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| In [[mathematics]], the '''Fubini–Study metric''' is a [[Kähler metric]] on [[projective Hilbert space]], that is, [[complex projective space]] '''CP'''<sup>''n''</sup> endowed with a [[Hermitian form]]. This [[Metric (mathematics)|metric]] was originally described in 1904 and 1905 by [[Guido Fubini]] and [[Eduard Study]].
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| A [[Hermitian form]] in (the vector space) '''C'''<sup>''n''+1</sup> 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. Equipped with a Fubini–Study metric, '''CP'''<sup>''n''</sup> is a [[symmetric space]]. The particular normalization on the metric depends on the application. In [[Riemannian geometry]], one uses a normalization so that the Fubini–Study metric simply relates to the standard metric on the [[N-sphere|(2''n''+1)-sphere]]. In [[algebraic geometry]], one uses a normalization making '''CP'''<sup>''n''</sup> a [[Hodge manifold]].
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| ==Construction==
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| The Fubini–Study metric arises naturally in the [[quotient space]] construction of [[complex projective space]].
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| Specifically, one may define '''CP'''<sup>''n''</sup> to be the space consisting of all complex lines in '''C'''<sup>''n''+1</sup>, i.e., the quotient of '''C'''<sup>''n''+1</sup>\{0} by the [[equivalence relation]] relating all complex multiples of each point together. This agrees with the quotient by the diagonal [[group action]] of the multiplicative group '''C'''<sup>*</sup> = '''C''' \ {0}:
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| :<math>\mathbf{CP}^n = \left\{ \mathbf{Z} = [Z_0,Z_1,\ldots,Z_n] \in {\mathbf C}^{n+1}\setminus\{0\}\, \right\} / \{ \mathbf{Z} \sim c\mathbf{Z}, c \in \mathbf{C}^* \}.</math>
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| This quotient realizes '''C'''<sup>''n''+1</sup>\{0} as a complex [[line bundle]] over the base space '''CP'''<sup>''n''</sup>. (In fact this is the so-called [[tautological bundle]] over '''CP'''<sup>''n''</sup>.) A point of '''CP'''<sup>''n''</sup> is thus identified with an equivalence class of (''n''+1)-tuples [''Z''<sub>0</sub>,...,''Z''<sub>''n''</sub>] modulo nonzero complex rescaling; the ''Z''<sub>''i''</sub> are called [[homogeneous coordinates]] of the point.
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| Furthermore, one may realize this quotient in two steps: since multiplication by a nonzero complex scalar ''z'' = ''R'' ''e''<sup>iθ</sup> can be uniquely thought of as the composition of a dilation by the modulus ''R'' followed by a counterclockwise rotation about the origin by an angle <math>\theta</math>, the quotient '''C'''<sup>''n''+1</sup> → '''CP'''<sup>''n''</sup> splits into two pieces.
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| :<math>\mathbf{C}^{n+1}\setminus\{0\} \stackrel{(a)}\longrightarrow S^{2n+1} \stackrel{(b)}\longrightarrow \mathbf{CP}^n</math>
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| where step (a) is a quotient by the dilation '''Z''' ~ ''R'''''Z''' for ''R'' ∈ '''R'''<sup>+</sup>, the multiplicative group of positive real numbers, and step (b) is a quotient by the rotations '''Z''' ~ ''e''<sup>iθ</sup>'''Z'''.
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| The result of the quotient in (a) is the real hypersphere ''S''<sup>2''n''+1</sup> defined by the equation |'''Z'''|<sup>2</sup> = |''Z''<sub>0</sub>|<sup>2</sup> + ... + |''Z''<sub>''n''</sub>|<sup>2</sup> = 1. The quotient in (b) realizes '''CP'''<sup>''n''</sup> = ''S''<sup>2''n''+1</sup>/''S''<sup>1</sup>, where ''S''<sup>1</sup> represents the group of rotations. This quotient is realized explicitly by the famous [[Hopf fibration]] ''S''<sup>1</sup> → ''S''<sup>2''n''+1</sup> → '''CP'''<sup>''n''</sup>, the fibers of which are among the [[great circles]] of <math>S^{2n+1}</math>.
