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Qm State Geometry

1 · Prerequisites

2 · Summary

This page develops the geometry of the framework's state space on the basis of the state and composition postulate adopted on qm-framework-and-quantities. The mathematical layer begins with the nonzero vectors, unit sphere, rays, ray space, normalization map and phase action of a Hilbert space (State vectors, unit vectors, rays, normalization and phases of a Hilbert space), whose elementary orbit structure is proved in Normalization, ray equivalence and the phase orbits of the unit sphere: unit representatives of one ray differ exactly by phase factors, and the ray space is the set of phase orbits of the unit sphere. The rank-one operators of the state-space calculus are then identified with the one-dimensional Hilbert projections, and the rays are shown to stand in bijection with them (Rays, unit vectors and rank-one Hilbert projections), which yields the density-operator picture of states, the pure and mixed states and the purity functional of The state space: density operators, pure and mixed states, and purity.

The state space itself is analysed as a subset of the self-adjoint trace-class operators: it is convex, closed under countable probability mixtures with grouping of consecutive blocks, and complete in the trace norm (The state space is convex, closed under countable mixtures, and complete in the trace norm); purity is the trace of the square, with value one exactly for the pure states and strictly less for the mixed states (Purity, spectral weights and the criterion for a pure state); and the extreme points of the state space are exactly the pure states (The extreme points of the state space are exactly the pure states).

The physical items read the state postulate in this vocabulary. The state space, the state assigned to a state vector and the pure, mixed and mixture terminology are fixed in The state space of the framework: preparations, pure rays and density operators; the pure-preparation assignment is proved well defined by the ray of its vector, phase invariant and injective, with the zero vector excluded (Pure preparations are well defined by their rays, and phase factors do not change their states); and the statistical mixtures, the spectral decomposition of every preparation into orthogonal pure preparations, and the equivalence of purity with extremality of the preparation set are established in Statistical mixtures, spectral decompositions and the extremal role of pure preparations. A closing scope audit (Scope audit of the state-geometry page: what is mathematics, what is adopted, and what is deferred) separates the conditional mathematics from the adopted state postulate, records the empirical neutrality of the page, and lists the deferred questions — outcome probabilities, measurement models, superselection and identical-particle sectors, and the photon reading of two-level spaces. The worked finite checks of the pair are collected on qm-state-geometry-examples.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

State vectors, unit vectors, rays, normalization and phases of a Hilbert space

Definition

The Hilbert space and its norm. Throughout this item $H \neq {0}$ is a nonzero complex Hilbert space, with pairing $⟨ \cdot, \cdot ⟩$ linear in the first argument and conjugate-linear in the second (Real and complex inner-product spaces and their induced length, Hilbert space), and with the induced norm $∥ ψ ∥ : = \sqrt{⟨ ψ, ψ ⟩}$ (The norm $∥ v ∥ = \sqrt{⟨ v, v ⟩}$ induced by a real or complex inner product, The induced length is a norm). A vector is an element of $H$ , and a unit vector is a vector $ψ$ with $∥ ψ ∥ = 1$ . The unit sphere of $H$ is $S (H) : = {ψ \in H : ∥ ψ ∥ = 1},$ and the set of nonzero vectors is $H^{\times} : = H ∖ {0}$ . The hypothesis $H \neq {0}$ says exactly that $H^{\times}$ is nonempty; then $S (H)$ is nonempty as well, since for every nonzero $ψ$ the vector $ψ / ∥ ψ ∥$ has norm one, because $∥ ψ ∥ > 0$ and $∥ ψ / ∥ ψ ∥ ∥ = ∣ ∥ ψ ∥^{- 1} ∣ ∥ ψ ∥ = 1$ .

Rays. For $ψ \in H^{\times}$ the ray of $ψ$ is $[ψ] : = {z ψ : z \in C, z \neq 0} \subseteq H^{\times} .$ A ray of $H$ is a subset of the form $[ψ]$ for some nonzero $ψ$ , and an element of a ray is called a representative of it. The set of rays $R (H) : = {[ψ] : ψ \in H^{\times}}$ is the ray space of $H$ . Two nonzero vectors $ψ, φ$ are ray-equivalent when they lie in a common ray, that is, when $φ = z ψ$ for some nonzero complex number $z$ . Ray equivalence is an equivalence relation on $H^{\times}$ : it is reflexive with $z = 1$ , symmetric with $z^{- 1}$ , and transitive by multiplying scalars. Its classes are exactly the rays, so $R (H)$ is the quotient set of $H^{\times}$ by this relation, and every nonzero vector lies in exactly one ray.

Normalization. The normalization map is $ν : H^{\times} ⟶ S (H), ν (ψ) : = \frac{ψ}{∥ ψ ∥},$ where the right-hand side is the scalar multiple $∥ ψ ∥^{- 1} ψ$ ; the displayed definition is meaningful because $∥ ψ ∥ > 0$ for $ψ \neq 0$ . The vector $ν (ψ)$ is the normalization of $ψ$ .

Phases. The phase group is $U (1) : = {ζ \in C : ∣ ζ ∣ = 1}$ (The complex numbers as $R [x] / (x^{2} + 1)$ , with the real embedding and imaginary unit $i$ , Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive), and a phase factor is an element of $U (1)$ . Two vectors $ψ, φ$ are phase-equivalent when $φ = ζ ψ$ for some phase factor $ζ$ . The phase action of $U (1)$ on the unit sphere is the map $U (1) \times S (H) ⟶ S (H), (ζ, ψ) ⟼ ζ ψ,$ which is well defined because $∥ ζ ψ ∥ = ∣ ζ ∣ ∥ ψ ∥ = ∥ ψ ∥$ for $ζ \in U (1)$ and $ψ \in S (H)$ . Ray equivalence allows multiplication by an arbitrary nonzero complex scalar, while phase equivalence allows multiplication by a scalar of modulus one only; within a given ray the two notions coincide exactly on representatives of equal norm, and the precise statements are proved in the normalization lemma of this page.

Scope. This definition fixes the mathematical objects of the state space of a Hilbert space and their elementary vocabulary: nonzero vectors, unit vectors, rays, the ray space, the normalization map and the phase action. It introduces no topology or measure on $R (H)$ , no dynamics and no physical interpretation, and it uses no choice principle. The identification of rays with preparations and of the unit sphere with their normalized representatives is the model assumption stated in the framework's state postulate, not part of this mathematical definition; the orbit structure of the phase action is analysed in the normalization lemma of this page.

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Normalization, ray equivalence and the phase orbits of the unit sphere

Statement

Let $H \neq {0}$ be a complex Hilbert space and use the notation of State vectors, unit vectors, rays, normalization and phases of a Hilbert space: the unit sphere $S (H)$ , the nonzero vectors $H^{\times}$ , the ray $[ψ]$ , the ray space $R (H)$ , the normalization map $ν$ , the phase group $U (1)$ and the phase action. Then:

(a) For every $ψ \in H^{\times}$ one has $ν (ψ) \in [ψ] \cap S (H)$ and $ν (ν (ψ)) = ν (ψ)$ ; and for every $z \in C$ with $z \neq 0$ , $ν (z ψ) = \frac{z}{∣ z ∣} ν (ψ) .$ Moreover $ν$ maps $H^{\times}$ onto $S (H)$ .

(b) For $ψ, φ \in H^{\times}$ : $[ψ] = [φ]$ if and only if $ν (φ) = ζ ν (ψ)$ for some phase factor $ζ \in U (1)$ ; the phase factor is then unique.

(c) For every $ψ \in S (H)$ the intersection of the ray of $ψ$ with the unit sphere is exactly the phase orbit of $ψ$ , $[ψ] \cap S (H) = {ζ ψ : ζ \in U (1)} = U (1) ψ .$ Consequently the fibres of the map $S (H) \to R (H)$ , $ψ \mapsto [ψ]$ , are exactly the phase orbits, and this map induces a bijection from the set of phase orbits on $S (H)$ onto $R (H)$ .

(d) For $ψ, φ \in S (H)$ : $[ψ] = [φ]$ if and only if $φ = ζ ψ$ for a unique $ζ \in U (1)$ ; in that case $ζ = ⟨ φ, ψ ⟩$ and $∣ ⟨ φ, ψ ⟩ ∣ = 1$ .

Facts & Assumptions

Given: A nonzero complex Hilbert space $H$ ; nonzero vectors $ψ, φ \in H^{\times}$ ; a nonzero scalar $z \in C$ ; unit vectors $u \in S (H)$ where stated.

[F1]

By State vectors, unit vectors, rays, normalization and phases of a Hilbert space: $S (H) = {ψ : ∥ ψ ∥ = 1}$ , $H^{\times} = H ∖ {0}$ , $[ψ] = {z ψ : z \neq 0}$ , $R (H) = {[ψ] : ψ \neq 0}$ , $ν (ψ) = ∥ ψ ∥^{- 1} ψ$ , $U (1) = {ζ : ∣ ζ ∣ = 1}$ , and the words ray-equivalent and phase-equivalent have the meanings fixed there.

[F2]

The inner-product norm satisfies $∥ ψ ∥ \geq 0$ with $∥ ψ ∥ = 0$ exactly for $ψ = 0$ , and $∥ z ψ ∥ = ∣ z ∣ ∥ ψ ∥$ for every scalar $z$ and vector $ψ$ (The induced length is a norm); the inner product is linear in the first argument, conjugate-linear in the second and conjugate symmetric (Real and complex inner-product spaces and their induced length).

[F3]

For complex numbers, $∣ z ∣ = 0$ exactly for $z = 0$ , and $∣ z w ∣ = ∣ z ∣ ∣ w ∣$ (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

[F4]

If $ζ \in U (1)$ and $ψ \in S (H)$ , then $∥ ζ ψ ∥ = ∣ ζ ∣ ∥ ψ ∥ = 1$ , so $ζ ψ \in S (H)$ ; this is the phase action of [F1] and uses [F2].

Proof

1.1givenF1F2F3algebra

[(a) normalization is well defined and lies in the ray] For $ψ \neq 0$ the vector $ν (ψ) = ∥ ψ ∥^{- 1} ψ$ is a nonzero complex multiple of $ψ$ , hence lies in $[ψ]$ ; its norm is $∥ ν (ψ) ∥ = ∣ ∥ ψ ∥^{- 1} ∣ ∥ ψ ∥ = ∥ ψ ∥^{- 1} ∥ ψ ∥ = 1$ by [F2] and [F3], so $ν (ψ) \in S (H)$ ; and $ν (ν (ψ)) = ν (ψ)$ because $ν (ψ)$ is a unit vector. For $z \neq 0$ the norm is $∥ z ψ ∥ = ∣ z ∣ ∥ ψ ∥$ by [F2], hence $ν (z ψ) = ∣ z ∣^{- 1} ∥ ψ ∥^{- 1} z ψ = (z / ∣ z ∣) ∥ ψ ∥^{- 1} ψ = (z / ∣ z ∣) ν (ψ)$ , using $z / ∣ z ∣ = ∣ z ∣^{- 1} z$ for the unit complex number $z / ∣ z ∣$ . Finally $ν$ is surjective because every $u \in S (H)$ satisfies $ν (u) = ∥ u ∥^{- 1} u = u$ .

2.1givenstep 1.1F1F2F3F4algebra

[(c) the unit representatives of a ray form one phase orbit] Let $u \in S (H)$ . If $v \in [u] \cap S (H)$ then $v = z u$ for some $z \neq 0$ , and $1 = ∥ v ∥ = ∣ z ∣ ∥ u ∥ = ∣ z ∣$ by [F2] and [F3]; hence $z \in U (1)$ and $v = z u \in U (1) u$ . Conversely if $ζ \in U (1)$ then $ζ u$ lies in $[u]$ by definition and in $S (H)$ by [F4]. Thus $[u] \cap S (H) = U (1) u$ .

2.2givenstep 1.1F1F2F3algebra

[(b) ray equivalence is phase equivalence of normalizations] If $[ψ] = [φ]$ , then by definition $φ = z ψ$ with $z \neq 0$ , so by step 1.1 $ν (φ) = ν (z ψ) = (z / ∣ z ∣) ν (ψ)$ with $z / ∣ z ∣ \in U (1)$ ; uniqueness of the phase factor holds because $ν (ψ) \neq 0$ . Conversely, if $ν (φ) = ζ ν (ψ)$ for some $ζ \in U (1)$ , then $φ = ∥ φ ∥ ν (φ) = ∥ φ ∥ ζ ∥ ψ ∥^{- 1} ψ$ is a nonzero complex multiple of $ψ$ , so $[φ] = [ψ]$ .

3.1givenstep 1.1step 2.1F1algebra

[(c) the induced map on phase orbits is a bijection] By step 2.1 the fibre of $ψ \mapsto [ψ]$ over a ray $R$ is ${u \in S (H) : [u] = R} = R \cap S (H)$ , which is a full phase orbit; and every phase orbit $U (1) u$ is such a fibre because $U (1) u \subseteq [u]$ . The map therefore descends to a well-defined injective map from the set of phase orbits onto $R (H)$ , and it is surjective because every ray contains the unit vector $ν (ψ)$ for any of its representatives $ψ$ by step 1.1.

4.1givenstep 2.2F1F2F3algebra∎

[(d) phase uniqueness for unit representatives] Let $ψ, φ \in S (H)$ with $[ψ] = [φ]$ . Then $φ = z ψ$ for some $z \neq 0$ , so $1 = ∥ φ ∥ = ∣ z ∣ ∥ ψ ∥ = ∣ z ∣$ by [F2] and [F3]; writing $ζ : = z \in U (1)$ gives $φ = ζ ψ$ , and if $ζ^{'} \in U (1)$ also satisfies $φ = ζ^{'} ψ$ , then $(ζ - ζ^{'}) ψ = 0$ with $ψ \neq 0$ , which forces $ζ = ζ^{'}$ by [F2] and [F3]. Moreover $⟨ φ, ψ ⟩ = ⟨ ζ ψ, ψ ⟩ = ζ ⟨ ψ, ψ ⟩ = ζ ∥ ψ ∥^{2} = ζ$ by linearity in the first argument, so $∣ ⟨ φ, ψ ⟩ ∣ = 1$ . The converse implication is the definition of ray equivalence.

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Rays, unit vectors and rank-one Hilbert projections

Statement

Assume the Axiom of Countable Choice, which is the choice hypothesis of the rank-one calculus Rank-one operator calculus. Let $H \neq {0}$ be a complex Hilbert space and let $T_{u, v} x = ⟨ x, v ⟩ u (x \in H)$ be the rank-one operator calculus of that item. Use the notation of State vectors, unit vectors, rays, normalization and phases of a Hilbert space for the unit sphere $S (H)$ , the nonzero vectors $H^{\times}$ , the ray $[ψ]$ , the ray space $R (H)$ and the normalization $ν$ . Then:

(a) For every $ψ \in H^{\times}$ the operator $T_{ψ, ψ}$ is a nonzero self-adjoint positive trace-class operator with $∥ T_{ψ, ψ} ∥_{1} = tr (T_{ψ, ψ}) = ∥ ψ ∥^{2}, T_{ψ, ψ} = ∥ ψ ∥^{2} T_{ν (ψ), ν (ψ)}, ⟨ T_{ψ, ψ} x, x ⟩ = ∣ ⟨ x, ψ ⟩ ∣^{2} (x \in H) .$

(b) For every $u \in S (H)$ the operator $T_{u, u}$ is the Hilbert orthogonal projection onto the one-dimensional linear subspace $C u$ ; in particular $T_{u, u} \neq 0$ , $T_{u, u}^{2} = T_{u, u}$ and $T_{u, u}^{*} = T_{u, u}$ . Conversely, if $M$ is a one-dimensional linear subspace of $H$ and $u \in S (H)$ satisfies $M = C u$ , then the Hilbert orthogonal projection $P_{M}$ onto $M$ equals $T_{u, u}$ . Thus the operators $T_{u, u}$ with $u \in S (H)$ are exactly the Hilbert orthogonal projections of $H$ with one-dimensional range.

(c) For nonzero $ψ, φ \in H^{\times}$ : $T_{ψ, ψ} = T_{φ, φ} ⟺ ∥ ψ ∥ = ∥ φ ∥ and [ψ] = [φ] .$ In particular, for $u, w \in S (H)$ one has $T_{u, u} = T_{w, w}$ if and only if $w = ζ u$ for some $ζ \in U (1)$ .

(d) The map $j : R (H) ⟶ S_{1} (H), j ([ψ]) : = T_{ν (ψ), ν (ψ)},$ is well defined and injective, and its image is exactly ${T_{u, u} : u \in S (H)}$ .

Facts & Assumptions

Given: A nonzero complex Hilbert space $H$ ; nonzero vectors $ψ, φ$ ; unit vectors $u, w$ ; the rank-one operators $T_{u, v}$ .

[F1]

Rank-one calculus (Rank-one operator calculus): $T_{u, v}$ is bounded with $∥ T_{u, v} ∥ = ∥ u ∥ ∥ v ∥$ , the assignment is linear in the first and conjugate-linear in the second variable, $T_{u, v}$ is trace class with $∥ T_{u, v} ∥_{1} = ∥ u ∥ ∥ v ∥$ and $tr (T_{u, v}) = ⟨ u, v ⟩$ , one has $T_{u, v}^{*} = T_{v, u}$ , for bounded $B$ one has $B T_{u, v} = T_{B u, v}$ and $T_{u, v} B = T_{u, B^{*} v}$ , for trace-class $S$ one has $S T_{u, v} = T_{S u, v}$ and $tr (S T_{x, y}) = ⟨ S x, y ⟩$ , and $(u, v) \mapsto T_{u, v}$ is trace-norm continuous.

[F2]

The state-space vocabulary (State vectors, unit vectors, rays, normalization and phases of a Hilbert space): $S (H)$ is the set of unit vectors, $H^{\times}$ the set of nonzero vectors, $[ψ] = {z ψ : z \neq 0}$ , $R (H)$ the set of rays, $ν (ψ) = ∥ ψ ∥^{- 1} ψ$ , and $U (1)$ the phase group.

[F3]

Normalization and phases (Normalization, ray equivalence and the phase orbits of the unit sphere): $ν (ψ) \in [ψ] \cap S (H)$ ; $[ψ] = [φ]$ if and only if $ν (φ) = ζ ν (ψ)$ for a unique $ζ \in U (1)$ ; and for $u, w \in S (H)$ , $[u] = [w]$ if and only if $w = ζ u$ with unique $ζ \in U (1)$ .

[F4]

Hilbert orthogonal projection (The Hilbert orthogonal projection onto a closed subspace): for a closed linear subspace $M$ of $H$ there is a unique map $P_{M} : H \to H$ with $P_{M} x \in M$ and $x - P_{M} x \in M^{⊥}$ for every $x$ ; a map with these two properties is therefore $P_{M}$ .

[F5]

The line spanned by a unit vector is closed (Linear combination of a finite list, and the span $span (S)$ as the smallest linear subspace containing $S$ , Normed subspace, A finite-dimensional normed subspace is closed): $C u = {z u : z \in C}$ is a linear subspace admitting the one-element basis $(u)$ , hence a finite-dimensional normed subspace of $H$ and therefore closed in $H$ .

[F6]

Norm, pairing and positivity (The induced length is a norm, Real and complex inner-product spaces and their induced length, Self-adjoint, positive, unitary and normal operators): $∥ v ∥ \geq 0$ with $∥ v ∥ = 0$ exactly for $v = 0$ , $∥ z v ∥ = ∣ z ∣ ∥ v ∥$ ; the pairing is linear in the first argument, conjugate-linear in the second and conjugate symmetric; and a bounded operator on any real or complex Hilbert space is positive when its quadratic form $⟨ T x, x ⟩$ is real and nonnegative for every $x$ .

