# Quantum state space

The quantum state space of a physical system is the set of all states the system can occupy, described mathematically as a complex [Hilbert space](https://www.edgepedia.org/hilbert-space) for pure states and as the convex set of density operators on that Hilbert space for general states. In the standard formulation, every physical system is associated with a Hilbert space, and the pure states of that system are unit vectors in it<sup>[1](https://www.cs.tufts.edu/~ablumer/QuantumLogic/principles.html)</sup>. The structure of this space, its identification of vectors that differ only by a phase, its convex geometry, and its behavior under composition, carries most of the distinctive predictions of quantum mechanics.

| Key fact | Detail |
|---|---|
| Pure states | Rays in projective Hilbert space \( P(\mathcal{H}) \); normalized vectors differing by a global phase \( e^{i\alpha} \) represent the same state<sup>[2](https://ir.soken.ac.jp/record/6647/files/A2402%E6%9C%AC%E6%96%87.pdf)</sup> |
| General states | Density operators: Hermitian, positive, unit-trace operators; the set of order-\( D \) density matrices is convex with \( D^2 - 1 \) real dimensions<sup>[3](https://chaos.if.uj.edu.pl/~karol/pdf/PMGDHZ12.pdf)</sup> |
| Purity test | A density operator is pure if and only if \( \rho^2 = \rho \); purity is measured by \( \mathrm{Tr}\,\rho^2 \)<sup>[4](https://web.gps.caltech.edu/~rls/book.pdf)</sup><sup> • </sup><sup>[5](https://lim.physics.ucdavis.edu/teaching/files/qi-notes-partiii.pdf)</sup> |
| One qubit | Pure states form the Bloch sphere \( \mathbb{CP}^1 \), the boundary of the Bloch ball of mixed states embedded in \( \mathbb{R}^3 \)<sup>[3](https://chaos.if.uj.edu.pl/~karol/pdf/PMGDHZ12.pdf)</sup><sup> • </sup><sup>[6](https://arxiv.org/abs/1112.2347)</sup> |
| Composition | The state space of a composite system is the tensor product \( H_A \otimes H_B \), and the state of the composite is not uniquely defined by the states of its components<sup>[7](https://plato.stanford.edu/entries/qm/)</sup> |
| Measurement | A projective measurement of observable \( O \) yields eigenvalue \( o_n \) with probability \( p_n = \langle\psi|P_n|\psi\rangle \), the Born rule, generalizing to \( p_n = \mathrm{Tr}\,P_n\rho \)<sup>[8](https://www.cpt.univ-mrs.fr/~verga/pages/AQ-principles.html)</sup> |
| Geometry | The pure-state manifold is a complex projective space with the unitary-invariant Fubini–Study geometry<sup>[9](https://arxiv.org/abs/quant-ph/9906086)</sup> |

## Pure states and rays

For an \( n \)-dimensional complex Hilbert space \( \mathcal{H} \), the normalized state vectors span the unit sphere \( S^{2n-1} \) in \( \mathbb{C}^n \). Physical states, however, are not individual vectors: a (projective) ray \( [|\psi\rangle] \) is an equivalence class of state vectors under \( |\psi\rangle \sim \lambda|\psi\rangle \) for any nonzero complex \( \lambda \), and for a normalized state \( (\langle\psi|\psi\rangle = 1) \) this reduces to identification up to a global phase, \( |\psi\rangle \sim e^{i\alpha}|\psi\rangle \) with \( \alpha \in \mathbb{R} \)<sup>[2](https://ir.soken.ac.jp/record/6647/files/A2402%E6%9C%AC%E6%96%87.pdf)</sup>. The projective Hilbert space \( P(\mathcal{H}) \) is the set of these rays, mathematically the projectivization of \( \mathcal{H} \)<sup>[2](https://ir.soken.ac.jp/record/6647/files/A2402%E6%9C%AC%E6%96%87.pdf)</sup>.

The phase identification has a physical basis: quantum symmetries act on physical rays, not on arbitrarily phased state vectors, so two vectors differing only by a phase are the same state for every purpose of prediction<sup>[10](https://qft.org/symmetry-gauge/symmetry-actions-redundancy/quantum-implementations-projective-actions-central-extensions/)</sup>. In practice, notation is simplified by writing a pure state as \( |\psi\rangle \in \mathcal{H} \), an abuse of notation made for convenience<sup>[2](https://ir.soken.ac.jp/record/6647/files/A2402%E6%9C%AC%E6%96%87.pdf)</sup>.

