potential: as Using homogeneous coordinates An expression is also possible in the notation of homogeneous coordinates, commonly used to describe projective varieties of algebraic geometry: Z = [Z0:...:Zn]. Formally, subject to suitably interpreting the expressions involved, one has 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: Now, this expression for ds2 apparently defines a tensor on the total space of the tautological bundle Cn+1\{0}. It is to be understood properly as a tensor on CPn by pulling it back along a holomorphic section σ of the tautological bundle of CPn. 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. The Kähler form of this metric is where the are the Dolbeault operators. The pullback of this is clearly independent of the choice of holomorphic section. The quantity log|Z|2 is the Kähler potential (sometimes called the Kähler scalar) of CPn. In bra-ket coordinate notation In quantum mechanics, the Fubini–Study metric is also known as the Bures metric. However, the Bures metric is typically defined in the notation of 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. The Fubini–Study metric may be written using the bra–ket notation commonly used in quantum mechanics. To explicitly equate this notation to the homogeneous coordinates given above, let where is a set of orthonormal basis vectors for Hilbert space, the are complex numbers, and is the standard notation for a point in the projective space in homogeneous coordinates. Then, given two points and in the space, the distance (length of a geodesic) between them is or, equivalently, in projective variety notation, Here, is the complex conjugate of . The appearance of in the denominator is a reminder that and likewise 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 0 to . The infinitesimal form of this metric may be quickly obtained by taking , or equivalently, to