Fubini–Study Geometry

The Fubini–Study geometry is the natural geometry of pure quantum states. It turns transition probabilities into distances on projective Hilbert space, so that two rays are close when they are hard to distinguish and far apart when they are nearly orthogonal.

This page uses the convention

$$d_{\rm FS}([\phi],[\psi]) = \arccos|\langle\phi|\psi\rangle|$$

for normalized representatives. Some authors use twice this distance. The factor matters in formulas for Bloch-sphere radii and quantum speed limits, so the convention will be stated explicitly whenever it matters.

Pulling this metric back to a restricted trial-state manifold produces the real tangent metric used by McLachlan variational dynamics. The projection equations and their conditioning belong there; this page owns the ambient pure-state distance geometry.

From Transition Probability to Distance

For normalized pure states, the transition probability is

$$P_{\psi\to\phi} = |\langle\phi|\psi\rangle|^2.$$

It depends only on the rays, not on the chosen phases of $\lvert\psi\rangle$ and $\lvert\phi\rangle$. Thus it is a projective invariant.

The Fubini–Study distance is defined by

$$\cos^2 d_{\rm FS} = P_{\psi\to\phi}, \qquad 0\le d_{\rm FS}\le\frac{\pi}{2}.$$

Equivalently,

$$d_{\rm FS}([\phi],[\psi]) = \arccos|\langle\phi|\psi\rangle|.$$

The limiting cases are physically transparent:

Relation between rays	Transition probability	Fubini–Study distance
identical	$1$	$0$
orthogonal	$0$	$\pi/2$
nearly identical	close to $1$	small

This is not a new postulate. It is the Born rule rewritten as geometry on the space of rays.

Infinitesimal Metric

Let $\lvert\psi\rangle$ be a normalized representative of a ray. A small change $\lvert d\psi\rangle$ contains both physical motion in ray space and an unphysical phase change. The phase direction is removed by projecting orthogonally to the ray:

$$\lvert d\psi_\perp\rangle = (I-\lvert\psi\rangle\langle\psi\rvert) \lvert d\psi\rangle.$$

The Fubini–Study line element is

$$ds_{\rm FS}^2 = \langle d\psi_\perp|d\psi_\perp\rangle.$$

Equivalently,

$$ds_{\rm FS}^2 = \langle d\psi|d\psi\rangle - |\langle\psi|d\psi\rangle|^2.$$

The subtraction removes the part of $\lvert d\psi\rangle$ that only changes the phase of the representative. If

$$\lvert d\psi\rangle = i\,d\alpha\,\lvert\psi\rangle,$$

then $ds_{\rm FS}=0$, as it must: the ray has not moved.

Coordinate-Free Meaning

The formula for $ds_{\rm FS}^2$ is invariant under a phase change of the representative,

$$\lvert\psi\rangle \mapsto e^{i\chi}\lvert\psi\rangle.$$

This invariance is essential. A metric on projective Hilbert space must assign the same distance no matter which phase convention is used to lift a ray to a vector.

The same metric can be expressed through projectors. For two nearby pure projectors,

$$\Pi_\psi = \lvert\psi\rangle\langle\psi\rvert, \qquad \Pi_{\psi+d\psi} = \Pi_\psi+d\Pi,$$

one has, with the convention of this page,

$$ds_{\rm FS}^2 = \frac12\operatorname{Tr}(d\Pi\,d\Pi).$$

The projector expression makes phase independence manifest and is often useful when comparing pure-state geometry with density-operator methods.

Geodesic Distance

The finite distance between two rays is the length of the shortest Fubini–Study path between them:

$$d_{\rm FS}([\phi],[\psi]) = \arccos|\langle\phi|\psi\rangle|.$$

The absolute value appears because the phases of representatives are arbitrary. One may choose phases so that $\langle\phi|\psi\rangle$ is real and nonnegative, and the shortest path lies in the two-dimensional complex subspace spanned by the two vectors.

If the rays are not orthogonal, a convenient geodesic representative is

$$\lvert\psi(s)\rangle = \frac{ \sin(d_{\rm FS}-s)\lvert\psi\rangle + \sin s\,\lvert\phi_{\parallel}\rangle } {\sin d_{\rm FS}}, \qquad 0\le s\le d_{\rm FS},$$

where $\lvert\phi_{\parallel}\rangle$ is the phase-adjusted representative of the final ray with positive overlap with $\lvert\psi\rangle$. This formula is meant as geometry, not as a claim that the system dynamically follows this path under a given Hamiltonian.

