μ)$, $(Δ, ν)$ be measure spaces and $\{τ_α\}_{α\in Ω}$, $\{ω_β\}_{β\in Δ}$ be 1-bounded continuous Parseval frames for a Hilbert space $\mathcal{H}$. Then we show that \begin{align} (1) \quad \quad \quad \quad \log (μ(Ω)ν(Δ))\geq S_τ(h)+S_ω(h)\geq -2 \log \left(\frac{1+\displaystyle \sup_{α\in Ω, β\in Δ}|\langleτ_α, ω_β\rangle|}{2}\right) , \quad \forall h \in \mathcal{H}_τ\cap \mathcal{H}_ω, \end{align} where \begin{align*} &\mathcal{H}_τ:= \{h_1 \in \mathcal{H}: \langle h_1 , τ_α\rangle \neq 0, α\in Ω\}, \quad \mathcal{H}_ω:= \{h_2 \in \mathcal{H}: \langle h_2, ω_β\rangle \neq 0, β\in Δ\},\\ &S_τ(h):= -\displaystyle\int\limits_Ω\left|\left \langle \frac{h}{\|h\|}, τ_α\right\rangle \right|^2\log \left|\left \langle \frac{h}{\|h\|}, τ_α\right\rangle \right|^2\,dμ(α), \quad \forall h \in \mathcal{H}_τ, \\ & S_ω(h):= -\displaystyle\int\limits_Δ\left|\left \langle \frac{h}{\|h\|}, ω_β\right\rangle \right|^2\log \left|\left \langle \frac{h}{\|h\|}, ω_β\right\rangle \right|^2\,dν(β), \quad \forall h \in \mathcal{H}_ω. \end{align*} We call Inequality (1) as \textbf{Continuous Deutsch Uncertainty Principle}. Inequality (1) improves the uncertainty principle obtained by Deutsch \textit{[Phys. Rev. Lett., 1983]}. We formulate Kraus conjecture for 1-bounded continuous Parseval frames. We also derive continuous Deutsch uncertainty principles for Banach spaces.
Let $\{f_j\}_{j=1}^n$ and $\{g_k\}_{k=1}^m$ be Parseval p-frames for a finite dimensional Banach space $\mathcal{X}$. Then we show that \begin{align} (1) \quad\quad\quad\quad \log (nm)\geq S_f (x)+S_g (x)\geq -p \log \left(\displaystyle\sup_{y \in \mathcal{X}_f\cap \mathcal{X}_g, \|y\|=1}\left(\max_{1\leq j\leq n, 1\leq k\leq m}|f_j(y)g_k(y)|\right)\right), \quad \forall x \in \mathcal{X}_f\cap \mathcal{X}_g, \end{align} where \begin{align*} &\mathcal{X}_f:= \{z\in \mathcal{X}: f_j(z)\neq 0, 1\leq j \leq n\}, \quad \mathcal{X}_g:= \{w\in \mathcal{X}: g_k(w)\neq 0, 1\leq k \leq m\},\\ &S_f (x):= -\sum_{j=1}^{n}\left|f_j\left(\frac{x}{\|x\|}\right)\right|^p\log \left|f_j\left(\frac{x}{\|x\|}\right)\right|^p, \quad S_g (x):= -\sum_{k=1}^{m}\left|g_k\left(\frac{x}{\|x\|}\right)\right|^p\log \left|g_k\left(\frac{x}{\|x\|}\right)\right|^p, \quad \forall x \in \mathcal{X}_g. \end{align*} We call Inequality (1) as \textbf{Functional Deutsch Uncertainty Principle}. For Hilbert spaces, we show that Inequality (1) reduces to the uncertainty principle obtained by Deutsch \textit{[Phys. Rev. Lett., 1983]}. We also derive a dual of Inequality (1).
Let $(Ω, μ)$, $(Δ, ν)$ be measure spaces. Let $(\{f_α\}_{α\in Ω}, \{τ_α\}_{α\in Ω})$ and $(\{g_β\}_{β\in Δ}, \{ω_β\}_{β\in Δ})$ be continuous p-Schauder frames for a Banach space $\mathcal{X}$. Then for every $x \in \mathcal{X}\setminus\{0\}$, we show that \begin{align} (1) \quad \quad \quad \quad μ(\operatorname{supp}(θ_f x))^\frac{1}{p} ν(\operatorname{supp}(θ_g x))^\frac{1}{q} \geq \frac{1}{\displaystyle\sup_{α\in Ω, β\in Δ}|f_α(ω_β)|}, \quad ν(\operatorname{supp}(θ_g x))^\frac{1}{p} μ(\operatorname{supp}(θ_f x))^\frac{1}{q}\geq \frac{1}{\displaystyle\sup_{α\in Ω, β\in Δ}|g_β(τ_α)|}. \end{align} where \begin{align*} &θ_f: \mathcal{X} \ni x \mapsto θ_fx \in \mathcal{L}^p(Ω, μ); \quad θ_fx: Ω\ni α\mapsto (θ_fx) (α):= f_α(x) \in \mathbb{K}, &θ_g: \mathcal{X} \ni x \mapsto θ_gx \in \mathcal{L}^p(Δ, ν); \quad θ_gx: Δ\ni β\mapsto (θ_gx) (β):= g_β(x) \in \mathbb{K} \end{align*} and $q$ is the conjugate index of $p$. We call Inequality (1) as \textbf{Functional Continuous Uncertainty Principle}. It improves the Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle obtained by K. Mahesh Krishna in [arXiv:2304.03324v1 [math.FA], 5 April 2023]. It also answers a question asked by Prof. Philip B. Stark to the author. Based on Donoho-Elad Sparsity Theorem, we formulate Measure Minimization Conjecture.
Let $(\{f_j\}_{j=1}^n, \{τ_j\}_{j=1}^n)$ and $(\{g_k\}_{k=1}^n, \{ω_k\}_{k=1}^n)$ be two p-orthonormal bases for a finite dimensional Banach space $\mathcal{X}$. If $ x \in \mathcal{X}\setminus\{0\}$ is such that $θ_fx$ is $\varepsilon$-supported on $M\subseteq \{1,\dots, n\}$ w.r.t. p-norm and $θ_gx$ is $δ$-supported on $N\subseteq \{1,\dots, n\}$ w.r.t. p-norm, then we show that \begin{align}\label{ME} (1) \quad \quad \quad \quad &o(M)^\frac{1}{p}o(N)^\frac{1}{q}\geq \frac{1}{\displaystyle \max_{1\leq j,k\leq n}|f_j(ω_k) |}\max \{1-\varepsilon-δ, 0\},\\ (2) \quad \quad \quad \quad&o(M)^\frac{1}{q}o(N)^\frac{1}{p}\geq \frac{1}{\displaystyle \max_{1\leq j,k\leq n}|g_k(τ_j) |}\max \{1-\varepsilon-δ, 0\},\label{ME2} \end{align} where \begin{align*} θ_f: \mathcal{X} \ni x \mapsto (f_j(x) )_{j=1}^n \in \ell^p([n]); \quad θ_g: \mathcal{X} \ni x \mapsto (g_k(x) )_{k=1}^n \in \ell^p([n]) \end{align*} and $q$ is the conjugate index of $p$. We call Inequalities (1) and (2) as \textbf{Functional Donoho-Stark Approximate Support Uncertainty Principle}. Inequalities (1) and (2) improve the finite approximate support uncertainty principle obtained by Donoho and Stark \textit{[SIAM J. Appl. Math., 1989]}.