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| ===As a metric quotient===
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| When a quotient is taken of a [[Riemannian manifold]] (or [[metric space]] in general), care must be taken to ensure that the quotient space is endowed with a [[Riemannian metric|metric]] that is well-defined. For instance, if a group ''G'' acts on a Riemannian manifold (''X'',''g''), then in order for the [[orbit space]] ''X''/''G'' to possess an induced metric, <math>g</math> must be constant along ''G''-orbits in the sense that for any element ''h'' ∈ ''G'' and pair of vector fields <math>X,Y</math> we must have ''g''(''Xh'',''Yh'') = ''g''(''X'',''Y'').
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| The standard [[Hermitian metric]] on '''C'''<sup>''n''+1</sup> is given in the standard basis by
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| :<math>ds^2 = d\mathbf{Z} \otimes d\overline{\mathbf{Z}} = dZ_0 \otimes d\overline{Z_0} + \cdots + dZ_n \otimes d\overline{Z_n}</math>
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| whose realification is the standard [[Euclidean metric]] on '''R'''<sup>2''n''+2</sup>. This metric is ''not'' invariant under the diagonal action of '''C'''<sup>*</sup>, so we are unable to directly push it down to '''CP'''<sup>n</sup> in the quotient. However, this metric ''is'' invariant under the diagonal action of ''S''<sup>1</sup> = U(1), the group of rotations. Therefore, step (b) in the above construction is possible once step (a) is accomplished.
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| The '''Fubini–Study metric''' is the metric induced on the quotient '''CP'''<sup>''n''</sup> = ''S''<sup>2''n''+1</sup>/''S''<sup>1</sup>, where <math>S^{2n+1}</math> carries the so-called "round metric" endowed upon it by ''restriction'' of the standard Euclidean metric to the unit hypersphere.
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| ===In local affine coordinates===
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| Corresponding to a point in '''CP'''<sup>''n''</sup> with homogeneous coordinates (''Z''<sub>0</sub>,...,''Z''<sub>''n''</sub>), there is a unique set of ''n'' coordinates (''z''<sub>1</sub>,…,''z''<sub>''n''</sub>) such that
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| :<math>[Z_0,\dots,Z_n] {\sim} [1,z_1,\dots,z_n],</math>
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| provided ''Z''<sub>0</sub> ≠ 0; specifically, ''z''<sub>''j''</sub> = ''Z''<sub>''j''</sub>/''Z''<sub>0</sub>. The (''z''<sub>1</sub>,…,''z''<sub>''n''</sub>) form an [[affine coordinates|affine coordinate system]] for '''CP'''<sup>''n''</sup> in the coordinate patch ''U''<sub>0</sub> = {''Z''<sub>0</sub> ≠ 0}. One can develop an affine coordinate system in any of the coordinate patches ''U''<sub>''i''</sub> = {''Z''<sub>''i''</sub> ≠ 0} by dividing instead by ''Z''<sub>''i''</sub> in the obvious manner. The ''n''+1 coordinate patches ''U''<sub>''i''</sub> cover '''CP'''<sup>''n''</sup>, and it is possible to give the metric explicitly in terms of the affine coordinates (''z''<sub>1</sub>,…,''z''<sub>''n''</sub>) on ''U''<sub>''i''</sub>. The coordinate derivatives define a frame <math>\{\partial_1,\ldots,\partial_n\}</math> of the holomorphic tangent bundle of '''CP'''<sup>''n''</sup>, in terms of which the Fubini–Study metric has Hermitian components
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| :<math>h_{i\bar{j}} = h(\partial_i,\bar{\partial}_j) = \frac{(1+|\mathbf{z}|^2)\delta_{i\bar{j}} - \bar{z}_i z_j}{(1+|\mathbf{z}|^2)^2}.</math>
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| where |'''z'''|<sup>2</sup> = ''z''<sub>1</sub><sup>2</sup>+...+''z''<sub>''n''</sub><sup>2</sup>. That is, the [[Hermitian matrix]] of the Fubini–Study metric in this frame is
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| :<math> \bigl(h_{i\bar{j}}\bigr) = \frac{1}{(1+|\mathbf{z}|^2)^2}
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| \left[
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| \begin{array}{cccc}
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| 1+|\mathbf{z}|^2 - |z_1|^2 & -\bar{z}_1 z_2 & \cdots & -\bar{z}_1 z_n \\
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| -\bar{z}_2 z_1 & 1 + |\mathbf{z}|^2 - |z_2|^2 & \cdots & -\bar{z}_2 z_n \\
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| \vdots & \vdots & \ddots & \vdots \\
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| -\bar{z}_n z_1 & -\bar{z}_n z_2 & \cdots & 1 + |\mathbf{z}|^2 - |z_n|^2
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| \end{array}
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| \right]
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| </math>
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| Note that each matrix element is unitary-invariant: the diagonal action <math>\mathbf{z} \mapsto e^{i\theta}\mathbf{z}</math> will leave this matrix unchanged.