Two-Level Example

For a qubit,

$$\mathbb P(\mathbb C^2) = \mathbb{CP}^1,$$

which is the Bloch sphere. A normalized representative may be written

$$\lvert\psi\rangle = \cos\frac{\theta}{2}\lvert0\rangle + e^{i\phi}\sin\frac{\theta}{2}\lvert1\rangle.$$

The Bloch vector is

$$\mathbf n = (\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta).$$

With the distance convention used here, the Fubini–Study line element becomes

$$ds_{\rm FS}^2 = \frac14 \left( d\theta^2 + \sin^2\theta\,d\phi^2 \right).$$

Thus $\mathbb{CP}^1$ is a sphere of radius $1/2$ in this convention. The ordinary angular separation $\gamma$ between two Bloch vectors satisfies

$$|\langle\phi|\psi\rangle|^2 = \frac{1+\mathbf n_\phi\cdot\mathbf n_\psi}{2} = \cos^2\frac{\gamma}{2},$$

so

$$d_{\rm FS} = \frac{\gamma}{2}.$$

Antipodal Bloch vectors correspond to orthogonal qubit states and have $d_{\rm FS}=\pi/2$.

Local Complex Coordinate

On the qubit chart where the coefficient of $\lvert0\rangle$ is nonzero, write

$$z = \frac{c_1}{c_0}, \qquad \lvert\psi(z)\rangle = \frac{\lvert0\rangle+z\lvert1\rangle} {\sqrt{1+|z|^2}}.$$

Then the Fubini–Study line element is

$$ds_{\rm FS}^2 = \frac{|dz|^2}{(1+|z|^2)^2}.$$

Using

$$z=e^{i\phi}\tan\frac{\theta}{2}$$

recovers the radius-$1/2$ sphere formula above. In higher-dimensional finite systems, similar local coordinates turn $\mathbb{CP}^{N-1}$ into a complex manifold with a natural Hermitian metric.

Relation to Energy Uncertainty

The metric speed is one side of the projective geometry. The complementary symplectic derivation of the same Schrödinger path is developed in Hamiltonian Flow on Projective Hilbert Space.

For Schrödinger evolution,

$$i\hbar\frac{d}{dt}\lvert\psi(t)\rangle = H\lvert\psi(t)\rangle,$$

the physical speed of the ray is

$$\frac{ds_{\rm FS}}{dt} = \frac{\Delta_\psi H}{\hbar},$$

where

$$(\Delta_\psi H)^2 = \langle H^2\rangle_\psi - \langle H\rangle_\psi^2.$$

This formula has a clean interpretation. The component of $H\lvert\psi\rangle$ proportional to $\lvert\psi\rangle$ only changes the phase of the vector representative. The component orthogonal to $\lvert\psi\rangle$ changes the ray. Energy uncertainty measures exactly the size of that orthogonal component.

An energy eigenstate has $\Delta_\psi H=0$, so its ray is stationary even though the vector accumulates a phase.

Quantum Speed Limit Preview

The length of any path in projective Hilbert space is at least the geodesic distance between its endpoints. Therefore Schrödinger evolution implies

$$\arccos|\langle\psi(0)|\psi(T)\rangle| \le \frac{1}{\hbar} \int_0^T dt\,\Delta_{\psi(t)}H.$$

For time-independent $\Delta H$, this gives the Mandelstam–Tamm form

$$T \ge \frac{\hbar}{\Delta H} \arccos|\langle\psi(0)|\psi(T)\rangle|.$$

For evolution to an orthogonal state,

$$T \ge \frac{\pi\hbar}{2\Delta H}.$$

This is a geometry statement about the minimum projective distance that must be traversed. Stronger or different speed limits can use additional assumptions, energy above the ground state, open-system metrics, or mixed-state geometry. This page only previews the pure-state Fubini–Study version.

Relation to Berry Geometry

The detailed bridge, including the quantum geometric tensor and the pullback of the Fubini–Study two-form to Berry curvature, is Relation to Berry Geometry.

The Fubini–Study metric is only part of the natural geometry of projective Hilbert space. The imaginary part of the Hilbert-space inner product gives a compatible symplectic form, and the phase bundle over ray space gives a natural Berry connection.

The metric answers:

$$\text{How far apart are the rays?}$$

The Berry connection answers:

$$\text{How does a phase convention twist along a path?}$$

Both structures come from the same Hilbert-space inner product, but they encode different physics. The metric controls distinguishability and speed; the connection controls holonomy and geometric phase.

Common Mistakes

• Forgetting the convention factor: some references use $2\arccos|\langle\phi|\psi\rangle|$ instead of $\arccos|\langle\phi|\psi\rangle|$.
• Treating the Bloch sphere as radius $1$ while using formulas normalized for radius $1/2$.
• Applying the Fubini–Study distance to mixed states without switching to an appropriate mixed-state metric.
• Confusing geodesic distance with the actual dynamical path length under a specific Hamiltonian.
• Forgetting that vector phase changes must have zero projective length.
• Thinking energy expectation $\langle H\rangle$ controls ray speed; it is energy uncertainty $\Delta H$ that controls the Fubini–Study speed.

