Projective Hilbert Space

Projective Hilbert space is the pure-state space after overall scale and global phase have been removed. The Core Formalism page Projective Hilbert Space gives the canonical quotient construction. This page uses that construction for dynamics: it treats rays as the points that actually move, vector phases as choices of lift, and Schrödinger evolution as a phase-free motion of projectors.

The main lesson is simple but easy to underuse. A state vector is a convenient coordinate on pure-state physics, not the physical point itself. Dynamics, distinguishability, Berry phases, and qubit geometry all become cleaner when one remembers that pure states live in the projective space $\mathbb P(\mathcal H)$.

When exact motion is restricted to a lower-dimensional manifold of rays, the pullback of this tangent geometry leads to the Time-Dependent Variational Principle. That page owns tangent projection, residuals, and reduced parameter dynamics.

Ray Space as State Space

Let $\mathcal H$ be a complex Hilbert space. Nonzero vectors that differ by a nonzero complex factor represent the same pure state:

$$\lvert\psi\rangle \sim \lambda\lvert\psi\rangle, \qquad \lambda\in\mathbb C^\times.$$

The projective Hilbert space is the quotient

$$\mathbb P(\mathcal H) = (\mathcal H\setminus\{0\})/\mathbb C^\times.$$

Equivalently, one may restrict to normalized representatives,

$$S(\mathcal H) = \{\lvert\psi\rangle: \langle\psi|\psi\rangle=1\},$$

and then quotient by phase:

$$\mathbb P(\mathcal H) = S(\mathcal H)/U(1).$$

For dynamics, the second form is often the most useful. The unit sphere $S(\mathcal H)$ is the space of normalized vector representatives. The physical pure-state path is its projection to $\mathbb P(\mathcal H)$.

Projectors as Phase-Free Representatives

A ray can be represented without choosing a phase by the rank-one projector

$$\Pi_\psi = \frac{\lvert\psi\rangle\langle\psi\rvert} {\langle\psi|\psi\rangle}.$$

This object is invariant under nonzero rescaling:

$$\Pi_{\lambda\psi} = \Pi_\psi, \qquad \lambda\neq0.$$

For normalized representatives, $\Pi_\psi=\lvert\psi\rangle\langle\psi\rvert$. This is often the cleanest bridge between pure-state geometry and the density-operator language:

$$\Pi_\psi^2=\Pi_\psi, \qquad \Pi_\psi^\dagger=\Pi_\psi, \qquad \operatorname{Tr}\Pi_\psi=1.$$

Thus pure states are precisely the rank-one density operators. The ray $[\psi]$ and the projector $\Pi_\psi$ contain the same physical pure-state information, but the projector makes phase redundancy invisible.

Why Dynamics Descends to Rays

The Schrödinger equation for a normalized vector is

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

If $H$ is shifted by a scalar multiple of the identity,

$$H' = H+cI,$$

then the vector solution changes by an overall phase:

$$\lvert\psi'(t)\rangle = e^{-ict/\hbar}\lvert\psi(t)\rangle.$$

The ray is unchanged. This is why absolute energy zero does not affect closed-system pure-state motion in projective Hilbert space; only differences and couplings that change the ray matter.

The projector equation makes this phase independence explicit. For a pure state,

$$\frac{d\Pi}{dt} = -\frac{i}{\hbar}[H,\Pi].$$

The same equation is the pure-state special case of the Liouville–von Neumann equation for density operators. It is often the best way to see that Schrödinger dynamics has a well-defined projection to $\mathbb P(\mathcal H)$.

Vertical and Horizontal Motion

Let $\lvert\psi(t)\rangle$ be a normalized curve. Differentiating $\langle\psi|\psi\rangle=1$ gives

$$\operatorname{Re}\langle\psi|\dot\psi\rangle=0.$$

The component

$$\lvert\dot\psi\rangle_{\rm vert} = \lvert\psi\rangle\langle\psi|\dot\psi\rangle$$

points along the phase direction. It changes the chosen representative but not the ray. The physically moving part is the horizontal component

$$\lvert\dot\psi\rangle_{\perp} = (I-\lvert\psi\rangle\langle\psi\rvert) \lvert\dot\psi\rangle,$$

which obeys

$$\langle\psi|\dot\psi_\perp\rangle=0.$$

For Schrödinger evolution,

$$\lvert\dot\psi\rangle_{\perp} = -\frac{i}{\hbar} \bigl(H-\langle H\rangle_\psi\bigr) \lvert\psi\rangle.$$

The subtracted expectation value removes the instantaneous phase rotation. In this form, the tangent vector depends only on the ray and on the Hamiltonian up to an additive scalar.

Tangent Vectors in Projector Form

The tangent direction at a ray can also be written as a variation of the projector. If $\lvert\delta\psi\rangle$ is a variation of a normalized representative, define

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

Then

$$\delta\Pi = \lvert\delta\psi_\perp\rangle\langle\psi\rvert + \lvert\psi\rangle\langle\delta\psi_\perp\rvert.$$

Adding a vertical phase variation $i\chi\lvert\psi\rangle$ to $\lvert\delta\psi\rangle$ leaves $\delta\Pi$ unchanged. This is the local version of the quotient: tangent vectors to projective Hilbert space are Hilbert-space variations modulo the phase direction.

