Towards Establishing a Connection Between Two-Level Quantum Systems and Physical Spaces

Abstract

This work seeks to make explicit the operational connection between the preparation of two-level quantum systems with their corresponding description (as states) in a Hilbert space. This may sound outdated, but we show there is more to this connection than common sense may lead us to believe. To bridge these two separated realms—the actual laboratory and the space of states—we rely on a paradigmatic mathematical object: the Hopf fibration. We illustrate how this connection works in practice with a simple optical setup. Remarkably, this optical setup also reflects the necessity of using two charts to cover a sphere. Put another way, our experimental realization reflects the bi-dimensionality of a sphere seen as a smooth manifold.

Appendix 1. Basic Set Theory and the Action of Groups on Arbitrary Sets

In this appendix, we briefly introduce a selection of topics concerning set theory: equivalence relations, equivalence classes, the action of a group on a set and related concepts. Although the points we address here are not new, our intention is to make the manuscript as self-consistent as possible. For an in-depth description, we direct the reader to [23, 40].

We start our overview by reviewing an alternative way of turning elements of a set into equivalent elements. Presumably, the first and foremost notion of equivalence is the one commonly connected with the concept of equality. Disguised by common sense, equality is a binary relation-a relation between two objects-which is reflexive (an object is equal to itself), symmetric (if an object is equal to another, then the latter is also equal to the former), and transitive (if an object is equal to a second, and this second to a third, then the original and the final objects also are so). What is crucial here is that equality, as we know it, is only a special case of a larger class of binary relations intended to capture and classify distinctive characteristics of a collection of objects. The next definitions rigorously address these points.

Definition 1

(Equivalence Relations). Let $\mathbb {X}$ be a non-empty set and $\sim \subset \mathbb {X} \times \mathbb {X}$ a relation. We say that $\sim$ is an equivalence relation whenever the three properties below hold true,

• i. $\sim$ is reflexive: $x \sim x$, $\forall x \in \mathbb {X}$.

• ii. $\sim$ is symmetric: $x \sim y \Rightarrow y \sim x$, $\forall x,y \in \mathbb {X}$

• iii. $\sim$ is transitive: if $x \sim y$ and $y \sim z$, then $x \sim z$, $\forall x,y,z \in \mathbb {X}$.

Remark

Any relation over a given set $\mathbb {X}$ is a subset R of $\mathbb {X} \times \mathbb {X}$. In this sense, R is a collection of ordered pairs (x, y) where $x,y \in \mathbb {X}$. Mainly because we want to express the relational aspect of R and also retain the parallel with the notion of equality, it is usual to write xRy instead of $(x,y) \in R$-the reader will certainly appreciate this change in notation.

Definition 2

(Equivalence Classes). Let $\sim$ be an equivalence relation over $\mathbb {X}$. Given an element y we define the following subsets of $\mathbb {X}$,

$$\begin{aligned}{}[y] := \{x\in \mathbb {X} \, \vert \, x \sim y \}. \end{aligned}$$

(27)

Because $y \sim y$, $y \in [y]$, for all elements of $\mathbb {X}$. Thus, these subsets are well-defined and we name them equivalence classes. In particular, for a given $y \in \mathbb {X}$, the subset [y] is called the equivalence class of y.

One of the main consequences of establishing an equivalence relation on a set resides in the theorem below. It guarantees that equivalence classes provide a partition of the set they are defined over—that is, a cover of the entire set formed by disjoint subsets.

Theorem 1

(Partition via Equivalence Classes). If $\sim$ is an equivalence relation on $\mathbb {X}$, then

$$\begin{aligned} 1) \, x \sim y \Rightarrow [x] = [y], \end{aligned}$$

(28)

$$\begin{aligned} 2) \, x \not \sim y \Rightarrow [x] \cap [y] = \emptyset , \end{aligned}$$

(29)

$$\begin{aligned} 3)\, \mathbb {X} = \bigcup _{x \in \mathbb {X}} [x]. \end{aligned}$$

(30)

Remark

A useful notation to the disjoint union in Eq. (30) is $\mathbb {X}/\sim$, as it denotes that the set has been partitioned across the many equivalence classes defined by the equivalence relation. For both an intuitive interpretation and formal demonstration, we direct the reader to [14].

Now we turn our attention to group actions on sets. Following the natural steps of introducing sets and relations, we could start with functions (or mappings), which are a special type of relation. However, we would like to do so bearing in mind that the elements of the set should transform guided by a group structure, retaining not only the symmetry properties but also the special transformations the latter usually conveys. Hence, we define the

Definition 3

(Action of a Group). Let $\mathcal {G}$ be a group and $\mathbb {X}$ be a non-empty set. The action of $\mathcal {G}$ on $\mathbb {X}$ is a map $\varphi : \mathcal {G} \times \mathbb {X} \rightarrow \mathbb {X}$, satisfying the following conditions,

• i. $\varphi (e,x) = x$, $\forall \,\, x \in \mathbb {X}$, where e stands for the group identity element.

• ii. For each $g \in \mathcal {G}$, $\varphi (g, \cdot ): \mathbb {X} \rightarrow \mathbb {X}$ is a bijection.

• iii. $\varphi (g_1, \varphi (g_2,x))=\varphi (g_1g_2,x)$, $\forall \,\, x \in \mathbb {x}$ and $\forall \,\, g_1, g_2 \in \mathcal {G}$.