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| Accordingly, the line element is given by
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| :<math>\begin{align}
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| ds^2 &= \frac{(1+|\mathbf{z}|^2)|d\mathbf{z}|^2 - (\bar{\mathbf{z}}\cdot d\mathbf{z})(\mathbf{z}\cdot d\bar{\mathbf{z}})}{(1+|\mathbf{z}|^2)^2}\\
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| &= \frac{(1+z_i\bar{z}^i)dz_jd\bar{z}^j - \bar{z}^j z_idz_jd\bar{z}^i}{(1+z_i\bar{z}^i)^2}.
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| \end{align}
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| </math>
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| In this last expression, the [[summation convention]] is used to sum over Latin indices ''i'',''j'' that range from 1 to ''n''.
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| The metric can be derived from the following Kähler potential:
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| :<math>
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| K=\ln(1+\delta_{ij^*}z^{i}\bar{z}^{j^*})
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| </math>
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| as
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| :<math>
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| g_{ij^*}=K_{ij^*}=\frac{\partial^{2}}{\partial z^{i}\partial \bar{z}^{j^*}}K
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| </math>
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| ===Homogeneous coordinates===
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| An expression is also possible in the homogeneous coordinates '''Z''' = [''Z''<sub>0</sub>,...,''Z''<sub>''n''</sub>]. Formally, subject to suitably interpreting the expressions involved, one has
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| :<math>\begin{align}
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| ds^2 &= \frac{|\mathbf{Z}|^2|d\mathbf{Z}|^2 - (\bar{\mathbf{Z}}\cdot d\mathbf{Z})(\mathbf{Z}\cdot d\bar{\mathbf{Z}})}{|\mathbf{Z}|^4}\\
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| &=\frac{Z_\alpha\bar{Z}^\alpha dZ_\beta d\bar{Z}^\beta - \bar{Z}^\alpha Z_\beta dZ_\alpha d\bar{Z}^\beta}{(Z_\alpha\bar{Z}^\alpha)^2}\\
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| &= \frac {2Z_{[\alpha}dZ_{\beta]} \overline{Z}^{[\alpha}\overline{dZ}^{\beta]}}
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| {\left( Z_\alpha \overline{Z}^\alpha \right)^2}.
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| \end{align}</math>
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| Here the summation convention is used to sum over Greek indices α β ranging from 0 to ''n'', and in the last equality the standard notation for the skew part of a tensor is used:
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| :<math>Z_{[\alpha}W_{\beta]} = \frac {1}{2} \left(
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| Z_{\alpha} W_{\beta} - Z_{\beta} W_{\alpha} \right).</math>
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| Now, this expression for d''s''<sup>2</sup> apparently defines a tensor on the total space of the tautological bundle '''C'''<sup>''n''+1</sup>\{0}. It is to be understood properly as a tensor on '''CP'''<sup>''n''</sup> by pulling it back along a holomorphic section σ of the tautological bundle of '''CP'''<sup>''n''</sup>. It remains then to verify that the value of the pullback is independent of the choice of section: this can be done by a direct calculation.
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| The Kähler form of this metric is, up to an overall constant normalization,
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| :<math>\omega = i\partial\overline{\partial}\log |\mathbf{Z}|^2</math>
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| the pullback of which is clearly independent of the choice of holomorphic section. The quantity log|'''Z'''|<sup>2</sup> is the Kähler scalar of '''CP'''<sup>''n''</sup>.
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| ===The ''n'' = 1 case ===
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| When ''n'' = 1, there is a diffeomorphism <math>S^2\cong \mathbb{CP}^1</math> given by [[stereographic projection]]. This leads to the "special" Hopf fibration ''S''<sup>1</sup> → ''S''<sup>3</sup> → ''S''<sup>2</sup>. When the Fubini–Study metric is written in coordinates on '''CP'''<sup>1</sup>, its restriction to the real tangent bundle yields an expression of the ordinary "round metric" of radius 1/2 (and [[Gaussian curvature]] 4) on ''S''<sup>2</sup>.