The set of all pure states acting on a \( D \)-dimensional Hilbert space is the complex projective space \( \mathbb{CP}^{D-1} \)<sup>[3](https://chaos.if.uj.edu.pl/~karol/pdf/PMGDHZ12.pdf)</sup>. This manifold carries the [Fubini–Study metric](https://www.edgepedia.org/fubini-study-metric), the unitary-invariant geometry under which the distance between states is unchanged when both are transformed by the same unitary operator<sup>[9](https://arxiv.org/abs/quant-ph/9906086)</sup>.

## Mixed states and density operators

**Density operators.** Describing a probabilistic mixture of pure states requires the density matrix formalism. A density operator \( \rho \) on \( \mathcal{H} \) is a positive semidefinite operator with unit trace, \( \mathrm{Tr}\,\rho = 1 \), which gives it a probabilistic interpretation<sup>[5](https://lim.physics.ucdavis.edu/teaching/files/qi-notes-partiii.pdf)</sup>. Any convex combination of projectors onto pure states, \( \rho = \sum_i p_i |\psi_i\rangle\langle\psi_i| \) with \( p_i \geq 0 \) and \( \sum_i p_i = 1 \), forms a mixed state<sup>[3](https://chaos.if.uj.edu.pl/~karol/pdf/PMGDHZ12.pdf)</sup>. The density operator was introduced by [John von Neumann](https://www.edgepedia.org/john-von-neumann) (and, in parallel, by L. Landau and F. Bloch) in 1927<sup>[11](https://citeseerx.ist.psu.edu/document?doi=74cb14b7e8894b9a281c4be2a3e4a2c9c776c0a2&repid=rep1&type=pdf)</sup>.

The distinction between pure and mixed states is read off the operator itself. A density operator is pure, and the state can be described by a ray, if \( \rho^2 = \rho \); otherwise the state is mixed<sup>[4](https://web.gps.caltech.edu/~rls/book.pdf)</sup>. Equivalently, the purity \( \mathrm{Tr}\,\rho^2 \) equals 1 for a pure state and is smaller for a mixed one<sup>[5](https://lim.physics.ucdavis.edu/teaching/files/qi-notes-partiii.pdf)</sup>. The completely mixed state on a \( d \)-dimensional space is \( \rho = I/d \)<sup>[5](https://lim.physics.ucdavis.edu/teaching/files/qi-notes-partiii.pdf)</sup>.

**Convex geometry.** The set \( Q_D \) of all density matrices of order \( D \) forms a convex set of \( D^2 - 1 \) real dimensions<sup>[3](https://chaos.if.uj.edu.pl/~karol/pdf/PMGDHZ12.pdf)</sup>. The projectors corresponding to the extremal points of this set are exactly the pure states, so the \( (2D-2) \)-dimensional pure-state manifold \( \mathbb{CP}^{D-1} \) sits inside the larger convex body<sup>[3](https://chaos.if.uj.edu.pl/~karol/pdf/PMGDHZ12.pdf)</sup>. For \( N = 2 \) this set is the Bloch ball embedded in \( \mathbb{R}^3 \)<sup>[6](https://arxiv.org/abs/1112.2347)</sup>.

## The Bloch ball for a qubit

A quantum system with a two-dimensional Hilbert space is called a qubit, and its general density matrix can be written

\[ \rho(\vec{P}) = \tfrac{1}{2}(\mathbb{I} + \vec{P} \cdot \vec{\sigma}), \]

where \( \vec{\sigma} \) is the vector of [Pauli matrices](https://www.edgepedia.org/pauli-matrices) and \( \vec{P} \) is a three-component vector of length \( |\vec{P}| \leq 1 \); pure states have \( |\vec{P}| = 1 \)<sup>[4](https://web.gps.caltech.edu/~rls/book.pdf)</sup>. The Bloch vectors for all qubit states collectively form the Bloch ball, whose surface, the [Bloch sphere](https://www.edgepedia.org/bloch-sphere), contains the Bloch vectors of all pure qubit states<sup>[12](https://journals.aps.org/prxquantum/abstract/10.1103/PRXQuantum.6.030202)</sup>.

The length of the Bloch vector representing a density matrix \( \rho \) is determined by its purity,

\[ |\mathbf{r}| = \sqrt{2\,\mathrm{Tr}(\rho^2) - 1}, \]

so mixed states have \( |\mathbf{r}| < 1 \), and the maximally mixed state \( \rho_{\mathrm{mix}} = \mathbb{I}/2 \) sits at the center of the ball with \( \mathbf{r} = 0 \)<sup>[12](https://journals.aps.org/prxquantum/abstract/10.1103/PRXQuantum.6.030202)</sup>. Any pure state can be parameterized as \( |\psi\rangle = \cos(\theta/2)|0\rangle + e^{i\phi}\sin(\theta/2)|1\rangle \), where \( \theta \) and \( \phi \) are the polar and azimuthal angles of its Bloch vector<sup>[12](https://journals.aps.org/prxquantum/abstract/10.1103/PRXQuantum.6.030202)</sup>. Unitary time evolution corresponds to rigid rotations of the Bloch sphere, which is one reason the representation is useful for visualizing qubit dynamics<sup>[12](https://journals.aps.org/prxquantum/abstract/10.1103/PRXQuantum.6.030202)</sup>.

## Composition and entanglement

When two physical systems interact, they form a composite system, and quantum mechanics supplies a rule for building the composite state space from the component spaces: the tensor product \( H_A \otimes H_B \)<sup>[7](https://plato.stanford.edu/entries/qm/)</sup>. If \( \{v_i^A\} \) and \( \{u_j^B\} \) are orthonormal bases, the pairs \( (v_i^A, u_j^B) \) form an orthonormal basis of the product space, with inner product \( \langle v_i^A \otimes u_m^B \,|\, v_j^A \otimes u_n^B \rangle = \langle v_i^A | v_j^A \rangle \langle u_m^B | u_n^B \rangle \)<sup>[7](https://plato.stanford.edu/entries/qm/)</sup>.

This construction has a consequence with no classical analog: although every vector in \( H_A \otimes H_B \) is a linear sum of product-form vectors, not every vector is itself of that form, and the state of a composite system is not uniquely defined by the states of its components<sup>[7](https://plato.stanford.edu/entries/qm/)</sup>. Facts about systems as wholes do not supervene on facts about their parts, a feature implicated in most of the theory's conceptual difficulties<sup>[7](https://plato.stanford.edu/entries/qm/)</sup>. For a bipartite system in a pure state \( |\psi\rangle \), the state of subsystem A is obtained by the partial trace over B, \( \rho_A = \mathrm{Tr}_B\,\rho \) with \( \rho = |\psi\rangle\langle\psi| \); mixed states can therefore be interpreted either as statistical mixtures or as subsystems of a larger system in a pure state<sup>[8](https://www.cpt.univ-mrs.fr/~verga/pages/AQ-principles.html)</sup>.

## Measurement on the state space

A projective measurement of a system in state \( |\psi\rangle \) is the projection \( P_n \) onto an eigenstate \( |o_n\rangle \) of the observable \( O \). The result is the eigenvalue \( o_n \) with probability given by the [Born rule](https://www.edgepedia.org/born-rule),

\[ p(o_n) = \langle\psi|P_n|\psi\rangle = |\langle o_n|\psi\rangle|^2, \]

and the post-measurement state collapses to \( |o_n\rangle \)<sup>[8](https://www.cpt.univ-mrs.fr/~verga/pages/AQ-principles.html)</sup>. The rule generalizes to mixed states: the probability of outcome \( n \) in state \( \rho \) is \( p_n = \mathrm{Tr}\,P_n\rho \), with the state updating to \( \rho_n = P_n\rho P_n / \mathrm{Tr}\,P_n\rho \)<sup>[8](https://www.cpt.univ-mrs.fr/~verga/pages/AQ-principles.html)</sup>.

