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Bloch sphere

In quantum mechanics, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit), named after the physicist Felix Bloch.

Quantum mechanics is mathematically formulated in Hilbert space or projective Hilbert space. The space of pure states of a quantum system is given by the one-dimensional subspaces of the corresponding Hilbert space (or the "points" of the projective Hilbert space). In a two-dimensional Hilbert space this is simply the complex projective line, which is a geometrical sphere.

The Bloch sphere is a unit 2-sphere, with each pair of antipodal points corresponding to mutually orthogonal state vectors. The north and south poles of the Bloch sphere are typically chosen to correspond to the standard basis vectors $|0\rangle$ and $|1\rangle$, respectively, which in turn might correspond e.g. to the spin-up and spin-down states of an electron. This choice is arbitrary, however. The points on the surface of the sphere correspond to the pure states of the system, whereas the interior points correspond to the mixed states. The Bloch sphere may be generalized to an n-level quantum system but then the visualization is less useful.

In optics, the Bloch sphere is also known as the Poincaré sphere and specifically represents different types of polarizations. See the Jones Vector for a detailed list of the 6 common polarization types and how they map on to the surface of this sphere.

The natural metric on the Bloch sphere is the Fubini–Study metric.

Definition

Given an orthonormal basis, any pure state $|\psi \rangle$ of a two-level quantum system can be written as a superposition of the basis vectors $|0\rangle$ and $|1\rangle$, where the coefficient or amount of each basis vector is a complex number. Since only the relative phase between the coefficients of the two basis vectors has any physical meaning, we can take the coefficient of $|0\rangle$ to be real and non-negative. We also know from quantum mechanics that the total probability of the system has to be one, so it must be that $\langle \psi ^{*}|\psi \rangle =1$, meaning $||\psi \rangle |^{2}=1$. Given this constraint, we can write $|\psi \rangle$ in the following representation:

$|\psi \rangle =\cos \left({\tfrac {\theta }{2}}\right)|0\rangle \,+\,e^{i\phi }\sin \left({\tfrac {\theta }{2}}\right)|1\rangle =\cos \left({\tfrac {\theta }{2}}\right)|0\rangle \,+\,(\cos \phi +i\sin \phi )\,\sin \left({\tfrac {\theta }{2}}\right)|1\rangle$

with $0\leq \theta \leq \pi$ and $0\leq \phi <2\pi$. Except in the case where $|\psi \rangle$ is one of the ket vectors $|0\rangle$ or $|1\rangle$ the representation is unique. The parameters $\theta \,$ and $\phi \,$, re-interpreted as spherical coordinates, specify a point ${\vec {a}}=(\sin \theta \cos \phi ,\;\sin \theta \sin \phi ,\;\cos \theta )$ on the unit sphere in $\mathbb {R} ^{3}$. For mixed states, any two-dimensional density operator $\rho$ can be expanded using the identity $I$ and the Hermitian, traceless Pauli matrices ${\vec {\sigma }}$:

$\rho ={\frac {1}{2}}\left(I+{\vec {a}}\cdot {\vec {\sigma }}\right)$,

where ${\vec {a}}\in \mathbb {R} ^{3}$ is called the Bloch vector of the system. The eigenvalues of $\rho$ are given by ${\frac {1}{2}}\left(1\pm |{\vec {a}}|\right)$. As density operators must be positive-semidefinite, we have $|{\vec {a}}|\leq 1$.

For pure states we must have

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

in accordance with the previous result. Hence 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.

A generalization for pure states

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

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

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 $|\psi \rangle$, the isotropy group of $|\psi \rangle$, (defined as the set of elements $g$ of U(n) such that $g|\psi \rangle =|\psi \rangle$) is isomorphic to the product group

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

In linear algebra terms, this can be justified as follows. Any $g$ of U(n) that leaves $|\psi \rangle$ invariant must have $|\psi \rangle$ 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 $|\psi \rangle$, 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

$A\mapsto e^{iA}$

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,

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

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.

The geometry of density operators

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. However, while the Bloch sphere parametrizes not only pure states but mixed states for 2-level systems, 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

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

In particular the orbit of A is isomorphic to

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

We note here that, in the literature, one can find non-Bloch type parametrizations of (mixed) states that do generalize to dimensions higher than 2.

See also

• Specific implementations of the Bloch sphere are enumerated under the qubit article.
• Gyrovector space

References

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• Dariusz Chruściński, "Geometric Aspect of Quantum Mechanics and Quantum Entanglement", Journal of Physics Conference Series, 39 (2006) pp. 9–16.
• Alain Michaud, "Rabi Flopping Oscillations" (2009). (A small animation of the bloch vector submitted to a resonant excitation.)
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