Proof

1.1givenF1F2F6algebra

[(a) trace, positivity and scaling] By [F1], $T_{ψ, ψ}$ is trace class with $∥ T_{ψ, ψ} ∥_{1} = ∥ ψ ∥^{2}$ and $tr (T_{ψ, ψ}) = ⟨ ψ, ψ ⟩ = ∥ ψ ∥^{2}$ , and it satisfies $T_{ψ, ψ}^{*} = T_{ψ, ψ}$ . For every $x \in H$ , $⟨ T_{ψ, ψ} x, x ⟩ = ⟨ ⟨ x, ψ ⟩ ψ, x ⟩ = ∣ ⟨ x, ψ ⟩ ∣^{2} \geq 0$ by [F6], so $T_{ψ, ψ}$ is positive under the general bounded-operator definition in [F6]; also $T_{ψ, ψ} ψ = ∥ ψ ∥^{2} ψ \neq 0$ , so $T_{ψ, ψ}$ is nonzero. For $z \neq 0$ the conjugate-linearity of [F1] gives $T_{z ψ, z ψ} = ∣ z ∣^{2} T_{ψ, ψ}$ ; substituting $z = ∥ ψ ∥^{- 1}$ and using $ν (ψ) = ∥ ψ ∥^{- 1} ψ$ gives $T_{ν (ψ), ν (ψ)} = ∥ ψ ∥^{- 2} T_{ψ, ψ}$ , that is the displayed identity after multiplying by $∥ ψ ∥^{2}$ .

2.1givenstep 1.1F1F4F5F6algebra

[(b) $T_{u, u}$ is the projection onto $C u$ ] Let $u \in S (H)$ . The line $C u$ is a closed linear subspace by [F5]. For every $x$ one has $T_{u, u} x = ⟨ x, u ⟩ u \in C u$ , and $⟨ x - T_{u, u} x, u ⟩ = ⟨ x, u ⟩ - ⟨ x, u ⟩ ∥ u ∥^{2} = 0$ by [F6], so $x - T_{u, u} x \in (C u)^{⊥}$ ; by the characterisation of [F4] applied to $M = C u$ , this says $T_{u, u} = P_{C u}$ , so $T_{u, u}$ is a Hilbert orthogonal projection and in particular is nonzero with $T_{u, u}^{2} = T_{u, u}$ and $T_{u, u}^{*} = T_{u, u}$ .

2.2givenstep 1.1F1F2F3F6algebra

[(c) equality of rank-one operators] Suppose first that $T_{ψ, ψ} = T_{φ, φ}$ with $ψ, φ \neq 0$ . Evaluating both sides at $ψ$ gives $∥ ψ ∥^{2} ψ = ⟨ ψ, ψ ⟩ ψ = T_{ψ, ψ} ψ = T_{φ, φ} ψ = ⟨ ψ, φ ⟩ φ$ by [F1] and [F6]; the left side is nonzero, so $⟨ ψ, φ ⟩ \neq 0$ and $φ = z ψ$ with $z = ∥ ψ ∥^{2} / ⟨ ψ, φ ⟩ \neq 0$ . Hence $φ$ and $ψ$ lie in the same ray, so $∥ φ ∥ = ∣ z ∣ ∥ ψ ∥$ by [F6]; moreover $T_{φ, φ} = T_{z ψ, z ψ} = ∣ z ∣^{2} T_{ψ, ψ} = T_{ψ, ψ}$ by conjugate-linearity and the hypothesis, and since $T_{ψ, ψ} \neq 0$ this forces $∣ z ∣^{2} = 1$ , hence $∥ φ ∥ = ∥ ψ ∥$ . Conversely, if $∥ ψ ∥ = ∥ φ ∥$ and $[ψ] = [φ]$ , then $φ = z ψ$ for some $z \neq 0$ , and taking norms gives $∣ z ∣ = 1$ ; then $T_{φ, φ} = T_{z ψ, z ψ} = ∣ z ∣^{2} T_{ψ, ψ} = T_{ψ, ψ}$ . For unit vectors this is the stated equivalence $T_{u, u} = T_{w, w} ⟺ w = ζ u$ , $ζ \in U (1)$ , by [F3].

3.1givenstep 2.1F1F4F6algebra

[(b) the converse identification] Let $M$ be a one-dimensional linear subspace and let $u \in S (H)$ with $M = C u$ . For arbitrary $x$ , the defining properties of $P_{M}$ in [F4] give $P_{M} x \in M = C u$ , so $P_{M} x = α u$ for a unique scalar $α$ , and $x - P_{M} x \in M^{⊥}$ , so $⟨ x - P_{M} x, u ⟩ = 0$ . Consequently $α = α ∥ u ∥^{2} = ⟨ α u, u ⟩ = ⟨ P_{M} x, u ⟩ = ⟨ x, u ⟩$ , and therefore $P_{M} x = ⟨ x, u ⟩ u = T_{u, u} x$ for every $x$ . Thus the Hilbert orthogonal projections with one-dimensional range are exactly the operators $T_{u, u}$ , $u \in S (H)$ .

4.1givenstep 2.1step 3.1step 2.2F1F2F3algebra∎

[(d) the ray-to-state map] If $[ψ] = [φ]$ , then by [F3] the unit representatives satisfy $ν (φ) = ζ ν (ψ)$ for some $ζ \in U (1)$ , whence $T_{ν (φ), ν (φ)} = T_{ζ ν (ψ), ζ ν (ψ)} = ∣ ζ ∣^{2} T_{ν (ψ), ν (ψ)} = T_{ν (ψ), ν (ψ)}$ by the conjugate-linearity of [F1]; thus $j$ is well defined. If $T_{ν (ψ), ν (ψ)} = T_{ν (φ), ν (φ)}$ , then step 2.2 applied to the unit vectors $ν (ψ), ν (φ)$ gives $ν (φ) = ζ ν (ψ)$ for some $ζ \in U (1)$ , so $[φ] = [ψ]$ by [F3]; thus $j$ is injective. Its image is ${T_{u, u} : u \in S (H)}$ by definition of $ν$ , and by steps 2.1 and 3.1 this set consists exactly of the one-dimensional Hilbert orthogonal projections of $H$ . This completes (a)-(d).

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

The state space: density operators, pure and mixed states, and purity

Definition

Density operators. Let $H \neq {0}$ be a complex Hilbert space. A density operator on $H$ is a self-adjoint positive trace-class operator $ρ$ with $tr ρ = 1$ , in the sense of Density operators and scalar spectral probability clause (a) (Trace class operator, Trace of a trace class operator, Self-adjoint, positive, unitary and normal operators). The state space of $H$ is the set of all density operators on $H$ and is denoted $D (H) : = {ρ : ρ is a density operator on H} .$ An element of $D (H)$ is called a state of $H$ in the mathematical sense fixed here.

Pure states. A state $ρ \in D (H)$ is pure when there is a unit vector $u \in S (H)$ with $ρ = T_{u, u}, T_{u, u} x = ⟨ x, u ⟩ u (x \in H),$ where $T_{u, u}$ is the rank-one operator of Rank-one operator calculus and $S (H)$ is the unit sphere of State vectors, unit vectors, rays, normalization and phases of a Hilbert space. By Rays, unit vectors and rank-one Hilbert projections the pure states are exactly the Hilbert orthogonal projections of $H$ with one-dimensional range, and the map $j : R (H) ⟶ D (H), j ([ψ]) : = T_{ν (ψ), ν (ψ)},$ on the ray space $R (H)$ , with $ν$ the normalization map, is well defined and injective with image the set of pure states. A state is mixed when it is not pure. For a nonzero vector $ψ$ the normalized rank-one operator of $ψ$ is $\hat{ρ} (ψ) : = \frac{T_{ψ, ψ}}{∥ ψ ∥^{2}} = \frac{T_{ψ, ψ}}{tr (T_{ψ, ψ})},$ which equals $T_{ν (ψ), ν (ψ)}$ ; thus $\hat{ρ} (ψ)$ is the pure state of the ray $[ψ]$ , and the vector $ψ$ enters only through its ray.

Purity. For $ρ \in D (H)$ the operator $ρ^{2}$ is self-adjoint, positive and trace class: it is self-adjoint by the adjoint algebra of The Hilbert-space adjoint of a bounded operator, positive because $⟨ ρ^{2} x, x ⟩ = ⟨ ρ x, ρ x ⟩ \geq 0$ , and trace class because $ρ$ is a bounded trace-class operator and the trace-class operators form a two-sided ideal (Trace class is a two sided Banach operator ideal clauses 1 and 2). The purity of $ρ$ is the real number $P (ρ) : = tr (ρ^{2}) \in R .$ The numerical range of the purity and the equivalence of purity with purity of the state are established by the purity criterion of this page; this definition records the object only.

Scope. This definition fixes the mathematical state space of a Hilbert space together with the pure states, the mixed states, the normalized rank-one operator of a nonzero vector and the purity functional. The cited rank-one calculus and the density-operator expansion assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )), and the definition is read under that assumption. It asserts no dynamics, no measurement, no probability law and no physical interpretation, and it introduces no new assumptions beyond the cited mathematics and that choice hypothesis. In particular, whether a given physical preparation is described by a pure or a mixed state is a question about the model's assignments, not about this definition.

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The state space is convex, closed under countable mixtures, and complete in the trace norm

Statement

Assume the Axiom of Countable Choice. Let $H \neq {0}$ be a complex Hilbert space and let $D (H)$ be the state space of The state space: density operators, pure and mixed states, and purity. Write $S_{1} (H)$ for the complex Banach space of trace-class operators on $H$ with the trace norm $∥ \cdot ∥_{1}$ , and $X : = {ρ \in S_{1} (H) : ρ^{*} = ρ}$ for the set of self-adjoint trace-class operators, regarded as a real vector space by restricting scalars from $C$ to $R$ (A field is a vector space over itself, and over any subfield $K \subseteq F$ every $F$ -vector space is a $K$ -vector space by restricting the scalars, Subfield: a subring of a field closed under inverses of its nonzero elements, and therefore a field with the restricted operations). Then:

(a) Finite mixtures. $X$ is a closed real-linear subspace of $S_{1} (H)$ and hence a real Banach space, and $D (H) \subseteq X$ is closed under finite real convex combinations, that is, it contains with any two of its points the whole real segment between them: for all $ρ, σ \in D (H)$ and every real $λ \in [0, 1]$ , $λ ρ + (1 - λ) σ \in D (H) .$ By induction every finite convex combination $\sum_{j = 1}^{n} λ_{j} ρ_{j}$ with $ρ_{j} \in D (H)$ , real $λ_{j} \geq 0$ and $\sum_{j} λ_{j} = 1$ is again an element of $D (H)$ .