Cross-Links

• Projective Hilbert Space explains rays, projectors, and tangent directions.
• Geometric Quantum Mechanics Overview places the metric beside the symplectic form and Hamiltonian flow.
• Hamiltonian Flow on Projective Hilbert Space derives the symplectic flow and quantum Poisson bracket.
• Relation to Berry Geometry relates the projective metric and symplectic form to the quantum metric and Berry curvature.
• Transition Probabilities gives the Born-rule invariant that becomes distance.
• Classical and Quantum Fisher Information uses four times this pure-state metric as the local quantum Fisher information and extends the construction to mixed states through Bures geometry.
• Bloch Sphere develops the two-level geometry.
• Berry Phase develops the phase-holonomy side of the same ray-space geometry.
• Adiabatic Theorem is one setting where metric separation, gaps, and Berry geometry meet.
• Formula Sheet collects the main dynamics formulas.

References

• J. P. Provost and G. Vallee, “Riemannian structure on manifolds of quantum states,” Communications in Mathematical Physics 76, 289, 1980.
• J. Anandan and Y. Aharonov, “Geometry of quantum evolution,” Physical Review Letters 65, 1697, 1990.
• T. W. B. Kibble, “Geometrization of quantum mechanics,” Communications in Mathematical Physics 65, 189, 1979.
• I. Bengtsson and K. Zyczkowski, Geometry of Quantum States, 2nd ed., Cambridge University Press, 2017.
• D. C. Brody and L. P. Hughston, “Geometric quantum mechanics,” Journal of Geometry and Physics 38, 19, 2001.
• L. Mandelstam and I. Tamm, “The uncertainty relation between energy and time in non-relativistic quantum mechanics,” Journal of Physics (USSR) 9, 249, 1945.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary ed., Cambridge University Press, 2010.

Exercises

1. Show that the infinitesimal Fubini–Study line element vanishes for pure phase motion.

Solution

Let

$$\lvert d\psi\rangle = i\,d\alpha\,\lvert\psi\rangle$$

with $\langle\psi|\psi\rangle=1$. Then

$$\langle d\psi|d\psi\rangle = d\alpha^2, \qquad \langle\psi|d\psi\rangle = i\,d\alpha.$$

Therefore

$$ds_{\rm FS}^2 = \langle d\psi|d\psi\rangle - |\langle\psi|d\psi\rangle|^2 = d\alpha^2-d\alpha^2 = 0.$$

The vector moved along the phase fiber, but the ray did not move.

2. Compute the Fubini–Study distance between orthogonal normalized states.

Solution

If $\langle\phi|\psi\rangle=0$, then

$$d_{\rm FS}([\phi],[\psi]) = \arccos 0 = \frac{\pi}{2}.$$

Orthogonal rays are maximally separated in this convention.

3. Derive the qubit relation $d_{\rm FS}=\gamma/2$.

Solution

For qubit pure states with Bloch vectors $\mathbf n_\psi$ and $\mathbf n_\phi$,

$$|\langle\phi|\psi\rangle|^2 = \frac{1+\mathbf n_\phi\cdot\mathbf n_\psi}{2}.$$

If $\gamma$ is the ordinary angle between the Bloch vectors, then

$$\mathbf n_\phi\cdot\mathbf n_\psi = \cos\gamma.$$

Hence

$$|\langle\phi|\psi\rangle|^2 = \frac{1+\cos\gamma}{2} = \cos^2\frac{\gamma}{2}.$$

Taking the positive square root and applying $d_{\rm FS}=\arccos|\langle\phi|\psi\rangle|$ gives

$$d_{\rm FS} = \frac{\gamma}{2},$$

with $0\le\gamma\le\pi$.

4. Show that Schrödinger evolution has Fubini–Study speed $\Delta H/\hbar$.

Solution

Schrödinger evolution gives

$$\lvert\dot\psi\rangle = -\frac{i}{\hbar}H\lvert\psi\rangle.$$

The Fubini–Study speed is

$$\left(\frac{ds_{\rm FS}}{dt}\right)^2 = \langle\dot\psi|\dot\psi\rangle - |\langle\psi|\dot\psi\rangle|^2.$$

Compute the two terms:

$$\langle\dot\psi|\dot\psi\rangle = \frac{1}{\hbar^2}\langle H^2\rangle_\psi, \qquad \langle\psi|\dot\psi\rangle = -\frac{i}{\hbar}\langle H\rangle_\psi.$$

Therefore

$$\left(\frac{ds_{\rm FS}}{dt}\right)^2 = \frac{1}{\hbar^2} \left( \langle H^2\rangle_\psi - \langle H\rangle_\psi^2 \right) = \frac{(\Delta_\psi H)^2}{\hbar^2}.$$

Taking the positive square root gives $ds_{\rm FS}/dt=\Delta_\psi H/\hbar$.

5. Use the speed formula to obtain the orthogonal-state Mandelstam–Tamm bound for constant $\Delta H$.

Solution

The projective path length obeys

$$L = \int_0^T dt\, \frac{\Delta H}{\hbar} = \frac{\Delta H}{\hbar}T$$

when $\Delta H$ is constant. Any path connecting orthogonal states must have length at least the geodesic distance $\pi/2$. Thus

$$\frac{\Delta H}{\hbar}T \ge \frac{\pi}{2},$$

or

$$T \ge \frac{\pi\hbar}{2\Delta H}.$$