Fubini-Study metric

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In mathematics, the Fubini-Study metric is a Kähler metric on projective Hilbert space, that is, complex projective space CPⁿ endowed with an inner product. In the context of quantum mechanics, this space is sometimes called the Bloch sphere; the Fubini-Study metric is the natural metric for the geometrization of quantum mechanics. Much of the peculiar behaviour of quantum mechanics, including quantum entanglement and the Berry phase effect, can be attributed to the peculiarities of the Fubini-Study metric.

An Hermitian form in (the vector space) Cⁿ⁺¹ defines a unitary subgroup U(n+1) in GL(n+1,C). A Fubini-Study metric is determined up to homothety (overall scaling) by invariance under such a U(n+1) action; thus it is homogeneous. By elementary linear algebra, any two Fubini-Study metrics are isometric under a projective automorphism of CPⁿ, so it is common to speak of "the" Fubini-Study metric.

1 Articulation
- 1.1 Case of n=1
2 Product metric
3 See also

[edit] Articulation

The metric may be defined either using the bra-ket notation commonly used in quantum mechanics, or the notation of projective varieties of algebraic geometry. To explicitly equate these two languages, let

$\vert \psi \rangle = \sum_{k=0}^n Z_k \vert e_k \rangle = [Z_0:Z_1:\ldots:Z_n]$

where $\{\vert e_k \rangle\}$ is a set of orthonormal basis vectors for Hilbert space, the $Z k$ are complex numbers, and $Z_\alpha = [Z_0:Z_1:\ldots:Z_n]$ is the standard notation for a point in the projective space $\mathbb{C}P^n$ in homogeneous coordinates (note that despite the name, these are not coordinates but are instead elements of the dual space). Then, given two points $\vert \psi \rangle = Z_\alpha$ and $\vert \phi \rangle = W_\alpha$ in the space, the distance between them is

$\gamma (\psi, \phi) = \arccos \sqrt \frac {\langle \psi \vert \phi \rangle \; \langle \phi \vert \psi \rangle } {\langle \psi \vert \psi \rangle \; \langle \phi \vert \phi \rangle}$

or, equivalently, in projective variety notation,

$\gamma (\psi, \phi) =\gamma (Z,W) = \arccos \sqrt {\frac {Z_\alpha \overline{W}^\alpha \; W_\beta \overline{Z}^\beta} {Z_\alpha \overline{Z}^\alpha \; W_\beta \overline{W}^\beta}}$

Here, $\overline{Z}^\alpha$ is the complex conjugate of $Z α$ . The appearance of $\langle \psi \vert \psi \rangle$ in the denominator is a reminder that $\vert \psi \rangle$ and likewise $\vert \phi \rangle$ were not normalized to unit length; thus the normalization is made explicit here. In Hilbert space, the metric can be rather trivially interpreted as the angle between two vectors; thus it is occasionally called the quantum angle. The angle is real-valued, and runs from zero to $π / 2$ .

The infinitessimal form of this metric may be quickly obtained by taking $\vert \phi \rangle = \vert \psi+\delta\psi \rangle$ , or equivalently, $W α = Z α + d Z α$ to obtain

$ds^2 = \frac{\langle \delta \psi \vert \delta \psi \rangle} {\langle \psi \vert \psi \rangle} - \frac {\langle \delta \psi \vert \psi \rangle \; \langle \psi \vert \delta \psi \rangle} {{\langle \psi \vert \psi \rangle}^2}$

or, equivalently,

$ds^2 = 2 \frac {Z_{[\alpha} dZ_{\beta]} \; \overline{Z}^{[\alpha} \overline{dZ}^{\beta]} } {\left( Z_\alpha \overline{Z}^\alpha \right)^2}$

Here, index commutator notation is used, so that

$Z_{[\alpha} W_{\beta]} = \frac {1}{2} \left( Z_{\alpha} W_{\beta} - Z_{\beta} W_{\alpha} \right)$

The last form is particularly suggestive, as it emphasizes that $Z [α W β]$ is a Grassmannian variety, specifically, the projective plane connecting the two projective points $Z α$ and $W β$ . In the language of quantum mechanics, it is the superposition of two states.

[edit] Case of n=1

In the case of n=1, this metric reduces to the ordinary metric on $\mathbb{C}P^1=\mathbb{C}\cup\infty = S^2$ ,

$ds^2= \frac{dz \; d\overline{z}} {\left(1+z^2\right)\left(1+\overline{z}^2\right)} = \frac{dx^2+dy^2}{ \left(1+r^2\right)^2 } = d\theta^2 + \sin^2 \theta \,d\varphi^2$

where $z = Z 1 / Z 2 = x + i y$ , and $r^2=z \overline{z} = x^2+y^2$ , while $\theta, \varphi$ are the usual spherical coordinates, given by projecting the sphere down to the complex plane, with $r\,\tan(\theta/2)=1$ and $\tan \varphi = y/x$ . That is, a single qubit is written notationally as

$|\psi \rangle = \cos \theta \, |0 \rangle + e^{i \varphi} \sin \theta \,|1 \rangle$

[edit] Product metric

The common notions of separability apply for the Fubini-Study metric. More precisely, the metric is separable on the natural product of projective spaces, the Segre embedding. That is, if $\vert\psi\rangle$ is a separable state, so that it can be written as $\vert\psi\rangle=\vert\psi_A\rangle\otimes\vert\psi_B\rangle$ , then the metric is the sum of the metric on the subspaces: