Projective Hilbert Space

• Projective Hilbert space is the quotient of nonzero Hilbert space vectors by scalar multiplication, representing pure quantum states without global phase.
• It features a rich geometric structure with the Fubini–Study metric, Kähler properties, and symplectic forms that model quantum dynamics and measurement.
• Its topology and symmetry groups underpin critical quantum concepts such as phase retrieval, quantum probability, and Hamiltonian evolution.

A projective Hilbert space is the quotient manifold of nonzero elements of a Hilbert space modulo the action of scalar multiplication, primarily used to treat quantum states and their geometric properties. Each point represents a physical pure state, abstracting away global phase, and the resulting space supports a rich Kähler, Riemannian, and symplectic structure. This geometry underpins fundamental aspects of quantum mechanics and quantum information, connecting measurement, state evolution, and probabilistic phenomena to projective geometry.

1. Formal Definitions and Basic Structure

Let $H$ denote a (real or complex) Hilbert space. The projective Hilbert space $P(H)$ comprises equivalence classes of nonzero vectors under the relation $x \sim \lambda x$, where $\lambda \in \mathbb{C}^\times$ (or $\lambda \in \mathbb{R}^\times$ for real $H$):

$P(H) = H^\times / \mathbb{C}^\times$

Alternatively, using the unit sphere $S(H) = \{ x \in H : \|x\| = 1 \}$ and the group $U(1)$ of complex phases, the construction yields

$P(H) = S(H)/U(1)$

Each point $[x]\in P(H)$ identifies the one-dimensional subspace $Span\{x\}\subset H$; in quantum physics, this is a pure state (Sontz, 23 Oct 2024, Neyshaburi et al., 9 Aug 2024, Alsing et al., 2023).

For finite dimension $n$ (e.g., $H\cong \mathbb{C}^n$), $P(H)$ is the complex projective space $\mathbb{CP}^{n-1}$, a smooth manifold of complex dimension $n-1$ and real dimension $2n-2$ (Pastorello, 2015).

2. Metric, Topology, and Symplectic Structure

Fubini–Study Metric

On $P(H)$ there exists a unique unitarily invariant Riemannian metric—the Fubini–Study metric. For representatives $x, y \in S(H)$,

$d_{FS}([x],[y]) = \arccos\, |\langle x, y \rangle|$

$$ds^2 = \langle dx, dx \rangle - |\langle x, dx \rangle|^2$$

This metric induces the quotient topology, and for finite dimension coincides with the standard topology on $\mathbb{CP}^{n-1}$ (Pastorello, 2015, Sontz, 23 Oct 2024, Neyshaburi et al., 9 Aug 2024, Alsing et al., 2023).

Kähler Structure

$P(H)$ supports a Kähler structure: its complex structure $J$, symplectic form $$\omega_{FS} = i \partial \bar{\partial} \log(1 + |w|^2)$$ (in homogeneous coordinates), and Fubini–Study metric $g$ interact compatibly. The associated symplectic form is non-degenerate and closed (Pastorello, 2015).

Topological Subtleties

The topology of $P(H)$ reflects phase retrieval properties. Given a measurement frame $\Phi = \{ \varphi_j \}$, the modulus-of-coefficients map

$$\mathcal{A}_\Phi([x]) = \{ |\langle x, \varphi_j \rangle | \}_{j}$$

is continuous under the quotient topology, and its injectivity up to global phase is equivalent to phase retrieval. In finite dimension, the initial topology generated by these functions coincides with the quotient topology precisely when $\Phi$ does phase retrieval (Neyshaburi et al., 9 Aug 2024).

3. Orthogonality, Projective Subspaces, and Quantum Logic

Orthogonality and subspace structure in projective Hilbert space are foundational in quantum logic and measurement theory. For $[x], [y] \in P(H)$,

$[x] \perp [y] \iff \langle x, y \rangle = 0$

Projective subspaces $S_V \subset P(H)$ correspond to equivalence classes of vectors in closed subspaces $V \subset H$, representing quantum events (propositions):

$S_V = \pi_V(V^\times) \subset P(H)$

The lattice of closed subspaces of $H$ is isomorphic to the lattice of projective events under the operations of intersection and closed span. Orthogonality in the projective space mirrors orthogonal projections in $H$ (Sontz, 23 Oct 2024).

4. Dynamical and Probabilistic Interpretation

Quantum Probability

Quantum probabilities are geometrically encoded in $P(H)$. Born’s rule for outcome probability in state $[x]$ for event $S_V$ reads

$$P_{[x]}(S_V) = \|\mathrm{P}_V x\|^2 = \cos^2 \theta([x], S_V)$$

where $\theta([x], S_V)$ is the angle from $[x]$ to $V$.

