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Kluwer Academic Publishers eBooks

Information and Distance in Hilbert Space

Lev B. Levitin, Tom Toffoli, Zac Walton

Published Feb 24, 2006• 1 citation
Publisher

Abstract

The main result of the paper can be summarized as follows. The number of distinguishable quantum states in a 2-dimensional Hilbert space is proportional to the square root of the number of identical copies of each state measured and to the total length of the angle intervals occupied by the state vectors. Surprisingly, it does not depend on the position of the arcs comprising the range on the unit circle. These results can be generalized for the N-dimensional Hilbert space of states of a quantum system.As in the 2-dimensional case, the unit sphere can be reduced to the non-negative orthant of the unit sphere in the real N-dimensional Euclidean space.It turns out [6] that the number of distinguishable states depends only on the area Ω of the domain on the unit sphere from which the states can be chosen, but does not depend on the shape and position of this domain. The optimal distribution is uniform over the domain in angular (polar) coordinates, and the number of distinguishable states is $$ W(n,\Omega ) = \Omega (cn)^{\tfrac{{N - 1}} {2}} $$ where c is a constant.

DOI
10.1007/0-306-47114-0_4

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The main result of the paper can be summarized as follows. The number of distinguishable quantum states in a 2-dimensional Hilbert space is proportional to the square root of the number of identical copies of each state measured and to the total length of the angle intervals occupied by the state vectors. Surprisingly, it does not depend on the position of the arcs comprising the range on the unit circle. These results can be generalized for the N-dimensional Hilbert space of states of a quantum system.As in the 2-dimensional case, the unit sphere can be reduced to the non-negative orthant of the unit sphere in the real N-dimensional Euclidean space.It turns out [6] that the number of distinguishable states depends only on the area Ω of the domain on the unit sphere from which the states can be chosen, but does not depend on the shape and position of this domain. The optimal distribution is uniform over the domain in angular (polar) coordinates, and the number of distinguishable states is W(n,Ω)=Ω(cn)N−12W(n,\Omega ) = \Omega (cn)^{\tfrac{{N - 1}} {2}}W(n,Ω)=Ω(cn)2N−1​ where c is a constant.
Understanding the geometry of state space is of fundamental interest in the field of quantum information, in particular to study entanglement. The states of a d dimensional system form a convex set in the d2−1 dimensional Euclidean space Rd2−1. For a two dimensional system this convex set is nothing but a solid unit sphere (Bloch sphere) with centre at the origin. The surface S of this unit sphere consists of all the pure states of the system. Every unitary transformation U ∈ SU(2) of the Hilbert space corresponds to an associated SO(3) rotation of this space. This kind of correspondence becomes much richer when we go to higher dimension. For example, in the d = 3 case the corresponding unitary group is the eight parameter group SU(3) whereas the rotation group in R is the 28 parameter group SO(8). Therefore, not every point on or inside the sphere S in R will correspond to a state of the system. So, not every rotation in R will correspond to a valid unitary transformation of the 3 dimensional Hilbert space. This gives us a fairly complex convex object as the set of all states. In order to gain insight into the structure of this state space, we study its two dimensional and three dimensional cross-sections. In particular, we describe in detail the discrete symmetries of a non-trivial 3-section of the convex set of states (density matrices) for three dimensional systems. This group turns out to be the same as that of a tetrahedron.
-dimensional Hilbert space, suppose a pure state |ψ⟩ is subjected to a complete orthogonal measurement. The measurement effectively maps |ψ⟩ to a point (p1,…,pd) in the appropriate probability simplex. It is a known fact-which depends crucially on the complex nature of the system's Hilbert space-that if |ψ⟩ is distributed uniformly over the unit sphere, then the resulting ordered set (p1,…,pd) is distributed uniformly over the probability simplex; that is, the resulting measure on the simplex is proportional to dp1⋯dpd-1. In this paper we ask whether there is some foundational significance to this uniform measure. In particular, we ask whether it is the optimal measure for the transmission of information from a preparation to a measurement in some suitably defined scenario. We identify a scenario in which this is indeed the case, but our results suggest that an underlying real-Hilbert-space structure would be needed to realize the optimization in a natural way.
Consider a quantum system prepared in state ψ, a unit vector in a d -dimensional Hilbert space. Let b 1 , ..., b d be an orthonormal basis and suppose that, with some probability 0 < p < 1, ψ 'collapses', i.e., gets replaced by b k (possibly times a phase factor) with Born's probability |〈 b k |ψ〉| 2 . The question we investigate is: how well can any quantum experiment on the system determine afterwards whether a collapse has occurred? The answer depends on how much is known about the initial vector ψ. We provide a number of different results addressing several variants of the question. In each case, no experiment can provide more than rather limited probabilistic information. In case ψ is drawn randomly with uniform distribution over the unit sphere in Hilbert space, no experiment performs better than a blind guess without measurement; that is, no experiment provides any useful information.
Discriminating between quantum states and channels is an important task in the field of quantum computation and quantum information theory. Metrics and statistical distances form the tools for various approaches proposed for this task. We analyse existing results to compare properties, main purpose, computability and advantages and disadvantages of each approach.This article is part of the theme issue 'Numerical analysis, spectral graph theory, orthogonal polynomials and quantum algorithms'.
We derive the two-valuedness and the angular momentum of a spin-1/2 from a rotation of 3-dimensional surface of a sphere existing in extra 4-dimensional space other than normal 3-dimensional space, in this paper. We will derive the two-valuedness of the spin as follows. We introduce 3-dimensional surface of a sphere S3 existing in extra 4-dimensional space (W, X, Y, Z) other than normal 3-dimensional space (x, y, z). We interpret the angle of rotation of the 3-sphere S3 as the phase of a wave function. We interpret the 3-sphere S3 as the absolute value of a wave function. We can express 3-sphere as the manifold with a constant sum of squares of the radius of two circles. When one circle's radius becomes the maximum, the other circle's radius becomes zero. Therefore, we can turn the circle inside out naturally. If we combine the circle turned inside out with the original circle, the manifold becomes a torus with a node. If we rotate the node of the torus by 360 degrees, we can turn the torus inside out. If we rotate the node of the torus 720 degrees, we can return the torus to the original state. This property is consistent with the property of the spin. We derive the angular momentum of the spin as follows. We make 3-dimensional solid sphere by removing one point from 3-sphere S3. On the other hand, we can make boundary like a 3-sphere S3 by removing one point from normal 3-dimensional space (x, y, z). We combine the boundaries of them. By repeating this, we can construct 3-dimensional helical space. The angle of rotation of the 3-sphere S3 is the angle of rotation of 3-dimensional helical space. On the other hand, we can interpret the angle of the rotation in the helical space as the coordinates (x, y, z) of the normal 3-dimensional space. Therefore, we can interpret the angular momentum of the 3-sphere S3 as the angular momentum of normal space.