(b) Countable mixtures. Let $(ρ_{n})_{n \in N} \subseteq D (H)$ and let $(w_{n})_{n \in N}$ be real weights with $w_{n} \geq 0$ and $\sum_{n} w_{n} = 1$ . Then the series $\sum_{n} w_{n} ρ_{n}$ converges absolutely in $∥ \cdot ∥_{1}$ to an element of $D (H)$ : the series converges in $X$ , its sum $T$ satisfies $T^{*} = T$ , $T \geq 0$ , $tr T = 1$ , and $∥ T ∥_{1} = tr T$ , with $∥ T - \sum_{n < K} w_{n} ρ_{n} ∥_{1} ⟶ 0 (K \to \infty) .$

(c) Grouping of consecutive blocks. In the situation of (b) let $0 = N_{0} < N_{1} < N_{2} < \dots$ be an increasing sequence with $N_{k} \to \infty$ , and for every $k \in N$ put $u_{k} : = \sum_{N_{k} \leq n < N_{k + 1}} w_{n}, σ_{k} : = \frac{1}{u_{k}} \sum_{N_{k} \leq n < N_{k + 1}} w_{n} ρ_{n},$ where it is assumed that $u_{k} > 0$ for every $k$ (blocks of zero weight are omitted). Then $σ_{k} \in D (H)$ for every $k$ , $\sum_{k} u_{k} = 1$ , and $\sum_{k} u_{k} σ_{k} = \sum_{n} w_{n} ρ_{n}$ in $∥ \cdot ∥_{1}$ , the series on the left converging absolutely.

(d) Closedness and completeness. $D (H)$ is closed in the Banach space $(X, ∥ \cdot ∥_{1})$ , and $D (H)$ is complete for the trace-norm metric.

Facts & Assumptions

Given: The complex Hilbert space $H \neq {0}$ ; density operators $ρ, σ, ρ_{n} \in D (H)$ ; real weights $λ, w_{n} \geq 0$ ; a block decomposition $N_{0} < N_{1} < \dots$ with $N_{0} = 0$ , $N_{k} \to \infty$ .

[F1]

Trace class is a Banach ideal (Trace class is a two sided Banach operator ideal): $S_{1} (H)$ is a linear subspace of $B (H)$ on which $∥ \cdot ∥_{1}$ is a norm and $(S_{1} (H), ∥ \cdot ∥_{1})$ is a Banach space; $∥ T ∥ \leq ∥ T ∥_{1}$ ; and $A T B$ is trace class with $∥ A T B ∥_{1} \leq ∥ A ∥ ∥ T ∥_{1} ∥ B ∥$ for bounded $A, B$ .

[F2]

Density operators (Density operators and scalar spectral probability clause (a), The state space: density operators, pure and mixed states, and purity): $ρ \in D (H)$ means $ρ^{*} = ρ$ , $ρ \geq 0$ (that is, $⟨ ρ x, x ⟩ \in [0, \infty)$ for all $x$ ) and $ρ$ is trace class with $tr ρ = 1$ .

[F3]

Positivity, norm and trace of a positive trace-class operator (Self-adjoint, positive, unitary and normal operators, Trace of a positive operator is the sum of its eigenvalues, Trace is absolutely convergent and basis independent): a positive operator satisfies $⟨ T x, x ⟩ \in [0, \infty)$ ; for a self-adjoint positive trace-class $T$ one has $tr T = ∥ T ∥_{1}$ ; the trace is linear and $∣ tr T ∣ \leq ∥ T ∥_{1}$ for every trace-class $T$ .

[F4]

Adjoints and operator-norm continuity (Hilbert-adjoint identities): $(a S + b T)^{*} = \overline{a} S^{*} + \overline{b} T^{*}$ for complex scalars $a, b$ ; $T^{* *} = T$ ; and $∥ T^{*} ∥ = ∥ T ∥$ . Thus $T \mapsto T^{*}$ is real-linear and isometric in the operator norm. By [F1], trace-norm convergence implies operator-norm convergence, so if trace-class $T_{m} \to T$ in trace norm and every $T_{m}$ is self-adjoint, then $T_{m}^{*} \to T^{*}$ and $T_{m} \to T$ in operator norm. Since $T_{m}^{*} = T_{m}$ for every $m$ , uniqueness of limits gives $T^{*} = T$ .

[F5]

Cauchy–Schwarz and its use (Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs): $∣ ⟨ y, x ⟩ ∣ \leq ∥ y ∥ ∥ x ∥$ ; hence $∣ ⟨ (T - S) x, x ⟩ ∣ \leq ∥ T - S ∥ ∥ x ∥^{2} \leq ∥ T - S ∥_{1} ∥ x ∥^{2}$ by [F1].

[F6]

Series in a Banach space and completeness (Series and absolute convergence in a normed space, Banach space, Convergence of a sequence in a metric space: $x_{k} \to x$ iff $d (x_{k}, x) \to 0$ in $R$ , Complete metric space: every Cauchy sequence converges in the space): an absolutely convergent series in a Banach space converges; a sequence converges to a limit $p$ when the distances tend to $0$ , and then every subsequence converges to the same limit.

[F7]

Closed subspaces of complete metric spaces (Closed subspaces of complete metric spaces are complete; the converse under countable choice clause 2): a closed subset of a complete metric space is complete; and the subspace metric is the restriction of the ambient metric.

[F8]

Convexity convention and restricted scalars (A field is a vector space over itself, and over any subfield $K \subseteq F$ every $F$ -vector space is a $K$ -vector space by restricting the scalars, Subfield: a subring of a field closed under inverses of its nonzero elements, and therefore a field with the restricted operations): a subset $C$ of a real or complex vector space is read as convex when $(1 - t) x + t y \in C$ for all $x, y \in C$ and all real $t$ with $0 \leq t \leq 1$ , coefficients always real; this is the reading fixed in clause (a) of the statement and used below. A complex vector space is a real vector space by restricting scalars along the subfield $R \subseteq C$ , with the same addition and scalar multiplication.

Proof

1.1givenF1F2F3F4F8algebra

[(a) finite mixtures are states] Let $ρ, σ \in D (H)$ and $λ \in [0, 1]$ real. The combination $λ ρ + (1 - λ) σ$ is trace class by [F1] and self-adjoint because $(λ ρ + (1 - λ) σ)^{*} = λ ρ^{*} + (1 - λ) σ^{*} = λ ρ + (1 - λ) σ$ by [F4] with real scalars; it is positive because $⟨ (λ ρ + (1 - λ) σ) x, x ⟩ = λ ⟨ ρ x, x ⟩ + (1 - λ) ⟨ σ x, x ⟩ \geq 0$ by [F2] and [F3]; and its trace is $λ tr ρ + (1 - λ) tr σ = λ + (1 - λ) = 1$ by the linearity of the trace in [F3]. Hence it is a density operator, so $D (H)$ is closed under finite real convex combinations in the reading of [F8]; the finite-combination statement follows by induction on the number of components.

1.2givenF1F3F6algebra

[(b) the mixture series converges in the trace norm] For every $n$ the operator $w_{n} ρ_{n}$ is self-adjoint by [F4] and positive, and it is trace class with $∥ w_{n} ρ_{n} ∥_{1} = w_{n} ∥ ρ_{n} ∥_{1} = w_{n} tr ρ_{n} = w_{n}$ by [F1] and [F3]; hence $\sum_{n} ∥ w_{n} ρ_{n} ∥_{1} = \sum_{n} w_{n} = 1 < \infty$ , so the series is absolutely convergent in the Banach space $S_{1} (H)$ by [F6] and has a trace-class sum $T$ ; moreover $∥ T - \sum_{n < K} w_{n} ρ_{n} ∥_{1} \to 0$ because the tail is a convergent series of norms of the omitted terms, by [F6]. The self-adjointness argument below places this sum in $X$ .

1.3givenF1F4F6F7F8algebra

[(a) $X$ is a real Banach space] If $T_{m} \in X$ and $∥ T_{m} - T ∥_{1} \to 0$ , then by [F1] and [F4], $∥ T^{*} - T ∥ \leq ∥ T^{*} - T_{m}^{*} ∥ + ∥ T_{m} - T ∥ = ∥ (T_{m} - T)^{*} ∥ + ∥ T_{m} - T ∥ = 2 ∥ T_{m} - T ∥ \leq 2 ∥ T_{m} - T ∥_{1} \to 0$ . Thus $T^{*} = T$ and $X$ is closed in $S_{1} (H)$ . It is a real-linear subspace because sums and real scalar multiples of self-adjoint operators are self-adjoint by [F4], and it is complete in the restricted trace-norm metric by [F7] applied to the complete space $S_{1} (H)$ of [F1].

2.1givenstep 1.1algebra

[(c) block members are states] Fix $k$ with $u_{k} > 0$ . The finite combination $σ_{k} = u_{k}^{- 1} \sum_{N_{k} \leq n < N_{k + 1}} w_{n} ρ_{n}$ is a convex combination of the operators $ρ_{n}$ , $N_{k} \leq n < N_{k + 1}$ , with nonnegative real coefficients $w_{n} / u_{k}$ summing to $1$ (the finite sum of the weights is $u_{k}$ ), so $σ_{k} \in D (H)$ by step 1.1 and induction.