More generally, a measurement is modeled as a unitary interaction between the system and an apparatus on the product space \( \mathcal{H}_S \otimes \mathcal{H}_A \), the von Neumann model of measurement. The resulting operators \( M_n \), called [Kraus operators](https://www.edgepedia.org/kraus-operators), satisfy \( \sum_n M_n^\dagger M_n = \mathbb{I} \), and the positive operators \( E_n = M_n^\dagger M_n \) form a complete set generalizing the projectors; in the quantum information literature this set is called a [positive operator-valued measure](https://www.edgepedia.org/positive-operator-valued-measure), or POVM<sup>[8](https://www.cpt.univ-mrs.fr/~verga/pages/AQ-principles.html)</sup>. A POVM can always be reduced to a projective measurement on a larger Hilbert space<sup>[13](https://www.sciencedirect.com/topics/mathematics/probability-amplitude)</sup>.

## Why the state space has this form

The Hilbert-space structure is not arbitrary. Work in the hidden-measurement approach shows that when the state space of a physical entity is assumed to be a complex Hilbert space and a principle of consistent interaction partitions the space of apparatus states, the normalized measure of apparatus states producing a fixed outcome agrees with the Born-rule probability<sup>[14](https://arxiv.org/abs/quant-ph/0212151)</sup>. Along a different line, starting from the representation of quantum states induced by a symmetric informationally complete measurement (SIC), key features of quantum-state space can be derived from the requirement that the Born rule arise as a simple modification of the law of total probability plus a limited number of additional assumptions of a Bayesian character<sup>[15](https://arxiv.org/abs/0912.4252)</sup>.

## References

1. [Principles of quantum mechanics (adapted from Jenann Ismael, Stanford Encyclopedia of Philosophy)](https://www.cs.tufts.edu/~ablumer/QuantumLogic/principles.html)
2. [Entanglement structure in quantum many-body systems, field theories, and holography (Soken dissertation, Appendix A)](https://ir.soken.ac.jp/record/6647/files/A2402%E6%9C%AC%E6%96%87.pdf)
3. [Z. Puchała et al., J. Phys. A: Math. Theor. 45, 415309 (2012)](https://chaos.if.uj.edu.pl/~karol/pdf/PMGDHZ12.pdf)
4. [J. Preskill, Lecture Notes on Quantum Information, Chapter 2: Foundations I, States and Ensembles](https://web.gps.caltech.edu/~rls/book.pdf)
5. [Quantum Information Theory Lecture Notes, UC Davis](https://lim.physics.ucdavis.edu/teaching/files/qi-notes-partiii.pdf)
6. [Geometry of the set of mixed quantum states: An apophatic approach, arXiv:1112.2347](https://arxiv.org/abs/1112.2347)
7. [Quantum Mechanics, Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/qm/)
8. [Quantum principles, Random physics (CPT Marseille lecture notes)](https://www.cpt.univ-mrs.fr/~verga/pages/AQ-principles.html)
9. [Geometric Quantum Mechanics, arXiv:quant-ph/9906086](https://arxiv.org/abs/quant-ph/9906086)
10. [Quantum Implementations, Projective Actions, and Central Extensions, QFT.org](https://qft.org/symmetry-gauge/symmetry-actions-redundancy/quantum-implementations-projective-actions-central-extensions/)
11. [A short introduction to the quantum (lecture notes, CiteSeerX)](https://citeseerx.ist.psu.edu/document?doi=74cb14b7e8894b9a281c4be2a3e4a2c9c776c0a2&repid=rep1&type=pdf)
12. [Practical Introduction to Benchmarking and Characterization of Quantum Computers, PRX Quantum 6, 030202 (2025)](https://journals.aps.org/prxquantum/abstract/10.1103/PRXQuantum.6.030202)
13. [Probability Amplitude, ScienceDirect Topics](https://www.sciencedirect.com/topics/mathematics/probability-amplitude)
14. [The Born rule from a consistency requirement on hidden measurements in complex Hilbert space, arXiv:quant-ph/0212151](https://arxiv.org/abs/quant-ph/0212151)
15. [A Quantum-Bayesian Route to Quantum-State Space, arXiv:0912.4252](https://arxiv.org/abs/0912.4252)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Quantum physics › Quantum mechanics › Quantum formalism and states › Quantum states and wave functions › State vectors and Hilbert-space states › Rays and projective Hilbert space*

*Initially written Sep 17, 2026 · Reviewed: Sep 19, 2026; Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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