Bloch sphere

Given an orthonormal basis, any pure state ${\displaystyle |\psi \rangle }$ of a two-level quantum system can be written as a superposition of the basis vectors ${\displaystyle |0\rangle }$ and ${\displaystyle |1\rangle }$, where the coefficient or amount of each of the two basis vector is a complex number. This means that the state is described by four real numbers. However only the relative phase between the coefficients of the two basis vectors has any physical meaning, so that there is redundancy in this description. We can take the coefficient of ${\displaystyle |0\rangle }$ to be real and non-negative. This allows the state to be described by only three real numbers, giving rise to the three dimensions of the Bloch sphere.

We also know from quantum mechanics that the total probability of the system has to be one:

${\displaystyle \langle \psi |\psi \rangle =1}$, or equivalently ${\displaystyle {\big \|}|\psi \rangle {\big \|}^{2}=1}$.

Given this constraint, we can write ${\displaystyle |\psi \rangle }$ using the following representation:

${\displaystyle |\psi \rangle =\cos \left(\theta /2\right)|0\rangle \,+\,e^{i\phi }\sin \left(\theta /2\right)|1\rangle =\cos \left(\theta /2\right)|0\rangle \,+\,(\cos \phi +i\sin \phi )\,\sin \left(\theta /2\right)|1\rangle }$, where ${\displaystyle 0\leq \theta \leq \pi }$ and ${\displaystyle 0\leq \phi <2\pi }$.

The representation is always unique, because, even though value of ${\displaystyle \phi }$ is not unique when ${\displaystyle |\psi \rangle }$ is one of the ket vectors (see Bra-ket notation) ${\displaystyle |0\rangle }$ or ${\displaystyle |1\rangle }$, the point represented by ${\displaystyle \theta }$ and ${\displaystyle \phi }$ is unique.

The parameters ${\displaystyle \theta \,}$ and ${\displaystyle \phi \,}$, re-interpreted in spherical coordinates as respectively the colatitude with respect to the z-axis and the longitude with respect to the x-axis, specify a point

${\displaystyle {\vec {a}}=(\sin \theta \cos \phi ,\;\sin \theta \sin \phi ,\;\cos \theta )=(u,v,w)}$

on the unit sphere in ${\displaystyle \mathbb {R} ^{3}}$.

For mixed states, one considers the density operator. Any two-dimensional density operator ρ can be expanded using the identity I and the Hermitian, traceless Pauli matrices ${\displaystyle {\vec {\sigma }}}$,

${\displaystyle {\begin{aligned}\rho &={\frac {1}{2}}\left(I+{\vec {a}}\cdot {\vec {\sigma }}\right)\\&={\frac {1}{2}}{\begin{pmatrix}1&0\\0&1\end{pmatrix}}+{\frac {a_{x}}{2}}{\begin{pmatrix}0&1\\1&0\end{pmatrix}}+{\frac {a_{y}}{2}}{\begin{pmatrix}0&-i\\i&0\end{pmatrix}}+{\frac {a_{z}}{2}}{\begin{pmatrix}1&0\\0&-1\end{pmatrix}}\\&={\frac {1}{2}}{\begin{pmatrix}1+a_{z}&a_{x}-ia_{y}\\a_{x}+ia_{y}&1-a_{z}\end{pmatrix}}\end{aligned}}}$,

where ${\displaystyle {\vec {a}}\in \mathbb {R} ^{3}}$ is called the Bloch vector.

It is this vector that indicates the point within the sphere that corresponds to a given mixed state. Specifically, as a basic feature of the Pauli vector, the eigenvalues of ρ are ${\displaystyle {\frac {1}{2}}\left(1\pm |{\vec {a}}|\right)}$. Density operators must be positive-semidefinite, so it follows that ${\displaystyle \left|{\vec {a}}\right|\leq 1}$.

For pure states, one then has

${\displaystyle \operatorname {tr} \left(\rho ^{2}\right)={\frac {1}{2}}\left(1+\left|{\vec {a}}\right|^{2}\right)=1\quad \Leftrightarrow \quad \left|{\vec {a}}\right|=1~,}$

in comportance with the above.[5]

As a consequence, the surface of the Bloch sphere represents all the pure states of a two-dimensional quantum system, whereas the interior corresponds to all the mixed states.