2.2givenstep 1.2F1F2F3F4F5algebra

[(b) the mixture sum is a state] Let $S_{K} : = \sum_{n < K} w_{n} ρ_{n}$ , the self-adjoint partial sums of step 1.2. Trace-norm convergence and [F1] give $∥ T - S_{K} ∥ \to 0$ , and operator-norm continuity of the adjoint [F4] gives $∥ T^{*} - T ∥ \leq ∥ T^{*} - S_{K}^{*} ∥ + ∥ S_{K} - T ∥ = 2 ∥ T - S_{K} ∥ \leq 2 ∥ T - S_{K} ∥_{1} \to 0$ . Hence $T$ is self-adjoint. It is positive because for every $x$ the numbers $⟨ S_{K} x, x ⟩ = \sum_{n < K} w_{n} ⟨ ρ_{n} x, x ⟩$ are nonnegative by [F2], while $∣ ⟨ (T - S_{K}) x, x ⟩ ∣ \leq ∥ T - S_{K} ∥_{1} ∥ x ∥^{2} \to 0$ by [F5], so $⟨ T x, x ⟩ \geq 0$ ; and $tr T = \lim_{K} \sum_{n < K} w_{n} = 1$ because $∣ tr T - tr S_{K} ∣ \leq ∥ T - S_{K} ∥_{1} \to 0$ by [F3] and the trace is linear. Thus $T \in D (H) \subseteq X$ , and since $S_{K} \in X$ this trace-norm convergence is also convergence in $X$ .

3.1givenstep 2.1step 1.2F3F6algebra

[(c) grouping of consecutive blocks] Let $T_{m} : = \sum_{n < m} w_{n} ρ_{n}$ be the partial sums of the $n$ -series; by regrouping finite sums in the real vector space $X$ one has, for every $K$ , $\sum_{k < K} u_{k} σ_{k} = \sum_{k < K} \sum_{N_{k} \leq n < N_{k + 1}} w_{n} ρ_{n} = \sum_{n < N_{K}} w_{n} ρ_{n} = T_{N_{K}} .$ Since $T_{m} \to T$ by step 1.2 and $N_{K} \to \infty$ , the sequence $(T_{N_{K}})_{K}$ is a subsequence of a convergent sequence in the metric space $X$ and therefore converges to $T$ as well by [F6]; the left series converges absolutely because $\sum_{k} ∥ u_{k} σ_{k} ∥_{1} = \sum_{k} u_{k} tr σ_{k} = \sum_{k} u_{k} = \sum_{n} w_{n} = 1$ by [F3] and step 2.1. Hence both series have the same sum $T$ .

4.1givenstep 1.3F2F3F5F7algebra∎

[(d) closedness and completeness] Let $ρ_{m} \in D (H)$ converge to $ρ \in X$ in the trace norm. Then $ρ$ is self-adjoint by definition of $X$ ; it is positive because $∣ ⟨ (ρ - ρ_{m}) x, x ⟩ ∣ \leq ∥ ρ - ρ_{m} ∥_{1} ∥ x ∥^{2} \to 0$ by [F5] while $⟨ ρ_{m} x, x ⟩ \geq 0$ ; and $tr ρ = \lim_{m} tr ρ_{m} = 1$ by [F3] and trace linearity. Hence $ρ \in D (H)$ , so $D (H)$ is closed in $X$ ; since $X$ is complete by step 1.3, $D (H)$ is complete for the restricted metric by [F7]. This proves (a)-(d).

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Purity, spectral weights and the criterion for a pure state

Statement

Assume the Axiom of Countable Choice. Let $H \neq {0}$ be a complex Hilbert space and let $ρ \in D (H)$ be a density operator (The state space: density operators, pure and mixed states, and purity, Density operators and scalar spectral probability). Let $(λ_{n})_{n \in J}, J a finite initial segment of N or all of N,$ be the positive eigenvalues of $ρ$ listed with multiplicity, so that $ρ = \sum_{n \in J} λ_{n} T_{u_{n}, u_{n}}$ for an orthonormal family $(u_{n})_{n \in J}$ of eigenvectors, $ρ u_{n} = λ_{n} u_{n}$ (Eigenvalues, eigenvectors, eigenspaces $E_{λ} (T) = \ker (T - λ I)$ , and the spectrum $σ_{F} (T)$ of an endomorphism). Then:

(a) $λ_{n} > 0$ for every $n \in J$ , $\sum_{n \in J} λ_{n} = 1$ , and $λ_{n} \leq 1$ for every $n \in J$ .

(b) $ρ^{2}$ is a self-adjoint positive trace-class operator, and its positive eigenvalues, counted with multiplicity, are exactly the numbers $λ_{n}^{2}$ with $n \in J$ ; consequently $tr (ρ^{2}) = \sum_{n \in J} λ_{n}^{2} \in (0, 1] .$

(c) The following four statements are equivalent: (i) $tr (ρ^{2}) = 1$ ; (ii) $J = {n_{0}}$ for a single index $n_{0}$ and $λ_{n_{0}} = 1$ ; (iii) $ρ$ is a pure state; (iv) $ρ^{2} = ρ$ .

(d) If $ρ$ is not pure, then $tr (ρ^{2}) < 1$ and $J$ contains at least two indices, so that $ρ$ has at least two positive eigenvalues counted with multiplicity.

Facts & Assumptions

Given: The Hilbert space $H \neq {0}$ ; a density operator $ρ$ ; its positive eigenvalues $λ_{n}$ with orthonormal eigenvectors $u_{n}$ , $n \in J$ .

[F1]

Density operators and their spectral expansion (Density operators and scalar spectral probability clause (a), The state space: density operators, pure and mixed states, and purity): $ρ$ is self-adjoint, positive and trace class with $tr ρ = 1$ ; and $ρ$ admits the expansion $ρ = \sum_{n \in J} λ_{n} T_{u_{n}, u_{n}}$ with weights $λ_{n} \geq 0$ , $\sum_{n} λ_{n} = 1$ , and $(u_{n})$ an orthonormal family of eigenvectors of $ρ$ , the weights being its positive eigenvalues counted with multiplicity, and conversely every such expansion determines $ρ$ .

[F2]

Spectral theorem for compact self-adjoint operators (Spectral theorem for compact self adjoint operators): the nonzero eigenvalues of the compact self-adjoint operator $ρ$ form a finite or countably infinite set of real numbers with finite-dimensional eigenspaces. Since $ρ$ is positive, each nonzero eigenvalue is positive. Let $Σ$ denote the set of distinct nonzero eigenvalues, let $E_{λ} : = \ker (ρ - λ I)$ , and let $P_{λ}$ be the orthogonal projection onto $E_{λ}$ for $λ \in Σ$ . The closed span $M$ of the union of these eigenspaces equals $(\ker ρ)^{⊥} = \overline{ran ρ}$ ; $H = M \oplus \ker ρ$ ; and $ρ x = \sum_{λ \in Σ} λ P_{λ} x$ in norm, where this sum is indexed by distinct eigenvalues, not by eigenvalues repeated with multiplicity.

[F3]

Trace of a positive trace-class operator (Trace of a positive operator is the sum of its eigenvalues): for a self-adjoint positive trace-class operator its trace is the sum of its positive eigenvalues counted with multiplicity, and this sum equals its trace norm.

[F4]

Trace-class ideal and norm bound (Trace class is a two sided Banach operator ideal): $ρ^{2} = ρ ρ$ is trace class because $ρ$ is bounded and trace class, and $∥ T ∥ \leq ∥ T ∥_{1}$ for trace-class $T$ .

[F5]

Adjoints and positivity of $ρ^{2}$ (Hilbert-adjoint identities, Self-adjoint, positive, unitary and normal operators, Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs): $ρ^{2}$ is self-adjoint with $(ρ^{2})^{*} = (ρ^{*})^{2} = ρ^{2}$ and positive because $⟨ ρ^{2} x, x ⟩ = ⟨ ρ x, ρ x ⟩ = ∥ ρ x ∥^{2} \geq 0$ ; and $∣ ⟨ y, x ⟩ ∣ \leq ∥ y ∥ ∥ x ∥$ .

[F6]

Orthonormal expansions (Fourier expansion in a Hilbert space, Orthonormal families, complete orthonormal systems and Hilbert bases): for a complete orthonormal family $(e_{i})$ of a closed subspace and $x$ in that subspace, $x = \sum_{i} ⟨ x, e_{i} ⟩ e_{i}$ in norm, and the coefficients are unique: if $\sum_{i} a_{i} e_{i} = x$ in the finite-subset net, then $a_{i} = ⟨ x, e_{i} ⟩$ for all $i$ .

[F7]

Pure states and rank-one operators (The state space: density operators, pure and mixed states, and purity, Rays, unit vectors and rank-one Hilbert projections): $ρ$ is pure exactly when $ρ = T_{u, u}$ for some $u \in S (H)$ ; and for such $u$ one has $T_{u, u} u = ∥ u ∥^{2} u = u$ and $T_{u, u}^{2} = T_{u, u}$ .

Proof

1.1givenF1F3algebra

[(a) the weights are positive and sum to one] By [F1] the $λ_{n}$ are exactly the positive eigenvalues with multiplicity, so $λ_{n} > 0$ for $n \in J$ ; the identity $\sum_{n \in J} λ_{n} = tr ρ = 1$ is [F1] together with [F3] applied to the positive trace-class operator $ρ$ ; since all terms are nonnegative and their sum is $1$ , each single term satisfies $λ_{n} \leq 1$ .

2.1givenstep 1.1F1F2F3F4F5F6algebra

[(b) the eigenvalues of $ρ^{2}$ ] By [F4] and [F5] the operator $ρ^{2}$ is self-adjoint, positive and trace class. For each distinct positive eigenvalue $λ$ of $ρ$ , [F1] lists exactly its finite multiplicity of orthonormal eigenvectors among $(u_{n})$ ; since an orthonormal family is linearly independent and $E_{λ}$ has that finite dimension by [F2], those vectors form an orthonormal basis of $E_{λ}$ . Their union is therefore a complete orthonormal family of $M$ by [F2]. If $ρ^{2} x = μ x$ with $μ > 0$ , then $x = μ^{- 1} ρ^{2} x \in ran ρ \subseteq M$ by [F2], so [F6] gives the norm-convergent expansion $x = \sum_{n \in J} a_{n} u_{n}$ , where $a_{n} = ⟨ x, u_{n} ⟩$ . The expansion in [F1] gives $ρ u_{n} = λ_{n} u_{n}$ ; applying the bounded operator $ρ$ twice to the finite partial sums of the expansion of $x$ yields $ρ^{2} x = \sum_{n \in J} λ_{n}^{2} a_{n} u_{n}$ in norm. By uniqueness of coefficients in [F6], $(λ_{n}^{2} - μ) a_{n} = 0$ for every $n$ . Thus the $μ$ -eigenspace of $ρ^{2}$ is exactly the closed span of those $u_{n}$ with $λ_{n}^{2} = μ$ : the reverse inclusion holds because each such $u_{n}$ is an eigenvector of $ρ^{2}$ , and the forward inclusion follows from the coefficient identity and completeness. Since $λ_{n} > 0$ , the positive eigenvalues of $ρ^{2}$ are exactly the numbers $λ_{n}^{2}$ with multiplicity. By [F3], $tr (ρ^{2}) = \sum_{n \in J} λ_{n}^{2}$ . The sum is positive because some $λ_{n} > 0$ by $\sum λ_{n} = 1$ , and it is at most $1$ because $λ_{n}^{2} \leq λ_{n}$ by step 1.1 and $\sum λ_{n} = 1$ .

