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arXiv:1710.07561

The Solution to the Frame Quantum Detection Problem

Sara Botelho-Andrade, Peter G. Casazza, Desai Cheng, Tin T. Tran

Published Oct 20, 2017• 1 citation

Abstract

We will give a complete solution to the frame quantum detection problem. We will solve both cases of the problem: the quantum injectivity problem and quantum state estimation problem. We will answer the problem in both the real and complex cases and in both the finite dimensional and infinite dimensional cases. Finite Dimensional Case: (1) We give two complete classifications of the sets of vectors which solve the injectivity problem - for both the real and complex cases. We also give methods for constructing them. (2) We show that the frames which solve the injectivity problem are open and dense in the family of all frames. (3) We show that the Parseval frames which give injectivity are dense in the Parseval frames. (4) We classify all frames for which the state estimation problem is solvable, and when it is not solvable, we give the best approximation to a solution. Infinite Dimensional Case: (1) We give a classification of all frames which solve the injectivity problem and give methods for constructing solutions. (2) We show that the frames solving the injectivity problem are neither open nor dense in all frames. (3) We give necessary and sufficient conditions for the state estimation problem to always have solutions for measurements in $\ell_1$ and show that there is no injective frame for which the state estimation problem is solvable for all measurements in $\ell_2$. (4) When the state estimation problem does not have an exact solution, we give the best approximation to a solution.

DOI: 10.48550/arxiv.1710.07561

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We will give a complete solution to the frame quantum detection problem. We will solve both cases of the problem: the quantum injectivity problem and quantum state estimation problem. We will answer the problem in both the real and complex cases and in both the finite dimensional and infinite dimensional cases. Finite Dimensional Case: (1) We give two complete classifications of the sets of vectors which solve the injectivity problem - for both the real and complex cases. We also give methods for constructing them. (2) We show that the frames which solve the injectivity problem are open and dense in the family of all frames. (3) We show that the Parseval frames which give injectivity are dense in the Parseval frames. (4) We classify all frames for which the state estimation problem is solvable, and when it is not solvable, we give the best approximation to a solution. Infinite Dimensional Case: (1) We give a classification of all frames which solve the injectivity problem and give methods for constructing solutions. (2) We show that the frames solving the injectivity problem are neither open nor dense in all frames. (3) We give necessary and sufficient conditions for the state estimation problem to always have solutions for measurements in

\ell_1

and show that there is no injective frame for which the state estimation problem is solvable for all measurements in

\ell_2

. (4) When the state estimation problem does not have an exact solution, we give the best approximation to a solution.

Some recent references have considered quantum detection problems in discrete and continuous frames. This paper examines the quantum detection problem with g-frame in finite dimensions. We will discuss two issues related to the problem at hand: the quantum injectivity problem and the solvability of the quantum state estimation problem. This work addresses the quantum detection problem in a broader context and provides multiple descriptions of quantum injective g-frames. Specifically, we also offer a quantitative stability outcome for g-frames.

A quantum injective frame is a frame whose measurements for density operators can be used as a distinguishing feature in a quantum system, and the frame quantum detection problem demands a characterization of all such frames. Very recently, the quantum detection problem for continuous as well as discrete frames in both finite and infinite dimensional Hilbert spaces received significant attention. The quantum detection problem pertaining to the characterization of informationally complete positive operator-valued measures (POVM) can be split into two cases: The quantum injectivity or state separability problem and the rang analysis or quantum state estimation problem. Building upon this notion, this note is aimed at the quantum detection problem for fusion frames. The injectivity of a family of vectors and a family of closed subspaces is characterized in terms of some operator equations in Hilbert–Schmidt and trace classes.

The main issues in quantum detection theory are the injectivity and state estimation problems, which recently linked and partially studied via frame theory on Hilbert spaces.Inspired by some recent researches and works on the quantum detection problem by (discrete) frames, continuous frames and fusion frames, in this note we examine the quantum detection problem for generalized frames (g-frames).In the theory of g-frames, instead of using the frame element itself, we use a family of bounded operators.We characterize injectivity problem by g-frames and with the help of these characterizations, we give some equivalent conditions for the injectivity problems.We answer the injectivity problems both for trace and Hilbert-Schmidt classes of operators.