Now, in $\mathbb {X}\times \mathbb {X}$ we introduce the following relation,

$$\begin{aligned} x \sim y \Leftrightarrow y = \varphi (g,x), \text{ for } \text{ some } g \in \mathcal {G}. \end{aligned}$$

(31)

We affirm that $\sim$ is an equivalence relation. In fact, $\sim$ is reflexive because i: $\varphi (e,x) = x$. The symmetry comes from

$$\begin{aligned} y = \varphi (g,x) \Rightarrow x = \varphi (g^{-1},y). \end{aligned}$$

Finally, the transitivity can also be promptly deduced. It suffices to note that if $y = \varphi (g_1,x)$ and $z = \varphi (g_2,y)$, then $z = \varphi (g_3,x)$, where $g_3 = g_2 g_1$.

With the equivalence relation defined by the action, we define the equivalence classes by

$$\begin{aligned}{}[x] := \{ y \in \mathbb {X} \, \vert \, y = \varphi (g,x) \text{ for } \text{ some } g \in \mathcal {G} \}. \end{aligned}$$

(32)

Due to its special role, an equivalence class of such type is called an orbit through x.

Crucially, according to the theorem 1 above, $\mathbb {X}$ is partitioned by its disjoint orbits. All in all, that is to say, starting from the action of a group over a set, we can partition that set across the orbits of each of its elements.

Next, we will present some examples of interest not only to our previous discussion per se but also to quantum mechanics in general.

Example 1

For our first example, we consider $\mathbb {X} = \mathbb {C}^2$ and $\mathcal {G} = \mathcal {U}(1) = \{ e^{i \alpha } \, \, \alpha \in \mathbb {R} \}$. Define the map

$$\begin{aligned} \varphi : \mathcal {U}(1) \times \mathbb {C}^2&\rightarrow \mathbb {C}^2 \\ (e^{i\alpha }, | \psi \rangle )&\mapsto | \psi \rangle ' = e^{i \alpha }| \psi \rangle . \end{aligned}$$

To check that $\varphi$ is indeed an action, we have

• i. $\mathcal {U}(1) \ni e =1 \Rightarrow \varphi (e, | \psi \rangle ) = | \psi \rangle$.

• ii. Given $\alpha \in \mathbb {R}$, $\varphi (e^{i \alpha }, \cdot ): \mathbb {C}^2 \rightarrow \mathbb {C}^2$ is a bijection. In effect,

  $$e^{i \alpha } | \psi _1 \rangle = e^{i \alpha } | \psi _2 \rangle \Rightarrow | \psi _1 \rangle = | \psi _2 \rangle$$

  which guarantees that the map is injective. In turn, we point out that any $| \psi \rangle \in \mathbb {C}^2$ is reached by $e^{-i\alpha }| \psi \rangle$ under $\varphi (e^{i \alpha }, \cdot )$. Thus, the map is surjective as well.

• iii. At last,

  $$\begin{aligned} \varphi (e^{i \alpha _1},e^{i \alpha _2} | \psi \rangle ) = e^{i(\alpha _1+ \alpha _2)}| \psi \rangle = \varphi (e^{i \alpha _1}e^{i \alpha _2}, | \psi \rangle ), \end{aligned}$$

  which concludes the proof.

The importance of the case in point lies on the fact that the orbits consists of indistinguishable states for two-level quantum systems as well as fibers in the Hopf fibration, as previously discussed in Section 3.

Example 2

Our second example is given by $\mathbb {X}= \mathbb {R}^n$ and $\mathcal {G} = \mathbb {R}^* = \mathbb {R}\smallsetminus \{0 \}$, where the group product is the usual multiplication of non-zero real numbers. Define

$$\begin{aligned} \varphi :\mathbb {R}^* \times \mathbb {R}^n&\rightarrow \mathbb {R}^n \\ (\lambda , \vec {v} )&\mapsto \lambda \vec {v}. \end{aligned}$$

It is not difficult to conclude that $\varphi$ is an action. In fact, we have,

• i. $\mathbb {R}^* \ni e = 1 \Rightarrow \varphi (1,\vec {v}) = \vec {v}$.

• ii. Given $\lambda \in \mathbb {R}^*$, $\varphi (\lambda , \cdot )$ is a bijection for

  $$\begin{aligned} \lambda \vec {v}_1=\lambda \vec {v}_2 \Rightarrow \vec {v}_1=\vec {v}_2, \end{aligned}$$

  which shows the injection. Moreover, any $\vec {v} \in \mathbb {R}^n$ can be obtained by applying $\varphi (\lambda , \cdot )$ to $\lambda ^{-1}\vec {v}$. Hence, the map is also surjective.

• iii. Finally, $\varphi (\lambda _1, \lambda _2 \vec {v}) = \varphi (\lambda _1 \lambda _2 \vec {v})$.

The classes, or orbits, here are straight lines crossing the origin, although $\vec {0} \ne [\vec {V}]$, for every non-zero vector $\vec {v}$ in $\mathbb {R}^n$. Their union forms what is called the projective space, named $\mathbb {R}\mathbb {P}^{n-1}$. The generalization for $\mathbb {C}^n$ and, accordingly, to $\mathbb {C}\mathbb {P}^{n-1}$ is straightforward. The value of the latter to quantum mechanics stems from its intrinsic connection to entanglement [50].