3.1givenstep 1.1step 2.1F1F3F7algebra

[(c) equivalences] Statement (i) holds, that is $\sum_{n} λ_{n}^{2} = 1 = \sum_{n} λ_{n}$ , if and only if $\sum_{n} λ_{n} (1 - λ_{n}) = 0$ ; since $0 \leq λ_{n} \leq 1$ by step 1.1, each term is nonnegative, so this happens exactly when $λ_{n} \in {0, 1}$ for every $n$ , and because $λ_{n} > 0$ on the index set $J$ and $\sum_{n \in J} λ_{n} = 1$ , exactly when $J$ consists of a single index $n_{0}$ with $λ_{n_{0}} = 1$ ; this is (ii). If (ii) holds, then the expansion of [F1] reads $ρ = T_{u_{n_{0}}, u_{n_{0}}}$ with $u_{n_{0}}$ a unit vector of the orthonormal family, so $ρ$ is pure, which is (iii), by [F7]. If $ρ$ is pure, say $ρ = T_{u, u}$ with $∥ u ∥ = 1$ , then $ρ^{2} = ρ$ by the product rule $T_{u, u} T_{u, u} = T_{T_{u, u} u, u} = T_{u, u}$ recalled in [F7], giving (iv). Finally, if $ρ^{2} = ρ$ and $ρ x = λ x$ , then $λ^{2} x = ρ^{2} x = ρ x = λ x$ , so $λ \in {0, 1}$ for the positive eigenvalues $n \in J$ ; since their sum is $1$ there is exactly one index $n_{0}$ with $λ_{n_{0}} = 1$ , which is (ii), and then $tr (ρ^{2}) = tr ρ = 1$ gives (i). The implications (i) $\Rightarrow$ (ii) $\Rightarrow$ (iii) $\Rightarrow$ (iv) $\Rightarrow$ (ii) show that all four statements are equivalent.

4.1givenstep 1.1step 2.1step 3.1algebra∎

[(d) non-pure states have at least two positive eigenvalues] If $ρ$ is not pure, then $tr (ρ^{2}) \neq 1$ by step 3.1, and $tr (ρ^{2}) \leq 1$ by step 2.1, so $tr (ρ^{2}) < 1$ . If $J$ had exactly one index $n_{0}$ , then $\sum_{n \in J} λ_{n} = λ_{n_{0}} = 1$ by step 1.1, and condition (ii) of step 3.1 would hold, making $ρ$ pure by that step, a contradiction; hence $J$ has at least two indices. This proves (a)-(d).

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

The extreme points of the state space are exactly the pure states

Statement

Assume the Axiom of Countable Choice. Let $H \neq {0}$ be a complex Hilbert space and let $D (H)$ be its state space (The state space: density operators, pure and mixed states, and purity), regarded as a subset of the real vector space $X$ of self-adjoint trace-class operators, closed under finite real convex combinations by The state space is convex, closed under countable mixtures, and complete in the trace norm. Call a state $ρ \in D (H)$ an extreme point of $D (H)$ when $ρ = (1 - t) σ + t τ, σ, τ \in D (H), 0 < t < 1 ⟹ σ = τ = ρ,$ holds for all real $t$ with $0 < t < 1$ and all $σ, τ \in D (H)$ . Then $ρ$ is an extreme point of $D (H)$ if and only if $ρ$ is a pure state.

Facts & Assumptions

Given: The Hilbert space $H \neq {0}$ ; a state $ρ \in D (H)$ ; states $σ, τ \in D (H)$ and a real $t \in (0, 1)$ with $ρ = (1 - t) σ + t τ$ where a decomposition is assumed.

[F1]

The state space and purity (The state space: density operators, pure and mixed states, and purity, Purity, spectral weights and the criterion for a pure state): a state is a self-adjoint positive trace-class operator of trace $1$ ; it is pure exactly when it equals $T_{u, u}$ for some unit vector $u$ ; for a state $ρ$ with positive eigenvalues $λ_{n}$ counted with multiplicity one has $\sum_{n} λ_{n} = 1$ , each $λ_{n} \in (0, 1]$ , $tr (ρ^{2}) \leq 1$ , and $ρ$ is mixed exactly when it has at least two positive eigenvalues.

[F2]

Convexity and countable mixtures (The state space is convex, closed under countable mixtures, and complete in the trace norm): $D (H)$ is a convex subset of the real vector space $X$ of self-adjoint trace-class operators, and every countable probability-weighted sum $\sum_{n} w_{n} ρ_{n}$ with $ρ_{n} \in D (H)$ , $w_{n} \geq 0$ , $\sum_{n} w_{n} = 1$ , converges in trace norm to an element of $D (H)$ .

[F3]

Spectral expansion of a state (Density operators and scalar spectral probability clause (a), Spectral theorem for compact self adjoint operators): a state $ρ$ equals $\sum_{n} λ_{n} T_{u_{n}, u_{n}}$ for its positive eigenvalues $λ_{n}$ counted with multiplicity and an orthonormal family of eigenvectors $u_{n}$ . For the spectral theorem, let $Σ$ be the set of distinct nonzero eigenvalues of $ρ$ , and for $λ \in Σ$ let $P_{λ}$ be the orthogonal projection onto $E_{λ} (ρ)$ . Then $ρ x = \sum_{λ \in Σ} λ P_{λ} x$ in norm, the sum being indexed by distinct eigenvalues; the closed span $M$ of those eigenspaces is $(\ker ρ)^{⊥} = \overline{ran ρ}$ , and $H = M \oplus \ker ρ$ . Positivity makes every member of $Σ$ positive, and the positive eigenvalues counted with multiplicity are exactly the $λ_{n}$ from the density expansion.

[F4]

Norm and trace bounds for states (Trace class is a two sided Banach operator ideal, Trace of a positive operator is the sum of its eigenvalues, Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs, The operator norm as the least bound and as the unit-sphere or unit-ball supremum): for trace-class $S$ one has $∥ S ∥ \leq ∥ S ∥_{1}$ ; for a positive trace-class $S$ one has $∥ S ∥_{1} = tr S$ ; hence $∥ σ ∥ \leq ∥ σ ∥_{1} = 1$ for every state $σ$ ; and $∣ ⟨ y, x ⟩ ∣ \leq ∥ y ∥ ∥ x ∥$ with equality if and only if $y, x$ are linearly dependent.

[F5]

Pure states and projections (Rays, unit vectors and rank-one Hilbert projections): for $u \in S (H)$ the operator $T_{u, u}$ is the Hilbert orthogonal projection onto $C u$ with $T_{u, u} u = u$ and $T_{u, u}^{2} = T_{u, u}$ , and for unit vectors $u, w$ one has $T_{u, u} = T_{w, w}$ if and only if $w = ζ u$ for some $ζ \in U (1)$ ; distinct rays therefore give distinct pure states.

[F6]

Extremality convention and real scalars (A field is a vector space over itself, and over any subfield $K \subseteq F$ every $F$ -vector space is a $K$ -vector space by restricting the scalars): a point $x$ of a set $K$ is read as extreme when every representation $x = (1 - t) y + t z$ with $y, z \in K$ and real $0 < t < 1$ forces $y = z = x$ ; this is the condition displayed in the statement and used below. The space $X$ is read as a real vector space by restricting scalars from $C$ to $R$ .

[F7]

Orthogonal projections (The Hilbert orthogonal projection onto a closed subspace): the Hilbert projection $P_{M}$ is characterised by $P_{M} x \in M$ and $x - P_{M} x \in M^{⊥}$ ; a one-dimensional subspace $C u$ with $∥ u ∥ = 1$ equals the range of $T_{u, u}$ by [F5].

Proof

1.1givenF1F4F5algebra

[(pure $\Rightarrow$ extreme) the quadratic forms are pinned to one] Assume $ρ = T_{u, u}$ with $u \in S (H)$ and $ρ = (1 - t) σ + t τ$ with $0 < t < 1$ . Then $⟨ ρ u, u ⟩ = ⟨ T_{u, u} u, u ⟩ = ⟨ u, u ⟩^{2} = ∥ u ∥^{2} = 1$ by [F5] and the definition of $T_{u, u}$ , so $1 = (1 - t) ⟨ σ u, u ⟩ + t ⟨ τ u, u ⟩$ . Every state $η \in D (H)$ satisfies $⟨ η u, u ⟩ \geq 0$ by positivity and $⟨ η u, u ⟩ \leq ∣ ⟨ η u, u ⟩ ∣ \leq ∥ η u ∥ ∥ u ∥ = ∥ η u ∥ \leq ∥ η ∥ \leq ∥ η ∥_{1} = 1$ by [F4]; a real convex combination with weights $1 - t, t > 0$ of two numbers $\leq 1$ equals $1$ only if both numbers equal $1$ . Hence $⟨ σ u, u ⟩ = ⟨ τ u, u ⟩ = 1$ .

