Uncertainty relations for characteristic functions

The notion of mutual unbiasedness for coarse-grained measurements of quantum continuous variable systems is considered. It is shown that while the procedure of "standard" coarse graining breaks the mutual unbiasedness between conjugate variables, this desired feature can be theoretically established and experimentally observed in periodic coarse graining. We illustrate our results in an optics experiment implementing Fraunhofer diffraction through a periodic diffraction grating, finding excellent agreement with the derived theory. Our results are an important step in developing a formal connection between discrete and continuous variable quantum mechanics.Introduction. The ability to measure a system in an infinite number of non-commuting bases distinguishes the quantum world from classical physics. Wave-particle duality and more generally the complementarity principle are directly rooted in this feature of quantum mechanics. Though one can measure a quantum system in several distinct bases, uncertainty relations limit the amount of information that can be obtained. It is well known that projection onto an eigenstate of one basis reduces the information that can be obtained through or inferred about subsequent measurement in a different basis. The information is minimum for mutually unbiased bases (MUBs), for which all outcomes of the second measurement are equally likely, so that total uncertainty is always substantial (the sharpest uncertainty relations [1]) and most insensitive to input states [2]. MUBs play an important role in complementarity [3], quantum cryptography [4] and quantum tomography [5,6], are useful for certifying quantum randomness [7], and for detecting quantum correlations such as entanglement [8][9][10] and steering [11][12][13][14][15][16][17][18].

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“…To show (2a), we represent [46] the probability associated with Ω l in terms of the position autocorrelation function [47]:…”

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confidence: 99%

Mutual Unbiasedness in Coarse-Grained Continuous Variables

et al. 2018

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“…Importantly, the case with

is always excluded, since in this case the PCG projectors describe commuting sets,

[ 138 , 139 , 140 ]. In other words a joint eigenstate of the product

existis for all k and l whenever

with

[ 153 ]. It is also interesting to note that using the periodicity definition from the PCG design (

), it is possible to write the unbiasedness condition given in Equation ( 48 ) in alternative, equivalent ways:

…”

Section: Realistic Coarse-grained Measurements Of Continuous Distrmentioning

confidence: 99%

Uncertainty Relations for Coarse–Grained Measurements: An Overview

Toscano

Tasca

Rudnicki

et al. 2018

Entropy

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Uncertainty relations involving complementary observables are one of the cornerstones of quantum mechanics. Aside from their fundamental significance, they play an important role in practical applications, such as detection of quantum correlations and security requirements in quantum cryptography. In continuous variable systems, the spectra of the relevant observables form a continuum and this necessitates the coarse graining of measurements. However, these coarse-grained observables do not necessarily obey the same uncertainty relations as the original ones, a fact that can lead to false results when considering applications. That is, one cannot naively replace the original observables in the uncertainty relation for the coarse-grained observables and expect consistent results. As such, a number of uncertainty relations that are specifically designed for coarse-grained observables have been developed. In recognition of the 90 th anniversary of the seminal Heisenberg uncertainty relation, celebrated last year, and all the subsequent work since then, here we give a review of the state of the art of coarse-grained uncertainty relations in continuous variable quantum systems, as well as their applications to fundamental quantum physics and quantum information tasks. Our review is meant to be balanced in its content, since both theoretical considerations and experimental perspectives are put on an equal footing.

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“…Following Ref. [11] we can construct the Gram matrix G for the following three vectors where |Ψ is an arbitrary state assumed pure for simplicity and without loss of generality, so that…”

Section: Spin-like Systemsmentioning

confidence: 99%

“…Since the operator E is not unitary, the change of k by −k is not trivial, so in order to follow the procedure in Ref. [11] we have to construct explicitly the two Gram matrices.…”

Section: Single-mode Phase and Numbermentioning

confidence: 99%

Phase–number uncertainty from Weyl commutation relations

Luis

Donoso

2017

Annals of Physics

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We derive suitable uncertainty relations for characteristics functions of phase and number variables obtained from the Weyl form of commutation relations. This is applied to finite-dimensional spinlike systems, which is the case when describing the phase difference between two field modes, as well as to the phase and number of a single-mode field. Some contradictions between the product and sums of characteristic functions are noted.

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Uncertainty relations for characteristic functions

Cited by 13 publications

References 83 publications

Mutual Unbiasedness in Coarse-Grained Continuous Variables

Mutual Unbiasedness in Coarse-Grained Continuous Variables

Uncertainty Relations for Coarse–Grained Measurements: An Overview

Phase–number uncertainty from Weyl commutation relations

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