The projector form is especially useful when comparing pure-state geometry with density-matrix dynamics. Mixed states live in a convex set of density operators; pure states form the nonlinear boundary subset of rank-one projectors.

Transition Probabilities Are Projective Data

For normalized representatives, the transition probability between two pure states is

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

This number depends only on the rays, since phase changes give

$$\lvert\psi\rangle\mapsto e^{i\alpha}\lvert\psi\rangle, \qquad \lvert\phi\rangle\mapsto e^{i\beta}\lvert\phi\rangle,$$

and therefore

$$|\langle\phi|\psi\rangle|^2 \mapsto |e^{i(\alpha-\beta)}\langle\phi|\psi\rangle|^2 = |\langle\phi|\psi\rangle|^2.$$

In projector language the same invariant is

$$\operatorname{Tr}(\Pi_\phi\Pi_\psi) = |\langle\phi|\psi\rangle|^2.$$

This is the seed of the Fubini–Study geometry: distance between pure states is ultimately distance as measured by transition probabilities. Orthogonal rays are perfectly distinguishable in a single ideal projective measurement; identical rays have transition probability one.

Two-Level Example

For a two-level system,

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

which is the Bloch sphere. On the chart where the coefficient of $\lvert0\rangle$ is nonzero, a ray may be represented by the complex coordinate

$$z = \frac{c_1}{c_0}, \qquad \lvert\psi\rangle \sim c_0\lvert0\rangle+c_1\lvert1\rangle.$$

A normalized representative is

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

Writing

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

gives the usual Bloch-sphere form

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

The global phase has disappeared. The relative phase remains as the azimuthal angle $\phi$. This is why the Bloch sphere is not the Hilbert space $\mathbb C^2$ itself; it is the projective pure-state space of a two-dimensional Hilbert space.

Relation to Quantum Information

Quantum information often represents pure states by normalized vectors because vectors are efficient for circuits, amplitudes, and tensor products. Projective Hilbert space explains which vector distinctions are physically redundant.

For a single qubit, pure states form $\mathbb{CP}^1$. For $n$ qubits, the Hilbert space is

$$\mathcal H = (\mathbb C^2)^{\otimes n}, \qquad \dim\mathcal H=2^n,$$

so the pure-state space is

$$\mathbb P(\mathcal H) = \mathbb{CP}^{2^n-1}.$$

This projective space is far larger than a product of single-qubit Bloch spheres. Entangled pure states are ordinary points of $\mathbb{CP}^{2^n-1}$ that cannot be written as product rays.

The warning is important: projective Hilbert space is not a vector space. One forms superpositions in $\mathcal H$ using representatives, then passes back to rays. That is why Hilbert-space vectors remain indispensable even though physical pure states are rays.

Relation to Fubini–Study Geometry

Projective Hilbert space is not merely a set of equivalence classes. The Hilbert-space inner product induces a natural metric, symplectic form, and complex structure on the pure-state space. Together these structures make finite-dimensional $\mathbb{CP}^{N-1}$ a Kähler manifold.

The most immediate metric fact is

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

for normalized representatives, in one common convention. Different communities sometimes use a distance differing by a factor of $2$, especially when comparing with a unit-radius Bloch sphere. The invariant content is the transition probability, not the convention-dependent scale.

The metric side is developed in Fubini–Study Geometry. The symplectic derivation of pure-state dynamics belongs to Hamiltonian Flow on Projective Hilbert Space. The broader picture is summarized in Geometric Quantum Mechanics Overview.

Relation to Berry Phase

The universal phase-bundle geometry and its pullback to parameter-space Berry data are connected explicitly in Relation to Berry Geometry.

The projection

$$S(\mathcal H)\to\mathbb P(\mathcal H)$$

has the structure of a $U(1)$ phase bundle. A curve of rays may be lifted to a curve of normalized vectors, but the lift is not unique. Choosing a phase convention is like choosing a gauge along the path.

If a ray follows a closed loop,

$$[\psi(T)]=[\psi(0)],$$

a chosen lift may return as

$$\lvert\psi(T)\rangle = e^{i\gamma}\lvert\psi(0)\rangle.$$

This does not contradict the redundancy of instantaneous global phase. The phase $\gamma$ compares a whole lifted path with its starting representative, and it can be observed through interference against another history. Berry phase is the canonical physical setting for this holonomy; see Berry Phase and U(1) Bundles and Quantum Phase.

Common Mistakes

• Treating a vector phase as physical data of a single pure state.
• Forgetting that relative phases inside a superposition remain physical even though global phase is removed.
• Confusing $\mathbb P(\mathcal H)$ with $\mathcal H$ itself.
• Thinking projective Hilbert space is a classical phase space of positions and momenta.
• Assuming projective Hilbert space is linear. It is a manifold of rays, not a vector space.
• Removing the Hilbert-space representative too early and then losing the ability to form superpositions or tensor products.
• Treating Berry phase as ordinary instantaneous global phase rather than holonomy of a lifted path.
• Forgetting that the pure-state projective picture does not by itself describe mixed states or open-system dynamics.