2.1givenstep 1.1F4algebra

[(pure $\Rightarrow$ extreme) the first component fixes $u$ ] For the state $σ$ we have the chain $1 = ⟨ σ u, u ⟩ \leq ∣ ⟨ σ u, u ⟩ ∣ \leq ∥ σ u ∥ ∥ u ∥ = ∥ σ u ∥ \leq ∥ σ ∥ \leq 1$ by [F4]; all inequalities are therefore equalities, so equality holds in the Cauchy–Schwarz inequality for the pair $(σ u, u)$ , and the equality case of [F4] gives $σ u = ζ u$ for a scalar $ζ$ because $u \neq 0$ . Taking the pairing with $u$ gives $1 = ⟨ σ u, u ⟩ = ζ ⟨ u, u ⟩ = ζ$ , so $σ u = u$ . The same argument gives $τ u = u$ .

3.1givenstep 2.1F1F3F5F6F7algebra

[(pure $\Rightarrow$ extreme) a state fixing $u$ equals $T_{u, u}$ ] Let $η \in D (H)$ with $η u = u$ . By [F3] its positive eigenvalues $λ_{n}$ , counted with multiplicity, satisfy $\sum_{n} λ_{n} = 1$ , and $u$ is an eigenvector with eigenvalue $1$ . If $m$ is the multiplicity of eigenvalue $1$ , then $1 = \sum_{n} λ_{n} = m + \sum_{λ_{n} \neq 1} λ_{n}$ with a nonnegative second sum, so $m = 1$ and there is no other positive eigenvalue. Thus $E_{1} (η)$ is one-dimensional and contains $u$ , so $E_{1} (η) = C u$ . By the distinct-eigenvalue decomposition in [F3], the closed span $M$ of the positive eigenspaces is now $E_{1} (η) = C u$ and $H = M \oplus \ker η$ ; hence $η$ is the orthogonal projection $P_{1}$ onto $C u$ . By [F7] and [F5] this projection is $T_{u, u}$ . Applying this to $σ$ and to $τ$ gives $σ = τ = T_{u, u} = ρ$ ; hence every representation of $ρ$ as a nontrivial convex combination is trivial, and $ρ$ is an extreme point by [F6].

4.1givenstep 3.1F1F2F3F5F6algebra

[(mixed $\Rightarrow$ not extreme) an explicit splitting] Assume now that $ρ$ is not pure. By [F1] the state $ρ$ has at least two positive eigenvalues; list them with multiplicity as $λ_{1} \geq λ_{2} \geq \dots > 0$ with orthonormal eigenvectors $u_{1}, u_{2}$ , so $λ_{2} > 0$ . Put $ε : = \frac{λ_{2}}{2} > 0, σ_{\pm} : = ρ \pm ε (T_{u_{1}, u_{1}} - T_{u_{2}, u_{2}}) .$ The expansion of [F3] for $ρ$ turns the displayed definitions into countable mixtures of the pure states $T_{u_{n}, u_{n}}$ with weights $λ_{1} \pm ε$ , $λ_{2} \mp ε$ , $λ_{3}, λ_{4}, \dots$ ; these weights are nonnegative because $λ_{2} - ε = ε > 0$ and $λ_{1} \pm ε \geq 0$ , and they sum to $\sum_{n} λ_{n} \pm ε \mp ε = 1$ ; hence $σ_{\pm} \in D (H)$ by [F2]. Moreover $ρ = \frac{1}{2} σ_{+} + \frac{1}{2} σ_{-}$ , and $σ_{+} - σ_{-} = 2 ε (T_{u_{1}, u_{1}} - T_{u_{2}, u_{2}})$ , which is nonzero because $u_{1}, u_{2}$ are orthogonal unit vectors and hence not phase multiples of one another, so that $T_{u_{1}, u_{1}} \neq T_{u_{2}, u_{2}}$ by [F5]. Thus the representation $ρ = \frac{1}{2} σ_{+} + \frac{1}{2} σ_{-}$ with $σ_{+} \neq σ_{-}$ is a nontrivial convex representation with $0 < \frac{1}{2} < 1$ , and $ρ$ is not an extreme point by [F6].

5.1step 3.1step 4.1∎

[Conclusion] Steps 1.1, 2.1 and 3.1 show that every pure state is extreme, and step 4.1 shows that every non-pure state is not extreme. Hence the extreme points of $D (H)$ are exactly the pure states.

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The state space of the framework: preparations, pure rays and density operators

Physical scope

The state-space vocabulary of the non-relativistic quantum framework in the declared massive-matter sector: given the state postulate's assignment of a nonzero separable complex Hilbert space and its identification of pure preparations with unit rays and general preparations with density operators, this item fixes the readings 'state space', 'state', 'pure state', 'mixed state', 'state assigned to a state vector' and 'purity', and the correspondence between pure preparations and pure states. It adds no physical assumption beyond the state postulate, asserts no dynamics, probability law, measurement model or empirical result, and excludes superselection restrictions, identical-particle sectors and all field or radiation content.

Definition

(a) State space. Adopt the state postulate State and composition postulate P1: Hilbert spaces, preparations and composite systems for a declared system of the declared massive-matter sector, with its assigned nonzero separable complex Hilbert space $H$ . The state space of the system in the model is the set $D (H) = {ρ : ρ is a density operator on H}$ of The state space: density operators, pure and mixed states, and purity: the self-adjoint positive trace-class operators on $H$ of trace one. A state of the system is an element of $D (H)$ ; this is the geometry vocabulary for the general preparations of clause (iii) of the state postulate. The model's adopted assignment of a state to each preparation is not produced by this definition; it is the content of that postulate.

(b) Pure and mixed states. A state is pure when it is one of the operators $T_{u, u}$ with $u \in S (H)$ , and mixed when it is not pure, as in The state space: density operators, pure and mixed states, and purity. By Rays, unit vectors and rank-one Hilbert projections the map $j : R (H) \to D (H), j ([ψ]) = T_{ν (ψ), ν (ψ)},$ is a bijection of the ray space onto the set of pure states; and by clause (ii) of the state postulate the pure preparations are the unit rays of $H$ , so the model's pure preparations are exactly its pure states.

(c) State assigned to a state vector. For a nonzero vector $ψ \in H^{\times}$ the state assigned to $ψ$ is the normalized rank-one operator $\hat{ρ} (ψ) = \frac{T_{ψ, ψ}}{∥ ψ ∥^{2}} = T_{ν (ψ), ν (ψ)} \in D (H),$ which by Rays, unit vectors and rank-one Hilbert projections depends only on the ray $[ψ]$ ; clause (ii) of the state postulate assigns the same operator to the pure preparation $[ψ]$ . The zero vector is excluded here because $∥ ψ ∥^{- 1}$ is defined only for $ψ \neq 0$ ; no state is assigned to it, in agreement with the state postulate.

(d) Mixtures and purity. A statistical mixture of preparations with weights is the trace-norm convergent sum of clause (iii) of the state postulate; its value is the corresponding convex combination of states. The purity of a state $ρ$ is $tr (ρ^{2})$ , and by Purity, spectral weights and the criterion for a pure state a state is pure exactly when its purity equals $1$ .

(e) Status. Clauses (a)-(d) record the readings of the state postulate in the geometry vocabulary of this page and fix the terminology used by the later pages of the category. The cited mathematical suppliers of the ray-to-state correspondence and of the purity criterion assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )), which is carried as a hypothesis of these readings. They adopt no assumption beyond the state postulate and that hypothesis, assert no dynamics, no probability law and no measurement model, and make no empirical claim: in particular, whether a given laboratory preparation is pure or mixed, and whether a given apparatus realizes a prescribed mixture, are physical identifications with uncertainty, governed by the preparation and apparatus assumptions of the framework, not consequences of this definition. Superselection restrictions and identical-particle symmetry sectors, where imposed, are separate assumptions by clause (v) of the state postulate and are not part of the state space fixed here.

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Pure preparations are well defined by their rays, and phase factors do not change their states

Physical scope

A model-internal well-definedness statement about the state assignment of the non-relativistic quantum framework: with the state postulate adopted, the pure preparation of a nonzero state vector is independent of the chosen representative of its ray, phase-equivalent unit representatives yield the same pure state, distinct rays yield distinct pure states, and the zero vector is assigned no preparation or state. The item makes no empirical claim, asserts no dynamics or probability law, and does not assert that any particular laboratory procedure prepares a ray.

Statement

Adopt the state and composition postulate P1 of the framework (State and composition postulate P1: Hilbert spaces, preparations and composite systems) and use the state-space vocabulary of The state space of the framework: preparations, pure rays and density operators. Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )), which is the choice hypothesis of the rank-one suppliers cited below. Let $H \neq {0}$ be the Hilbert space assigned to a declared system, let $ψ, φ \in H^{\times}$ be nonzero state vectors, and let $[ψ], [φ]$ be their rays. Then:

(a) Representative independence. If $φ = z ψ$ for some nonzero $z \in C$ , then the normalized rank-one operators agree, $\frac{T_{φ, φ}}{∥ φ ∥^{2}} = \frac{T_{ψ, ψ}}{∥ ψ ∥^{2}} = T_{ν (ψ), ν (ψ)},$ and this operator is a pure state. In particular the density operator associated with a pure preparation depends only on the ray $[ψ]$ and not on the representative used to name it.

(b) Phase invariance and injectivity. If $u, w \in S (H)$ are unit representatives of the same ray, then $w = ζ u$ for a unique $ζ \in U (1)$ and $T_{w, w} = T_{u, u}$ ; conversely, if two pure preparations $[ψ] \neq [φ]$ are distinct, then their assigned pure states are distinct. Hence the assignment $[ψ] \mapsto T_{ν (ψ), ν (ψ)}$ is a bijection from the set of pure preparations onto the set of pure states.

(c) Exclusion of the zero vector. The zero vector is not a pure preparation of the model: it has no unit representative, the normalization map is undefined at it, the operator $T_{0, 0} = 0$ has trace $0$ and is not a density operator, and the state postulate assigns pure preparations only to unit rays of vectors of norm one. Consequently no pure preparation and no state is assigned to $0$ .

Facts & Assumptions

Given: The adopted state postulate P1 with its Hilbert-space assignment $H$ to the declared system; nonzero state vectors $ψ, φ$ and unit representatives $u, w$ ; the ray $[ψ]$ and normalization $ν$ of State vectors, unit vectors, rays, normalization and phases of a Hilbert space.