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| Namely, if ''z'' = ''x'' + i''y'' is the standard affine coordinate chart on the [[Riemann sphere]] '''CP'''<sup>1</sup> and ''x'' = ''r'' cosθ, ''y'' = ''r'' sinθ are polar coordinates on '''C''', then a routine computation shows
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| :<math>ds^2= \frac{\operatorname{Re}(dz \otimes d\overline{z})}{\left(1+|z|^2\right)^2}
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| = \frac{dx^2+dy^2}{ \left(1+r^2\right)^2 }
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| = \frac{1}{4}(d\phi^2 + \sin^2 \phi \,d\theta^2)
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| = \frac{1}{4} ds^2_{us}
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| </math>
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| where <math>ds^2_{us}</math> is the round metric on the unit 2-sphere. Here φ, θ are "mathematician's [[spherical coordinates]]" on ''S''<sup>2</sup> coming from the stereographic projection ''r'' tan(φ/2) = 1, tanθ = ''y''/''x''. (Many physics references interchange the roles of φ and θ.)
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| ==Curvature properties==
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| In the ''n'' = 1 special case, the Fubini–Study metric has constant scalar curvature identically equal to 4, according to the equivalence with the 2-sphere's round metric (which given a radius ''R'' has scalar curvature <math>1/R^2</math>). However, for ''n'' > 1, the Fubini–Study metric does not have constant curvature. Its sectional curvature is instead given by the equation<ref>Sakai, T. ''Riemannian Geometry'', Translations of Mathematical Monographs No. 149 (1995), American Mathematics Society.</ref>
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| :<math>K(\sigma) = 1 + 3\langle JX,Y \rangle^2</math>
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| where <math>\{X,Y\} \in T_p \mathbf{CP}^n</math> is an orthonormal basis of the 2-plane σ, ''J'' : ''T'''''CP'''<sup>''n''</sup> → ''T'''''CP'''<sup>''n''</sup> is the [[linear complex structure|complex structure]] on '''CP'''<sup>''n''</sup>, and <math>\langle \cdot , \cdot \rangle</math> is the Fubini–Study metric.
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| A consequence of this formula is that the sectional curvature satisfies <math>1 \leq K(\sigma) \leq 4</math> for all 2-planes <math>\sigma</math>. The maximum sectional curvature (4) is attained at a [[holomorphic]] 2-plane — one for which ''J''(σ) ⊂ σ — while the minimum sectional curvature (1) is attained at a 2-plane for which ''J''(σ) is orthogonal to σ. For this reason, the Fubini–Study metric is often said to have "constant ''holomorphic'' sectional curvature" equal to 4.
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| This makes '''CP'''<sup>''n''</sup> a (non-strict) [[quarter-pinched sphere theorem|quarter pinched manifold]]; a celebrated theorem shows that a strictly quarter-pinched [[simply connected]] ''n''-manifold must be homeomorphic to a sphere.
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| The Fubini–Study metric is also an [[Einstein metric]] in that it is proportional to its own [[Ricci tensor]]: there exists a constant λ such that for all ''i'',''j'' we have
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| :<math>Ric_{ij} = \lambda g_{ij}</math>.
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| This implies, among other things, that the Fubini–Study metric remains unchanged up to a scalar multiple under the [[Ricci flow]]. It also makes '''CP'''<sup>''n''</sup> indispensable to the theory of [[general relativity]], where it serves as a nontrivial solution to the vacuum [[Einstein field equations]].