Sequential and conditional probabilities (Wigner's rule) and state collapse are given by chains of projective projections and normed transitions (Sontz, 23 Oct 2024).

Hamiltonian Dynamics

Quantum evolution on $P(H)$ proceeds via Hamiltonian flow. A Hermitian operator $A$ yields a function

$f_A([x]) = \langle x, A x \rangle$

This function generates a Hamiltonian vector field

$X_{f_A}([x]) = -i [A, p_x]$

where $p_x = |x\rangle\langle x| / \langle x|x \rangle$.

Quantum dynamics (Schrödinger flow) in projective space:

$\dot{p}_x(t) = -i [H, p_x(t)]$

Thus, projective Hilbert space serves as a bona fide phase space for quantum systems with Hamiltonian flows defined by the Fubini–Study symplectic structure (Pastorello, 2015).

5. Group Actions, Automorphisms, and Reconstruction Principles

Projective Hilbert spaces admit rich symmetry groups. The full group of bijections preserving orthogonality (for real $H$) is the projective orthogonal group $\mathrm{PO}(H) = \mathrm{O}(H)/\{\pm 1\}$, or for complex $H$, the projective unitary group $\mathrm{PU}(n)$ in finite dimension.

The automorphism group is characterized by actions transitive on orthogonal complements and conjugacy of subgroups fixing pairs; under homogeneously transitive and divisible transitivity conditions, an orthoset arises from a projective Hermitian space, and in particular, additional symmetry constraints single out real Hilbert spaces (Vetterlein, 2021).

A summary of the reconstruction principle is given in (Vetterlein, 2021):

Step	Construction	Consequence
1	Form complete ortholattice from orthogonality relation	Ortholattice atoms are projective points
2	Show modular, atomistic, irreducible ortholattice (via symmetry)	Equivalent to subspaces of Hermitian vector space
3	Projective points correspond to $1$-dim subspaces	Automorphism group matches unitary group
4	Divisible transitivity $\to$ commutative division ring	Positive-definite quadratic form, field $\subseteq \mathbb{R}$
5	Final identification	Standard real projective Hilbert space structure

6. Phase Retrieval, Quantum Information, and Statistical Geometry

Phase retrieval investigates whether modulus measurements $\{ |\langle x, \varphi_j \rangle | \}$ uniquely determine the pure state $[x]$. This is centrally a geometric property of the topology and metric on $P(H)$. Metrics relevant include Fubini–Study, Bures–Wasserstein, and frame-induced distances; in finite dimension with sufficient measurement richness, these are all bi-Lipschitz equivalent, and the topology is compact, metrizable (Neyshaburi et al., 9 Aug 2024).

In dimension two, $P(\mathbb{C}^2)\cong S^2$ (the Bloch sphere), with the Fubini–Study distance corresponding to half the Euclidean angle. Injective measurement schemes correspond to frames satisfying the complement/full-spark property in the real case (Neyshaburi et al., 9 Aug 2024).

7. Geometric Constraints on Quantum Evolution

Kinematic invariants in projective Hilbert space—including quantum speed, acceleration, curvature, and torsion—impose fundamental bounds on system evolution. The instantaneous speed in the Fubini–Study metric is proportional to the energy uncertainty,

$v(t) = \Delta H / \hbar$

The acceleration $a(t)$ is bounded above by the variance of the time derivative of the Hamiltonian:

$a(t)^2 \leq \mathrm{Var}(\dot{H}(t))$

with curvature $\kappa$ and torsion $\tau$ expressible via covariant projective derivatives of the generator $h(t)$. These geometric constraints govern trade-offs in quantum control, speed, and robustness, establishing ultimate limits for state manipulations (Alsing et al., 2023).

• Sontz, Quantum Probability Geometrically Realized in Projective Space (Sontz, 23 Oct 2024)
• Vetterlein, Transitivity and homogeneity of orthosets and the real Hilbert spaces (Vetterlein, 2021)
• Arabyani et al., Topological structure of projective Hilbert spaces associated with phase retrieval vectors (Neyshaburi et al., 9 Aug 2024)
• Chruściński et al., A geometric approach to quantum control in projective Hilbert spaces (Pastorello, 2015)
• Alsing & Cafaro, Upper limit on the acceleration of a quantum evolution in projective Hilbert space (Alsing et al., 2023)