Cross-Links

• Core Projective Hilbert Space gives the canonical quotient construction.
• Rays and Global Phase gives the first-pass physical interpretation.
• Transition Probabilities explains the Born-rule invariant $|\langle\phi|\psi\rangle|^2$.
• Density Operators gives the mixed-state setting that contains pure projectors as a special case.
• Schrödinger Picture reviews the vector evolution whose phase-free projection is used here.
• Liouville–von Neumann Equation gives the density-operator dynamics containing $\dot\Pi=-(i/\hbar)[H,\Pi]$.
• Bloch Sphere develops the two-level example.
• Fubini–Study Geometry develops the natural distance, metric, and quantum-speed interpretation of ray space.
• Hamiltonian Flow on Projective Hilbert Space derives the symplectic flow generated by energy expectation.
• Relation to Berry Geometry explains how eigenstate families pull the universal phase bundle back to parameter space.
• Geometric Quantum Mechanics Overview places projective Hilbert space inside the metric, symplectic, and Hamiltonian-flow formulation.

References

• P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford University Press, 1958.
• J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.
• T. W. B. Kibble, “Geometrization of quantum mechanics,” Communications in Mathematical Physics 65, 189, 1979.
• A. Ashtekar and T. A. Schilling, “Geometrical formulation of quantum mechanics,” in On Einstein’s Path, Springer, 1999.
• J. P. Provost and G. Vallee, “Riemannian structure on manifolds of quantum states,” Communications in Mathematical Physics 76, 289, 1980.
• 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.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary ed., Cambridge University Press, 2010.

Exercises

1. Show that the projector equation follows from Schrödinger evolution.

Solution

For a normalized state, let

$$\Pi = \lvert\psi\rangle\langle\psi\rvert.$$

The Schrödinger equation gives

$$\lvert\dot\psi\rangle = -\frac{i}{\hbar}H\lvert\psi\rangle, \qquad \langle\dot\psi\rvert = \frac{i}{\hbar}\langle\psi\rvert H.$$

Therefore

$$\dot\Pi = \lvert\dot\psi\rangle\langle\psi\rvert + \lvert\psi\rangle\langle\dot\psi\rvert = -\frac{i}{\hbar}H\Pi + \frac{i}{\hbar}\Pi H.$$

Thus

$$\dot\Pi = -\frac{i}{\hbar}[H,\Pi].$$

2. Verify that adding a scalar to the Hamiltonian does not change the ray motion.

Solution

Let $H'=H+cI$. The vector equation becomes

$$i\hbar\frac{d}{dt}\lvert\psi'\rangle = (H+cI)\lvert\psi'\rangle.$$

If $\lvert\psi(t)\rangle$ solves the equation for $H$, then

$$\lvert\psi'(t)\rangle = e^{-ict/\hbar}\lvert\psi(t)\rangle$$

solves the equation for $H'$. The two vectors differ only by phase at each time, so they define the same ray.

In projector form,

$$[H+cI,\Pi] = [H,\Pi]+c[I,\Pi] = [H,\Pi],$$

so $\dot\Pi$ is unchanged directly.

3. Compute the projective coordinate $z$ for a qubit state.

Solution

For

$$\lvert\psi\rangle = \frac{\sqrt3}{2}\lvert0\rangle + \frac{i}{2}\lvert1\rangle,$$

the chart coordinate with $c_0\neq0$ is

$$z = \frac{c_1}{c_0} = \frac{i/2}{\sqrt3/2} = \frac{i}{\sqrt3}.$$

Thus $|z|=\tan(\theta/2)=1/\sqrt3$, so $\theta=\pi/3$, and the relative phase is $\phi=\pi/2$.

4. Show that the transition probability is the trace of two pure projectors.

Solution

For normalized states,

$$\Pi_\psi = \lvert\psi\rangle\langle\psi\rvert, \qquad \Pi_\phi = \lvert\phi\rangle\langle\phi\rvert.$$

Then

$$\Pi_\phi\Pi_\psi = \lvert\phi\rangle \langle\phi|\psi\rangle \langle\psi\rvert.$$

Taking the trace gives

$$\operatorname{Tr}(\Pi_\phi\Pi_\psi) = \langle\phi|\psi\rangle \operatorname{Tr}(\lvert\phi\rangle\langle\psi\rvert) = \langle\phi|\psi\rangle \langle\psi|\phi\rangle = |\langle\phi|\psi\rangle|^2.$$

5. Explain why projective Hilbert space is not enough to replace Hilbert space in calculations.

Solution

Projective Hilbert space removes the physically redundant global scale and phase of a pure state. That is correct for identifying physical pure-state points.

However, many calculations require operations in the vector space $\mathcal H$: forming superpositions, applying linear operators, taking tensor products, and computing amplitudes before taking absolute squares. Projective Hilbert space records the equivalence classes of nonzero vectors, but it is not itself a vector space. In practice one uses Hilbert-space representatives for calculation and projective language to remember which representative choices are gauge-like.