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| ==In quantum mechanics==
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| In [[quantum mechanics]], the Fubini–Study metric is also known as the [[Bures metric]].<ref name=facchi>Paolo Facchi, Ravi Kulkarni, V. I. Man'ko, Giuseppe Marmo, E. C. G. Sudarshan, Franco Ventriglia "[http://arxiv.org/abs/1009.5219 Classical and Quantum Fisher Information in the Geometrical Formulation of Quantum Mechanics]" (2010), ''Physics Letters'' '''A 374''' pp. 4801. DOI: 10.1016/j.physleta.2010.10.005</ref> However, the Bures metric is typically defined in the notation of [[mixed state (physics)|mixed states]], whereas the exposition below is written in terms of a [[pure state]]. The real part of the metric is (four times) the [[Fisher information metric]].<ref name=facchi/>
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| The Fubini–Study metric may be written 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
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| :<math>\vert \psi \rangle = \sum_{k=0}^n Z_k \vert e_k \rangle = [Z_0:Z_1:\ldots:Z_n]</math>
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| where <math>\{\vert e_k \rangle\}</math> is a set of [[orthonormal]] [[basis vector]]s for [[Hilbert space]], the <math>Z_k</math> are complex numbers, and <math>Z_\alpha = [Z_0:Z_1:\ldots:Z_n]</math> is the standard notation for a point in the projective space <math>\mathbb{C}P^n</math> in [[homogeneous coordinates]]. Then, given two points <math>\vert \psi \rangle = Z_\alpha</math> and <math>\vert \phi \rangle = W_\alpha</math> in the space, the distance between them is
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| :<math>\gamma (\psi, \phi) = \arccos
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| \sqrt \frac {\langle \psi \vert \phi \rangle \;
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| \langle \phi \vert \psi \rangle }
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| {\langle \psi \vert \psi \rangle \;
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| \langle \phi \vert \phi \rangle}
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| </math>
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| or, equivalently, in projective variety notation,
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| :<math>\gamma (\psi, \phi) =\gamma (Z,W) =
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| \arccos \sqrt {\frac
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| {Z_\alpha \overline{W}^\alpha \; W_\beta \overline{Z}^\beta}
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| {Z_\alpha \overline{Z}^\alpha \; W_\beta \overline{W}^\beta}}.
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| </math>
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| Here, <math>\overline{Z}^\alpha</math> is the [[complex conjugate]] of <math>Z_\alpha</math>. The appearance of <math>\langle \psi \vert \psi \rangle</math> in the denominator is a reminder that <math>\vert \psi \rangle</math> and likewise <math>\vert \phi \rangle</math> 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 <math>\pi/2</math>.
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| The infinitesimal form of this metric may be quickly obtained by taking <math>\phi = \psi+\delta\psi</math>, or equivalently, <math>W_\alpha = Z_\alpha + dZ_\alpha</math> to obtain
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| :<math>ds^2 = \frac{\langle \delta \psi \vert \delta \psi \rangle}
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| {\langle \psi \vert \psi \rangle} -
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| \frac {\langle \delta \psi \vert \psi \rangle \;
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| \langle \psi \vert \delta \psi \rangle}
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| {{\langle \psi \vert \psi \rangle}^2}.
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| </math>
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| In the context of [[quantum mechanics]], '''CP'''<sup>1</sup> is called the [[Bloch sphere]]; the Fubini–Study metric is the natural [[metric (mathematics)|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.
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| ==Product metric==
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| 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 <math>\vert\psi\rangle</math> is a [[separable state]], so that it can be written as <math>\vert\psi\rangle=\vert\psi_A\rangle\otimes\vert\psi_B\rangle</math>, then the metric is the sum of the metric on the subspaces:
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| :<math>ds^2 = {ds_A}^2+{ds_B}^2</math>
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| where <math>{ds_A}^2</math> and <math>{ds_B}^2</math> are the metrics, respectively, on the subspaces ''A'' and ''B''.
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| ==See also==
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| * [[Non-linear sigma model]]
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| * [[Kaluza–Klein theory]]
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| * [[Arakelov height]]
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| ==References==
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| {{reflist}}
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| *{{Citation | authorlink=Arthur Besse |last1=Besse | first1=Arthur L. | title=Einstein manifolds | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10 | isbn=978-3-540-15279-8 | year=1987 | pages=xii+510}}
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| * {{citation | first1=D.C.|last1=Brody |first2=L.P.|last2=Hughston | title=Geometric Quantum Mechanics| journal=Journal of Geometry and Physics | year=2001 | volume=38 | pages=19–53 | doi=10.1016/S0393-0440(00)00052-8 |arxiv = quant-ph/9906086 |bibcode = 2001JGP....38...19B }}
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| * {{citation | first1=P. |last1=Griffiths | authorlink1=Phillip Griffiths | first2=J.|last2=Harris| authorlink2=Joe Harris (mathematician)|title=Principles of Algebraic Geometry | series=Wiley Classics Library | publisher=Wiley Interscience | year=1994 | isbn=0-471-05059-8 | pages=30–31 }}
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| * {{springer|id=F/f041860|title=Fubini–Study metric|first=A.L.|last=Onishchik|year=2001}}.
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| ==External links==
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| {{DEFAULTSORT:Fubini-Study metric}}
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| [[Category:Projective geometry]]
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| [[Category:Complex manifolds]]
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| [[Category:Symplectic geometry]]
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| [[Category:Structures on manifolds]]
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| [[Category:Quantum mechanics]]
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