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arXiv:1705.09139

The Maccone-Pati uncertainty relation

Jonas Maziero

Published May 25, 2017• 1 citation
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Abstract

The existence of incompatible observables constitutes one of the most prominent characteristics of quantum mechanics (QM) and can be revealed and formalized through uncertainty relations. The Heisenberg-Robertson-Schrödinger uncertainty relation (HRSUR) was proved at the dawn of quantum formalism and is ever-present in the teaching and research on QM. Notwithstanding, the HRSUR possess the so called triviality problem. That is to say, the HRSUR yields no information about the possible incompatibility between two observables if the system was prepared in a state which is an eigenvector of one of them. After about 85 years of existence of the HRSUR, this problem was solved recently by Lorenzo Maccone and Arun K. Pati. In this article, we start doing a brief discussion of general aspects of the uncertainty principle in QM and recapitulating the proof of HRSUR. Afterwards we present in simple terms the proof of the Maccone-Pati uncertainty relation, which can be obtained basically via the application of the parallelogram law and Cauchy-Schwarz inequality.

DOI
10.48550/arxiv.1705.09139

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The existence of incompatible observables constitutes one of the most prominent characteristics of quantum mechanics (QM) and can be revealed and formalized through uncertainty relations. The Heisenberg-Robertson-Schrödinger uncertainty relation (HRSUR) was proved at the dawn of quantum formalism and is ever-present in the teaching and research on QM. Notwithstanding, the HRSUR possess the so called triviality problem. That is to say, the HRSUR yields no information about the possible incompatibility between two observables if the system was prepared in a state which is an eigenvector of one of them. After about 85 years of existence of the HRSUR, this problem was solved recently by Lorenzo Maccone and Arun K. Pati. In this article, we start doing a brief discussion of general aspects of the uncertainty principle in QM and recapitulating the proof of HRSUR. Afterwards we present in simple terms the proof of the Maccone-Pati uncertainty relation, which can be obtained basically via the application of the parallelogram law and Cauchy-Schwarz inequality.
Recently, Maccone and Pati [Phys. Rev. Lett. {\bf 113}, 260401 (2014)] derived few inequalities among variances of incompatible operators which they called stronger uncertainty relations, stronger than Heisenberg-Robertson or Schrodinger uncertainty relations. Here we generalize their study to get infinite number of such inequalities and propose that only one of them may be the correct uncertainty relation between incompatible operators. We get back well known uncertainty relations of Heisenberg-Robertson and Schrodinger under certain limits. We also reexamine the conclusions of Maccone and Pati and comment on their work.
The Heisenberg-Robertson uncertainty relation quantitatively expresses the impossibility of jointly sharp preparation of incompatible observables. However, it does not capture the concept of incompatible observables because it can be trivial even for two incompatible observables. We experimentally demonstrate that the new stronger uncertainty relations proposed by Maccone and Pati [Phys. Rev. Lett. 113, 260401 (2014)] relating to the sum of variances are valid in a state-dependent manner and that the lower bound is guaranteed to be nontrivial when two observables are incompatible on the state of the system being measured. The behavior we find agrees with the predictions of quantum theory and obeys the new uncertainty relations even for the special states which trivialize the Heisenberg-Robertson relation. We realize a direct measurement model and perform an experimental investigation of the strengthened relations.
Abstract In this article, I show how the Aharonov–Vaidman identity A∣ψ⟩=⟨A⟩∣ψ⟩+ΔA∣ψA⊥⟩A\left| \psi \right\rangle = \left\langle A \right\rangle \left| \psi \right\rangle + \Delta A \left| \psi ^{\perp }_A \right\rangle A∣ψ⟩=⟨A⟩∣ψ⟩+ΔA​ψA⊥​⟩ A ψ = A ψ + Δ A ψ A ⊥ can be used to prove relations between the standard deviations of observables in quantum mechanics. In particular, I review how it leads to a more direct and less abstract proof of the Robertson uncertainty relation ΔAΔB≥12∣⟨[A,B]⟩∣\Delta A \Delta B \ge \frac{1}{2} \left| \left\langle [A,B] \right\rangle \right| ΔAΔB≥21​∣⟨[A,B]⟩∣ Δ A Δ B ≥ 1 2 [ A , B ] than the textbook proof. I discuss the relationship between these two proofs and show how the Cauchy–Schwarz inequality can be derived from the Aharonov–Vaidman identity. I give Aharonov–Vaidman based proofs of the Maccone–Pati uncertainty relations and show how the Aharonov–Vaidman identity can be used to handle propagation of uncertainty in quantum mechanics. Finally, I show how the Aharonov–Vaidman identity can be extended to mixed states and discuss how to generalize the results to the mixed case.
A existência de observáveis incompatíveis em mecânica quântica (MQ) é uma das suas características mais notáveis e pode ser revelada e formalizada através de relações de incerteza. A relação de incerteza de Heisenberg-Robertson-Schrödinger (RIHRS) foi provada nos primórdios do formalismo quântico e está unipresente tanto no ensino quanto na pesquisa em MQ. Todavia, a RIHRS possui o chamado problema da trivialidade, ou seja, ela não fornece informação alguma sobre a possível incompatibilidade entre dois observáveis se o estado no qual o sistema foi preparado for autovetor de um deles. Este problema foi resolvido recentemente por Lorenzo Maccone e Arun K. Pati, depois de aproximadamente 85 anos de existência da RIHRS. Neste artigo começamos fazendo uma breve discussão sobre aspectos gerais do princípio de incerteza em MQ e recapitulando a prova da RIHRS. Na sequência apresentamos de forma simples a prova da relação de incerteza de Maccone-Pati, que pode ser obtida usando basicamente a lei dos paralelogramos e a desigualdade de Cauchy-Schwarz.
In quantum mechanics, the variance-based Heisenberg-type uncertainty relations are a series of mathematical inequalities posing the fundamental limits on the achievable accuracy of the state preparations. In contrast, we construct and formulate two quantum uncertainty equalities, which hold for all pairs of incompatible observables and indicate the new uncertainty relations recently introduced by L. Maccone and A. K. Pati [Phys. Rev. Lett. 113, 260401 (2014)]. In fact, we obtain a series of inequalities with hierarchical structure, including the Maccone-Pati's inequalities as a special (weakest) case. Furthermore, we present an explicit interpretation lying behind the derivations and relate these relations to the so-called intelligent states. As an illustration, we investigate the properties of these uncertainty inequalities in the qubit system and a state-independent bound is obtained for the sum of variances. Finally, we apply these inequalities to the spin squeezing scenario and its implication in interferometric sensitivity is also discussed.