Injective continuous frames and quantum detections
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Banach J. Math. Anal. https://doi.org/10.1007/s43037-020-00086-7 ORIGINAL PAPER
Injective continuous frames and quantum detections Deguang Han1 · Qianfeng Hu2 · Rui Liu2 Received: 27 February 2020 / Accepted: 23 July 2020 © Tusi Mathematical Research Group (TMRG) 2020
Abstract A quantum injective frame is a frame that can be used to distinguish density operators (states) from their frame measurements, and the frame quantum detection problem asks to characterize all such frames. This problem was recently settled in Botelho-Andrade et al. (Springer Proc Math Stat 255:337–352, 2017) and BotelhoAndrade et al. (J Fourier Anal Appl 25:2268–2323, 2019) mainly for finite or infinite but discrete frames. In this paper, we consider the continuous frame version of the quantum detection problem. Instead of using the frame element itself, we use discrete representations of continuous frames to obtain several versions of characterizations for quantum injective continuous frames. With the help of these characterizations, we also examine the issues involving constructions and stability of continuous quantum injective frames. In particular, we show that injective continuous frames are stable for finite-dimensional Hilbert spaces but unstable for infinitedimensional cases. Keywords Quantum detection · Quantum injectivity · Continuous frame · Positive operator-valued measure Mathematics Subject Classification 42C15 · 46L10 · 47A05 · 42C99 · 46C10 · 47B38
Communicated by Joseph Ball. * Rui Liu [email protected] Deguang Han [email protected] Qianfeng Hu [email protected] 1
Department of Mathematics, University of Central Florida, Orlando, USA
2
School of Mathematical Sciences and LPMC, Nankai University, Tianjin, China
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D. Han et al.
1 Introduction The quantum detection problem with discrete frame coefficient measurements was recently settled by Botelho-Andrade et al. for both finite and infinite dimensional Hilbert spaces in [7, 9], where the characterization was given in terms the spanning properties of some derived sequences from the frame vectors. Naturally, we would wonder how much of the results from [7, 9] is still valid for other type of frames, for instance, continuous frames. While it is reasonable to expect a similar type of characterizations in terms of the range space of a continuous frame, it seems unpractical to verify the injectivity by performing uncountably many number of operations. The purpose of this paper is to present a similar type of characterization for injective continuous frames in terms of their discrete representations that were introduced in [17]. Constructions of injective continuous frames and their stability will also be discussed. Let us first recall some backgrounds and basics about frames and the quantum detection problem. The notion of discrete frames was first introduced by Duffin and Shaeffer [14], and it allows (like basis) stable but not necessarily unique decomposition of arbitrary element into the expression of the frame element. Later, motiv
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