Linear constraint systems (LCS) provide a compact algebraic language for nonlocal games and state-independent contextuality. In the binary case the Mermin--Peres square shows that inconsistent LCS can be solved by promoting variables to local Pauli operators, whereas a series of no-go results excludes analogous Pauli-, Clifford- and monomial unitary-based constructions at odd prime modulus. We first isolate the reason for this difference: for tensor-product quantum solutions, the obstruction to a classical solution of a LCS over

\mathbb{Z}_n

decomposes into obstruction classes carried by the individual factors. As a consequence, a genuinely global obstruction cannot arise solely by tensoring locally unobstructed systems. The argument is specific to odd modulus; at even modulus the reordering phase need not vanish, as in Pauli-based examples. Given this distinction, we instead associate LCS to finite arrangements of rank-one projectors and resolutions of the identity in finite-dimensional vector spaces. Quantum solvability for such LCS is automatic and the search for a quantum-classical gap rests entirely on proving classical unsatisfiability of the underlying modular incidence problem of the arrangement. By relating such LCS with group-valued frame functions, we identify a family of examples in the work of [Harding, Jager, and Smith, Int. J. Theor. Phys. 44, 539 (2005)]. To establish the (non)existence of classically unsatisfiable LCS over

\mathbb{Z}_n

with quantum solutions in all dimensions

d

and for all

n\in\mathbb{N}

, we further construct explicit examples for the cases

d=n

prime. Our main result thus positively resolves the existence problem of such LCS beyond the binary case: such systems exist if and only if

d\geq 3

and

\gcd(n,d)>1

. Moreover, we establish the existence of LCS with a quantum-classical gap over finite fields if and only if

d\geq 3

and

p\mid d

.

A quantum injective frame is a frame that can be used to distinguish quantum systems by measuring density operators. The quantum detection problem aims to characterize all such frames. Re-cently, there has been significant attention on the problem of quantum detection in both finite and infinite dimensional Hilbert spaces, for both continuous and discrete frames. This paper aims to present various characterizations for injective continuous g-frames. The work solves the quantum detection problem in a more general context and provides characterizations for a broader class of injective continuous g-frames.

Solving linear systems of equations is a common problem that arises both on its own and as a subroutine in more complex problems: given a matrix

A

and a vector $\stackrel{\ensuremath{\rightarrow}}{b}$, find a vector $\stackrel{\ensuremath{\rightarrow}}{x}$ such that $A\stackrel{\ensuremath{\rightarrow}}{x}=\stackrel{\ensuremath{\rightarrow}}{b}$. We consider the case where one does not need to know the solution $\stackrel{\ensuremath{\rightarrow}}{x}$ itself, but rather an approximation of the expectation value of some operator associated with $\stackrel{\ensuremath{\rightarrow}}{x}$, e.g., ${\stackrel{\ensuremath{\rightarrow}}{x}}^{\ifmmode\dagger\else\textdagger\fi{}}M\stackrel{\ensuremath{\rightarrow}}{x}$ for some matrix

M

. In this case, when

A

is sparse, $N\ifmmode\times\else\texttimes\fi{}N$ and has condition number $\ensuremath{\kappa}$, the fastest known classical algorithms can find $\stackrel{\ensuremath{\rightarrow}}{x}$ and estimate ${\stackrel{\ensuremath{\rightarrow}}{x}}^{\ifmmode\dagger\else\textdagger\fi{}}M\stackrel{\ensuremath{\rightarrow}}{x}$ in time scaling roughly as $N\sqrt{\ensuremath{\kappa}}$. Here, we exhibit a quantum algorithm for estimating ${\stackrel{\ensuremath{\rightarrow}}{x}}^{\ifmmode\dagger\else\textdagger\fi{}}M\stackrel{\ensuremath{\rightarrow}}{x}$ whose runtime is a polynomial of

\mathrm{log}﻿(N)

and $\ensuremath{\kappa}$. Indeed, for small values of $\ensuremath{\kappa}$ [i.e.,

\mathrm{poly}\mathrm{log}﻿(N)

], we prove (using some common complexity-theoretic assumptions) that any classical algorithm for this problem generically requires exponentially more time than our quantum algorithm.