A geometric proof of the spectral theorem for unbounded self-adjoint operators

Leinfelder, Herbert

doi:10.1007/bf01420484

Cited by 22 publications

(3 citation statements)

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“…But although some theoretical proposals [44], and even devised experimental arrangements [45][46][47], are based on continuous variables (for a review, see [48]), historical breakthroughs [20][21][22][23][24][25] and recent loophole-free measurements [26][27][28][29]49] consider discrete observables. Second, the spectrum theorem for self-adjoint operators -relevant to determine suitable basis for |Ψ -holds true in very general terms [50]. However, to establish solid grounded properties of transformations (similar to those presented in Sec.…”

Section: A Necessary Condition For Epr States: the Observables Are Pa...mentioning

confidence: 96%

Correlations in the EPR States Observables

Orsini,

Oliveira,

Luz

2023

Preprint

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The identification and physical interpretation of quantum correlations, more recently explored beyond entanglement, is not always a simple task. Two assumptions that, at least in principle, should lead to a relatively large dispersion for the observables of a quantum bipartite formed by systems I and II are to assume that; (a) all the possible observables describing the composite are potentially equally probable as outcomes of measurements; and (b) there cannot be concurrence (positive reinforcement) between any of the observables within a given system, meaning all their corresponding operators do not commute. The so-called EPR states are known to observe (a). Here we demonstrate in general terms that they also verify (b). As examples, we discuss three-level systems. Given the character of (a) and (b), one may expect the CHSH correlation for qubits to naturally violate its associated Bell’s inequality (i.e., it would yield values greater than 2) when applied to EPR states. Surprisingly, we show the CHSH contradicts such prediction. This finding emphasizes the subtleties of correlations in quantum mechanics, where the conceivable dispersive interdependence of EPR states observables results counterintuitively in a more limited range of values for the CHSH correlation, not surpassing the nonlocality threshold of 2.

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Section: A Necessary Condition For Epr States: the Observables Are Pa...mentioning

confidence: 96%

Correlations in the EPR States Observables

Orsini,

Oliveira,

Luz

2023

Preprint

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show abstract

“…Albeit some theoretical proposals [ 53 ] and devised experimental arrangements [ 54 , 55 , 56 ] are based on continuous variables (for a review, see [ 57 ]), historical breakthroughs [ 21 , 22 , 23 , 24 , 25 , 26 ] and recent loophole-free measurements [ 27 , 28 , 29 , 30 , 58 ] consider discrete observables. Second, the spectrum theorem for self-adjoint operators—relevant for determining the suitable basis for

—holds true very generally [ 59 ]. However, establishing solidly grounded properties of transformations (similar to those presented in Section 3.1 ) between continuous bases may require additional technicalities; this goes far beyond the scope of this contribution.…”

Section: A Necessary Condition For Epr States: the Observables Are Pa...mentioning

confidence: 99%

Correlations in the EPR State Observables

Orsini,

Oliveira,

da Luz

2024

Entropy

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The identification and physical interpretation of arbitrary quantum correlations are not always effortless. Two features that can significantly influence the dispersion of the joint observable outcomes in a quantum bipartite system composed of systems I and II are: (a) All possible pairs of observables describing the composite are equally probable upon measurement, and (b) The absence of concurrence (positive reinforcement) between any of the observables within a particular system; implying that their associated operators do not commute. The so-called EPR states are known to observe (a). Here, we demonstrate in very general (but straightforward) terms that they also satisfy condition (b), a relevant technical fact often overlooked. As an illustration, we work out in detail the three-level systems, i.e., qutrits. Furthermore, given the special characteristics of EPR states (such as maximal entanglement, among others), one might intuitively expect the CHSH correlation, computed exclusively for the observables of qubit EPR states, to yield values greater than two, thereby violating Bell’s inequality. We show such a prediction does not hold true. In fact, the combined properties of (a) and (b) lead to a more limited range of values for the CHSH measure, not surpassing the nonlocality threshold of two. The present constitutes an instructive example of the subtleties of quantum correlations.

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“…We will not attempt to give a proof of the spectral theorem here; for a convenient approach, see [40] (cf. also [2,Section 26.3] and [65,Theorems 7.14,7.17]).…”

Section: A Linear Operator T Is a Function From D(t ) To H With The Pmentioning

confidence: 99%

Topics from spectral theory of differential operators

Hinz¹

2004

Spectral Theory of Schrödinger Operators

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A geometric proof of the spectral theorem for unbounded self-adjoint operators

Cited by 22 publications

References 5 publications

Correlations in the EPR States Observables

Correlations in the EPR States Observables

Correlations in the EPR State Observables

Topics from spectral theory of differential operators

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