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arXiv:1209.1225 · Journal of Mathematical Physics

On Pauli pairs

Stanislav Shkarin

Published Mar 1, 2013• 1 citation
arXivHTMLPDF

Abstract

The state of a system in classical mechanics can be uniquely reconstructed if we know the positions and the momenta of all its parts. In 1958 Pauli has conjectured that the same holds for quantum mechanical systems. The conjecture turned out to be wrong. In this paper we provide a new set of examples of Pauli pairs, being the pairs of quantum states indistinguishable by measuring the spatial location and momentum. In particular, we construct a new set of spatially localized Pauli pairs.

DOI
10.1063/1.4794087

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The state of a system in classical mechanics can be uniquely reconstructed if we know the positions and the momenta of all its parts. In 1958 Pauli has conjectured that the same holds for quantum mechanical systems. The conjecture turned out to be wrong. In this paper we provide a new set of examples of Pauli pairs, being the pairs of quantum states indistinguishable by measuring the spatial location and momentum. In particular, we construct a new set of spatially localized Pauli pairs.
In classical mechanics, we can uniquely reconstruct the state of each particle by measuring its spatial location and momentum. In his 1958 paper, W. Pauli, one of the founders of quantum mechanics, conjectured that the same should be true in the quantum case as well: that every quantum state can be uniquely determined by measuring spatial location and momentum. This conjecture was disproven: there are pairs of physically different states that cannot be distinguished if we only measure special location and momentum. A natural question is: how frequent are such pairs? In this paper, we show that almost all quantum states can be uniquely reconstructed by measuring spatial location and momentum. Thus, in practice, Pauli's conjecture is true.
In this paper we consider the problem to determine a unique quantum state from given distribution of position (density) and momentum density of particles, namely the so called Pauli problem in quantum physics. In the first part, we will review the method of wave function lifting developed in [ 4 , 5 ] to construct a complex wave function $ \psi\in H^s({{\mathbf R}}^d) $, $ s = 1, 2 $, associated to given density and momentum density. The second part is focused on the dynamical version of the wave function lifting, namely we study the relation between solutions to quantum fluid models and wave functions solving the nonlinear Schrödinger equation. The uniqueness of the lifted wave function is essentially related to the structure of vacuum regions of the position density.
In experiments it is often required that we determine an unknown quantum state produced by a source. Quantum Tomography accomplishes this through repeated measurements on the state. This process requires 4^n measurements for an n-qubit system. However, if it is known that the state is pure, fewer measurements are required. Many such results already exist, but they either use non-local measurements or they have some chance of failing. Our goal is to obtain the smallest set of Pauli operators which can uniquely determine any pure state among all states. This involves ensuring the complement of the span of the measurements must have two positive and two negative eigenvalues. This is nontrivial since there are few relationships between the eigenvalues of a set of operators and the eigenvalues their real linear combinations. We obtain the lower bound of 30 Paulis necessary for 3-qubit tomography which improves on any known result.
A complete description of the quantum state of a physical system is the fundamental knowledge necessary to statistically predict the outcome of measurements. In turning this statement around, Wolfgang Pauli raised already in 1933 the question, whether an unknown quantum state could be uniquely determined by appropriate measurements--a problem that has gained new relevance in recent years. In order to harness the prospects of quantum computing, secure communication, teleportation, and the like, the development of techniques to accurately control and measure quantum states has now become a matter of practical as well as fundamental interest. However, there is no general answer to Pauli's very basic question, and quantum state reconstruction algorithms have been developed and experimentally demonstrated only for a few systems so far. This thesis presents a novel experimental method to measure the unknown and generally mixed quantum state for an angular momentum of arbitrary magnitude. The (2F + 1) x (2F + 1) density matrix describing the quantum state is hereby completely determined from a set of Stern-Gerlach measurements with (4F + 1) different orientations of the quantization axis. This protocol is implemented for laser cooled Cesium atoms in the 6S₁/₂(F = 4) hyperfine ground state manifold, and is applied to a number of test states prepared by optical pumping and Larmor precession. A comparison of the input and the measured states shows successful reconstructions with fidelities of about 0.95.
Identical quantum subsystems can possess a property which does not have any classical counterpart: indistinguishability. As a long-debated phenomenon, identical particles’ indistinguishability has been shown to be at the heart of various fundamental physical results. When concerned with the spatial degree of freedom, identical constituents can be made indistinguishable by overlapping their spatial wave functions via appropriately defined spatial deformations . By the laws of quantum mechanics, any measurement designed to resolve a quantity which depends on the spatial degree of freedom only and performed on the regions of overlap is not able to assign the measured outcome to one specific particle within the system. The result is an entangled state where the measured property is shared between the identical constituents. In this work, we present a coherent formalization of the concept of deformation in a general N -particle scenario, together with a suitable measure of the degree of indistinguishability. We highlight the basic differences with non-identical particles scenarios and discuss the inherent role of spatial deformations as entanglement activators within the spatially localized operations and classical communication operational framework. This article is part of the theme issue ‘Identity, individuality and indistinguishability in physics and mathematics’.
We define and study the notion of quantum polarity, which is a kind of geometric Fourier transform between sets of positions and sets of momenta. Extending previous work of ours, we show that the orthogonal projections of the covariance ellipsoid of a quantum state on the configuration and momentum spaces form what we call a dual quantum pair. We thereafter show that quantum polarity allows solving the Pauli reconstruction problem for Gaussian wavefunctions. The notion of quantum polarity exhibits a strong interplay between the uncertainty principle and symplectic and convex geometry and our approach could therefore pave the way for a geometric and topological version of quantum indeterminacy. We relate our results to the Blaschke-Santaló inequality and to the Mahler conjecture. We also discuss the Hardy uncertainty principle and the less-known Donoho-Stark principle from the point of view of quantum polarity.