The Bloch vector ${\displaystyle {\vec {a}}=(u,v,w)}$ can be represented in the following basis, with reference to the density operator ${\displaystyle \rho }$:[6]

${\displaystyle u=\rho _{10}+\rho _{01}=2\operatorname {Re} (\rho _{01})}$

${\displaystyle v=i(\rho _{01}-\rho _{10})=2\operatorname {Im} (\rho _{10})}$

${\displaystyle w=\rho _{00}-\rho _{11}}$

where

${\displaystyle \rho ={\begin{pmatrix}\rho _{00}&\rho _{01}\\\rho _{10}&\rho _{11}\end{pmatrix}}={\frac {1}{2}}{\begin{pmatrix}1+w&u-iv\\u+iv&1-w\end{pmatrix}}.}$

This basis is often used in laser theory, where ${\displaystyle w}$ is known as the population inversion.[7] In this basis, the numbers ${\displaystyle u,v,w}$ are the expectations of the three Pauli matrices ${\displaystyle X,Y,Z}$, allowing one to identify the three coordinates with x y and z axes.

Consider an n-level quantum mechanical system. This system is described by an n-dimensional Hilbert space H_n. The pure state space is by definition the set of 1-dimensional rays of H_n.

Theorem. Let U(n) be the Lie group of unitary matrices of size n. Then the pure state space of H_n can be identified with the compact coset space

${\displaystyle \operatorname {U} (n)/(\operatorname {U} (n-1)\times \operatorname {U} (1)).}$

To prove this fact, note that there is a natural group action of U(n) on the set of states of H_n. This action is continuous and transitive on the pure states. For any state ${\displaystyle |\psi \rangle }$, the isotropy group of ${\displaystyle |\psi \rangle }$, (defined as the set of elements ${\displaystyle g}$ of U(n) such that ${\displaystyle g|\psi \rangle =|\psi \rangle }$) is isomorphic to the product group

${\displaystyle \operatorname {U} (n-1)\times \operatorname {U} (1).}$

In linear algebra terms, this can be justified as follows. Any ${\displaystyle g}$ of U(n) that leaves ${\displaystyle |\psi \rangle }$ invariant must have ${\displaystyle |\psi \rangle }$ as an eigenvector. Since the corresponding eigenvalue must be a complex number of modulus 1, this gives the U(1) factor of the isotropy group. The other part of the isotropy group is parametrized by the unitary matrices on the orthogonal complement of ${\displaystyle |\psi \rangle }$, which is isomorphic to U(n − 1). From this the assertion of the theorem follows from basic facts about transitive group actions of compact groups.

The important fact to note above is that the unitary group acts transitively on pure states.

Now the (real) dimension of U(n) is n². This is easy to see since the exponential map

${\displaystyle A\mapsto e^{iA}}$

is a local homeomorphism from the space of self-adjoint complex matrices to U(n). The space of self-adjoint complex matrices has real dimension n².

Corollary. The real dimension of the pure state space of H_n is 2n − 2.

In fact,

${\displaystyle n^{2}-\left((n-1)^{2}+1\right)=2n-2.\quad }$

Let us apply this to consider the real dimension of an m qubit quantum register. The corresponding Hilbert space has dimension 2^m.

Corollary. The real dimension of the pure state space of an m-qubit quantum register is 2^m+1 − 2.

Bloch sphere centered at the origin of ${\displaystyle \mathbb {R} ^{3}}$. A pair of points on it, ${\displaystyle \left|\uparrow \right\rangle }$ and ${\displaystyle \left|\downarrow \right\rangle }$ have been chosen as a basis. Mathematically they are orthogonal even though graphically the angle between them is π. In ${\displaystyle \mathbb {R} ^{3}}$ those points have coordinates (0,0,1) and (0,0,−1). An arbitrary spinor ${\displaystyle \left|\nearrow \right\rangle }$ on the Bloch sphere is representable as a unique linear combination of the two basis spinors, with coefficients being a pair of complex numbers; call them α and β. Let their ratio be ${\displaystyle u={\beta \over \alpha }}$, which is also a complex number ${\displaystyle u_{x}+iu_{y}}$. Consider the plane z = 0, the equatorial plane of the sphere, as it were, to be a complex plane and that the point u is plotted on it as ${\displaystyle (u_{x},u_{y},0)}$. Project point u stereographically onto the Bloch sphere away from the South Pole — as it were — (0,0,−1). The projection is onto a point marked on the sphere as ${\displaystyle \left|\nearrow \right\rangle }$.

Given a pure state

${\displaystyle \alpha \left|\uparrow \right\rangle +\beta \left|\downarrow \right\rangle =\left|\nearrow \right\rangle }$

where ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are complex numbers which are normalized so that

${\displaystyle |\alpha |^{2}+|\beta |^{2}=\alpha ^{*}\alpha +\beta ^{*}\beta =1}$

and such that ${\displaystyle \langle \downarrow |\uparrow \rangle =0}$ and ${\displaystyle \langle \downarrow |\downarrow \rangle =\langle \uparrow |\uparrow \rangle =1}$, i.e., such that ${\displaystyle \left|\uparrow \right\rangle }$ and ${\displaystyle \left|\downarrow \right\rangle }$ form a basis and have diametrically opposite representations on the Bloch sphere, then let

${\displaystyle u={\beta \over \alpha }={\alpha ^{*}\beta \over \alpha ^{*}\alpha }={\alpha ^{*}\beta \over |\alpha |^{2}}=u_{x}+iu_{y}}$

be their ratio.

