Bloch sphere

In quantum mechanics, the '''Bloch sphere''' (also known as the Poincaré sphere in optics) is a geometrical representation of the [[pure state]] space of a 2-level quantum system. Alternatively, it is the pure state space of a 1 [[qubit]] quantum register. The Bloch sphere is actually geometrically a sphere and the correspondence between elements of the Bloch sphere and [[pure states]] can be explicitly given. [[Image:Bloch.png|thumb|Bloch sphere]] To show this correspondence, consider the qubit description of the Bloch sphere; any [[pure state]] ψ can be written as a complex superposition of the [[ket notation|ket vector]]s $|0\; \backslash rangle$ and $|1\; \backslash rangle$; moreover since global phase factors do not affect physical state, we can take the representation so that the coefficient of $|0\; \backslash rangle$ is real and non-negative. Thus ψ has a representation as :$|\backslash psi\; \backslash rangle\; =\; \backslash cos\; \backslash theta\; \backslash ,\; |0\; \backslash rangle\; +\; e^\{i\; \backslash phi\}\; \backslash sin\; \backslash theta\; \backslash ,|1\; \backslash rangle$ with : The representation is unique except in the case ψ is one of the ket vectors $|0\; \backslash rangle$ or $|1\; \backslash rangle$ The parameters φ and θ uniquely specify a point on the unit sphere of euclidean space '''R'''3, namely the point whose coordinates (''x'',''y'',''z'') are : In this representation $|0\; \backslash rangle$ is mapped into (0,0,1) and $|1\; \backslash rangle$ is mapped into (0,0,-1). The interior of the Bloch sphere, the open Bloch ball, represents the [[mixed states]] of a single qubit. The $\backslash vec\{r\}=(x,y,z)$ co-ordinates of a state represent the expectation values of the $\backslash sigma\_\{x,y,z\}$ operators respectively. This is conveniently expressed by, :$\backslash rho=\backslash frac\{1\}\{2\}(\backslash mathbb\{I\}+\backslash vec\{r\}\backslash cdot\backslash vec\{\backslash sigma\})$ where $\backslash mathbb\{I\}$ is the 2x2 identity matrix, and $\backslash vec\{r\}.\backslash vec\{\backslash sigma\}=\backslash sum\_\{j=x,y,z\}r\_j\backslash sigma\_j$. A convex combination of pure states $\backslash \{\backslash hat\{\backslash vec\{r\}\}\_j\backslash \}$ with weights $p\_j$ gives a [[mixed state]] with Bloch vector $\backslash vec\{r\}=\backslash sum\_j\; p\_j\; \backslash hat\{\backslash vec\{r\}\}\_j$. ==Generalization == 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 to the compact coset space :$\backslash operatorname\{U\}(n)\; /(\backslash operatorname\{U\}(n-1)\; \backslash times\; \backslash operatorname\{U\}(1)).$ 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 ψ, the fixed point set of ψ, (defined as the set of elements ''g'' of U(''n'') such that ''g'' ψ = ψ) is isomorphic to the product group :$\backslash operatorname\{U\}(n-1)\; \backslash times\; \backslash operatorname\{U\}(1).$ 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 ''n''2. This is easy to see since the exponential map :$A\; \backslash mapsto\; e^\{i\; A\}$ 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 ''n''2. '''Corollary'''. The real dimension of the [[pure state]] space of ''H''''n'' is 2''n'' − 2. In fact, :$n^2\; -\; ((n-1)^2\; +1)\; =\; 2\; n\; -\; 2.\; \backslash quad$ 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 matrix|density operators]]. The topological description is complicated by the fact that the unitary group does not act transitively on [[density matrix|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 ''n''1, ...,''n''''k''. Then the group of unitary operators ''V'' such that ''V A V''* = ''A'' is isomorphic (as a Lie group) to :$\backslash operatorname\{U\}(n\_1)\; \backslash times\; \backslash cdots\; \backslash times\; \backslash operatorname\{U\}(n\_k).$ In particular the orbit of ''A'' is isomorphic to :$\backslash operatorname\{U\}(n)/(\backslash operatorname\{U\}(n\_1)\; \backslash times\; \backslash cdots\; \backslash times\; \backslash operatorname\{U\}(n\_k)).$ {{Template:FromWikipedia}} [[Category:Quantum States]] [[Category:Handbook of Quantum Information]]