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arXiv:1210.6043 · Physical Review A

Uncertainty relations and indistinguishable particles

Cael L. Hasse

Published Dec 4, 2012• 1 citation
arXivHTMLPDF

Abstract

We show that for fermion states, measurements of any two finite outcome particle quantum numbers (e.g., spin) are not constrained by a minimum total uncertainty. We begin by defining uncertainties in terms of the outputs of a measurement apparatus. This allows us to compare uncertainties between multiparticle states of distinguishable and indistinguishable particles. Entropic uncertainty relations are derived for both distinguishable and indistinguishable particles. We then derive upper bounds on the minimum total uncertainty for bosons and fermions. These upper bounds apply to any pair of particle quantum numbers and depend only on the number of particles $N$ and the number of outcomes $n$ for the quantum numbers. For general $N$, these upper bounds necessitate a minimum total uncertainty much lower than that for distinguishable particles. The fermion upper bound on the minimum total uncertainty for $N$ an integer multiple of $n$ is zero. Our results show that uncertainty limits derived for single particle observables are valid only for particles that can be effectively distinguished. Outside this range of validity, the apparent fundamental uncertainty limits can be overcome.

DOI
10.1103/physreva.86.062101

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We show that for fermion states, measurements of any two finite outcome particle quantum numbers (e.g., spin) are not constrained by a minimum total uncertainty. We begin by defining uncertainties in terms of the outputs of a measurement apparatus. This allows us to compare uncertainties between multiparticle states of distinguishable and indistinguishable particles. Entropic uncertainty relations are derived for both distinguishable and indistinguishable particles. We then derive upper bounds on the minimum total uncertainty for bosons and fermions. These upper bounds apply to any pair of particle quantum numbers and depend only on the number of particles NNN and the number of outcomes nnn for the quantum numbers. For general NNN, these upper bounds necessitate a minimum total uncertainty much lower than that for distinguishable particles. The fermion upper bound on the minimum total uncertainty for NNN an integer multiple of nnn is zero. Our results show that uncertainty limits derived for single particle observables are valid only for particles that can be effectively distinguished. Outside this range of validity, the apparent fundamental uncertainty limits can be overcome.
Learning physical properties of a quantum system is essential for the developments of quantum technologies. However, Heisenberg’s uncertainty principle constrains the potential knowledge one can simultaneously have about a system in quantum theory. Aside from its fundamental significance, the mathematical characterization of this restriction, known as ‘uncertainty relation’, plays important roles in a wide range of applications, stimulating the formation of tighter uncertainty relations. In this work, we investigate the fundamental limitations of variance-based uncertainty relations, and introduce several ‘near optimal’ bounds for incompatible observables. Our results consist of two morphologically distinct phases: lower bounds that illustrate the uncertainties about measurement outcomes, and the upper bound that indicates the potential knowledge we can gain. Combining them together leads to an uncertainty interval, which captures the essence of uncertainties in quantum theory. Finally, we have detailed how to formulate lower bounds for product-form variance-based uncertainty relations by employing entropic uncertainty relations, and hence built a link between different forms of uncertainty relations.
The uncertainty principle is a fundamental principle in quantum physics. It implies that the measurement outcomes of two incompatible observables cannot be predicted simultaneously. In quantum information theory, this principle can be expressed in terms of entropic measures. M. Berta et al. [Nat. Phys. 6, 659 (2010)] have indicated that uncertainty bound can be altered by considering a particle as a quantum memory correlating with the primary particle. In this article, we obtain a lower bound for entropic uncertainty in the presence of a quantum memory by adding an additional term depending on the Holevo quantity and mutual information. We conclude that our lower bound will be tightened with respect to that of Berta et al. when the accessible information about measurements outcomes is less than the mutual information about the joint state. Some examples have been investigated for which our lower bound is tighter than Berta et al.'s lower bound. Using our lower bound, a lower bound for the entanglement of formation of bipartite quantum states has been obtained, as well as an upper bound for the regularized distillable common randomness.
Quantum measurements are inherently probabilistic and quantum theory often forbids precisely predicting the outcomes of simultaneous measurements. This phenomenon is captured and quantified through uncertainty relations. Although studied since the inception of quantum theory, the problem of determining the possible expectation values of a collection of quantum measurements remains, in general, unsolved. By constructing a close connection between observables and graph theory, we derive uncertainty relations valid for any set of dichotomic observables. For arbitrary observables, we obtain an upper bound on the sum of the square of their expectation values. We furthermore show how this bound behaves when the observables are imprecisely calibrated. As applications, our results can be straightforwardly used to formulate entropic uncertainty relations, separability criteria, and entanglement witnesses.
The well-known Robertson–Schrödinger uncertainty relations have state-dependent lower bounds, which are trivial for certain states. We present a general approach to deriving tight state-independent uncertainty relations for qubit measurements that completely characterise the obtainable uncertainty values. This approach can give such relations for any number of observables, and we do so explicitly for arbitrary pairs and triples of qubit measurements. We show how these relations can be transformed into equivalent tight entropic uncertainty relations. More generally, they can be expressed in terms of any measure of uncertainty that can be written as a function of the expectation value of the observable for a given state.
How much of the uncertainty in predicting measurement outcomes for noncommuting quantum observables is genuinely quantum mechanical? We provide a natural decomposition of the total entropic uncertainty of two noncommuting observables into a classical component and an intrinsically quantum mechanical component. We show that the total quantum component in a state is never lower or upper bounded by any state-independent quantities, but instead admits ``purity-based'' lower bounds that generalize entropic formulations such as the Maassen-Uffink relation. These relations reveal a nontrivial interplay between quantum and classical randomness in any finite-dimensional state.
The well-known Robertson–Schrödinger uncertainty relations have state-dependent lower bounds, which are trivial for certain states. We present a general approach to deriving tight state-independent uncertainty relations for qubit measurements that completely characterise the obtainable uncertainty values. This approach can give such relations for any number of observables, and we do so explicitly for arbitrary pairs and triples of qubit measurements. We show how these relations can be transformed into equivalent tight entropic uncertainty relations. More generally, they can be expressed in terms of any measure of uncertainty that can be written as a function of the expectation value of the observable for a given state.