If the Bloch sphere is thought of as being embedded in ${\displaystyle \mathbb {R} ^{3}}$ with its center at the origin and with radius one, then the plane z = 0 (which intersects the Bloch sphere at a great circle; the sphere's equator, as it were) can be thought of as an Argand diagram. Plot point u in this plane — so that in ${\displaystyle \mathbb {R} ^{3}}$ it has coordinates ${\displaystyle (u_{x},u_{y},0)}$.

Draw a straight line through u and through the point on the sphere that represents ${\displaystyle \left|\downarrow \right\rangle }$. (Let (0,0,1) represent ${\displaystyle \left|\uparrow \right\rangle }$ and (0,0,−1) represent ${\displaystyle \left|\downarrow \right\rangle }$.) This line intersects the sphere at another point besides ${\displaystyle \left|\downarrow \right\rangle }$. (The only exception is when ${\displaystyle u=\infty }$, i.e., when ${\displaystyle \alpha =0}$ and ${\displaystyle \beta \neq 0}$.) Call this point P. Point u on the plane z = 0 is the stereographic projection of point P on the Bloch sphere. The vector with tail at the origin and tip at P is the direction in 3-D space corresponding to the spinor ${\displaystyle \left|\nearrow \right\rangle }$. The coordinates of P are

${\displaystyle P_{x}={2u_{x} \over 1+u_{x}^{2}+u_{y}^{2}}}$

${\displaystyle P_{y}={2u_{y} \over 1+u_{x}^{2}+u_{y}^{2}}}$

${\displaystyle P_{z}={1-u_{x}^{2}-u_{y}^{2} \over 1+u_{x}^{2}+u_{y}^{2}}}$.

Note: mathematically the Bloch sphere for a two-spinor state can be considered to be a Riemann sphere or a complex 2-dimensional projective Hilbert space, denotable as ${\displaystyle \mathbb {P} \mathbf {H} ^{2}}$. The complex 2-dimensional Hilbert space ${\displaystyle \mathbf {H} ^{2}}$ (of which ${\displaystyle \mathbb {P} \mathbf {H} ^{2}}$ is a projection) is a representation space of SO(3).[8]

Formulations of quantum mechanics in terms of pure states are adequate for isolated systems; in general quantum mechanical systems need to be described in terms of density operators. The Bloch sphere parametrizes not only pure states but mixed states for 2-level systems. The density operator describing the mixed-state of a 2-level quantum system (qubit) corresponds to a point inside the Bloch sphere with the following coordinates:

${\displaystyle \left(\sum p_{i}x_{i},\sum p_{i}y_{i},\sum p_{i}z_{i}\right),}$

where ${\displaystyle p_{i}}$ is the probability of the individual states within the ensemble and ${\displaystyle x_{i},y_{i},z_{i}}$ are the coordinates of the individual states (on the surface of Bloch sphere). The set of all points on and inside the Bloch sphere is known as the Bloch ball.

For states of higher dimensions there is difficulty in extending this to mixed states. The topological description is complicated by the fact that the unitary group does not act transitively on density operators. The orbits moreover are extremely diverse as follows from the following observation:

Theorem. Suppose A is a density operator on an n level quantum mechanical system whose distinct eigenvalues are μ₁, ..., μ_k with multiplicities n₁, ..., n_k. Then the group of unitary operators V such that V A V* = A is isomorphic (as a Lie group) to

${\displaystyle \operatorname {U} (n_{1})\times \cdots \times \operatorname {U} (n_{k}).}$

In particular the orbit of A is isomorphic to

${\displaystyle \operatorname {U} (n)/\left(\operatorname {U} (n_{1})\times \cdots \times \operatorname {U} (n_{k})\right).}$

It is possible to generalize the construction of the Bloch ball to dimensions larger than 2, but the geometry of such a "Bloch body" is more complicated than that of a ball.[9]

A useful advantage of the Bloch sphere representation is that the evolution of the qubit state is describable by rotations of the Bloch sphere. The most concise explanation for why this is the case is that the lie algebra for the group of unitary and hermitian matrices ${\displaystyle SU(2)}$ is isomorphic to the lie algebra of the group of three dimensional rotations ${\displaystyle SO(3)}$.[10]

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