[F1]

The state postulate (State and composition postulate P1: Hilbert spaces, preparations and composite systems clauses (i)-(iii)): the model assigns to the system a nonzero separable complex Hilbert space $H$ ; a pure preparation is a unit ray $[ψ] = {z ψ : z \neq 0}$ with $∥ ψ ∥ = 1$ , two nonzero vectors describing the same pure preparation exactly when they differ by a nonzero complex factor; a general preparation is a density operator; the zero vector is not a state and no preparation is assigned to it; and the rank-one operator $T_{ψ, ψ}$ is the density operator associated with the ray of the unit vector $ψ$ .

[F2]

State-space vocabulary (The state space of the framework: preparations, pure rays and density operators, The state space: density operators, pure and mixed states, and purity): the state assigned to a nonzero vector is the normalized rank-one operator $T_{ψ, ψ} / ∥ ψ ∥^{2} = T_{ν (ψ), ν (ψ)}$ , the pure states are the operators $T_{u, u}$ with $u \in S (H)$ , and $D (H)$ consists of the density operators of trace one.

[F3]

Rank-one scaling and equality (Rays, unit vectors and rank-one Hilbert projections, Rank-one operator calculus clauses (a)-(c)): $T_{z ψ, z ψ} = ∣ z ∣^{2} T_{ψ, ψ}$ for $z \neq 0$ ; $T_{ψ, ψ} = ∥ ψ ∥^{2} T_{ν (ψ), ν (ψ)}$ ; $T_{ψ, ψ} = T_{φ, φ}$ holds exactly when $∥ ψ ∥ = ∥ φ ∥$ and $[ψ] = [φ]$ ; and for unit $u, w$ , $T_{u, u} = T_{w, w}$ holds exactly when $w = ζ u$ for some $ζ \in U (1)$ .

[F4]

Normalization and phases (Normalization, ray equivalence and the phase orbits of the unit sphere): $ν (ψ) \in [ψ] \cap S (H)$ , $[ψ] = [φ]$ holds exactly when $ν (φ) = ζ ν (ψ)$ for a unique $ζ \in U (1)$ , and for unit $u, w$ one has $[u] = [w]$ exactly when $w = ζ u$ with unique $ζ \in U (1)$ .

[F5]

Norms (The induced length is a norm): $∥ z ψ ∥ = ∣ z ∣ ∥ ψ ∥$ , $∥ ψ ∥ = 0$ exactly for $ψ = 0$ , and $∥ ψ ∥ \geq 0$ . In particular $∥ 0 ∥ = 0$ , so the scalar $∥ ψ ∥^{- 1}$ used in the normalization is defined only for nonzero $ψ$ .

[F6]

Rank-one trace, including zero (Rank-one operator calculus clauses (b)-(c)): for all $x, y \in H$ , including zero, $T_{x, y}$ is trace class and $tr (T_{x, y}) = ⟨ x, y ⟩$ ; its defining formula gives $T_{0, 0} = 0$ .

Proof

1.1givenF1F2F3F5algebra

[(a) representative independence] Let $φ = z ψ$ with $z \neq 0$ . By [F3] one has $T_{φ, φ} = T_{z ψ, z ψ} = ∣ z ∣^{2} T_{ψ, ψ}$ , and by [F5] the squared norms satisfy $∥ φ ∥^{2} = ∣ z ∣^{2} ∥ ψ ∥^{2}$ with $∣ z ∣^{2} \neq 0$ ; dividing the first identity by the second gives $T_{φ, φ} / ∥ φ ∥^{2} = T_{ψ, ψ} / ∥ ψ ∥^{2}$ , and the second identity of [F3] identifies the common value as $T_{ν (ψ), ν (ψ)}$ , which is a pure state by [F2]. Thus the operator depends only on the ray.

1.2givenF1F3F4algebra

[(b) phase representatives give the same state] Let $u, w \in S (H)$ lie in the same ray. By [F4] one has $w = ζ u$ for a unique $ζ \in U (1)$ ; then $T_{w, w} = T_{ζ u, ζ u} = ∣ ζ ∣^{2} T_{u, u} = T_{u, u}$ by [F3]. Hence multiplying a unit representative by a phase factor does not change the assigned pure state, in agreement with clause (ii) of [F1], which identifies the pure preparation only up to a nonzero complex factor.

2.1givenstep 1.2F2F3F4algebra

[(b) injectivity of the assignment] Suppose two pure preparations $[ψ], [φ]$ have the same assigned pure state, that is $T_{ν (ψ), ν (ψ)} = T_{ν (φ), ν (φ)}$ . Since $ν (ψ), ν (φ) \in S (H)$ , [F3] gives $ν (φ) = ζ ν (ψ)$ for some $ζ \in U (1)$ , and [F4] then gives $[φ] = [ψ]$ . Hence the assignment is injective, and it is surjective onto the set of pure states by [F2]; so it is a bijection from the pure preparations onto the pure states.

3.1givenstep 1.1F1F2F5F6algebra∎

[(c) the zero vector] By [F5], $∥ 0 ∥ = 0$ , so the normalization map $ν (ψ) = ∥ ψ ∥^{- 1} ψ$ is undefined at $ψ = 0$ because the scalar $∥ ψ ∥^{- 1}$ does not exist; the ray of $0$ would be ${z \cdot 0 : z \neq 0} = {0}$ , which contains no unit vector, so clause (ii) of [F1] assigns no pure preparation to it. Moreover $T_{0, 0} = 0$ has $tr (T_{0, 0}) = ⟨ 0, 0 ⟩ = 0$ by [F6], so it is not a density operator of trace one and hence not an element of $D (H)$ by [F2]; clause (iii) of [F1] likewise assigns no general preparation to $0$ . This proves (c) and completes (a)-(c).

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Statistical mixtures, spectral decompositions and the extremal role of pure preparations

Physical scope

Model-internal statements about preparations of the non-relativistic quantum framework under the state postulate: statistical mixtures of preparations with probability weights are preparations; every preparation has a spectral decomposition as a countable mixture of pairwise orthogonal pure preparations; and the pure preparations are exactly the extreme points of the preparation set, so every mixed preparation is a nontrivial mixture of two preparations. The weights are classical preparation probabilities, not amplitudes. The item asserts no apparatus realization of any mixture, no measurement or probability law, no dynamics and no empirical result.

Statement

Adopt the state and composition postulate P1 of the framework (State and composition postulate P1: Hilbert spaces, preparations and composite systems) and use the state-space vocabulary of The state space of the framework: preparations, pure rays and density operators. Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )), which is the choice hypothesis of the spectral and mixture suppliers cited below. Let $H \neq {0}$ be the Hilbert space assigned to a declared system. Then:

(a) Mixtures. Let $(ρ_{n})$ be a finite or countable family of preparations with real weights $w_{n} \geq 0$ satisfying $\sum_{n} w_{n} = 1$ . Then the statistical mixture $\sum_{n} w_{n} ρ_{n}$ of clause (iii) of P1 converges in trace norm to a preparation, namely to the density operator $T = \sum_{n} w_{n} ρ_{n}$ of The state space is convex, closed under countable mixtures, and complete in the trace norm. The mixture is affine in its data: a term of zero weight may be added or removed without changing $T$ , and splitting one weight between two copies of the same preparation, $w ρ = w_{1} ρ + w_{2} ρ$ with $w_{1} + w_{2} = w$ , leaves $T$ unchanged. If the index set is partitioned into consecutive finite blocks with total weights $u_{k} > 0$ and block mixtures $σ_{k}$ , then $\sum_{k} u_{k} σ_{k} = \sum_{n} w_{n} ρ_{n}$ in trace norm, so mixtures may be regrouped.

(b) Spectral decomposition. Every preparation $ρ$ is a countable statistical mixture of pairwise orthogonal pure preparations: there exist a finite or countable orthonormal family $(u_{n})$ of unit vectors and positive weights $λ_{n}$ with $\sum_{n} λ_{n} = 1$ such that $ρ = \sum_{n} λ_{n} T_{u_{n}, u_{n}},$ the sum converging in trace norm, and the weights $λ_{n}$ are exactly the positive eigenvalues of $ρ$ counted with multiplicity.

(c) Extremality and nontrivial decompositions. A preparation is pure if and only if its density operator is an extreme point of the state space $D (H)$ in the sense of the condition displayed in The extreme points of the state space are exactly the pure states; consequently every mixed preparation $ρ$ admits a representation as a statistical mixture of two preparations in a nontrivial way: if $λ_{1} \geq λ_{2} > 0$ are two positive eigenvalues of $ρ$ with orthonormal eigenvectors $u_{1}$ and $u_{2}$ , then with $ε : = λ_{2} / 2$ the operators $σ_{\pm} : = ρ \pm ε (T_{u_{1}, u_{1}} - T_{u_{2}, u_{2}})$ are preparations with $ρ = \frac{1}{2} σ_{+} + \frac{1}{2} σ_{-}$ and $σ_{+} \neq σ_{-}$ .

Facts & Assumptions

Given: The adopted state postulate P1, the assigned Hilbert space $H \neq {0}$ , preparations $ρ, ρ_{n} \in D (H)$ , weights $w_{n} \geq 0$ with $\sum_{n} w_{n} = 1$ , and where required two positive eigenvalues $λ_{1} \geq λ_{2} > 0$ of $ρ$ with orthonormal eigenvectors $u_{1}, u_{2}$ .

[F1]

The state postulate (State and composition postulate P1: Hilbert spaces, preparations and composite systems): pure preparations are the unit rays of $H$ ; general preparations are the density operators on $H$ ; a statistical mixture of preparations $ρ_{n}$ with weights $w_{n} \geq 0$ , $\sum_{n} w_{n} = 1$ , is the trace-norm convergent sum $ρ = \sum_{n} w_{n} ρ_{n}$ , whose weights are a classical preparation uncertainty and not amplitudes; and the density operator associated with a unit ray $[ψ]$ is $T_{ψ, ψ}$ .

[F2]

State-space vocabulary and mixtures (The state space of the framework: preparations, pure rays and density operators, The state space is convex, closed under countable mixtures, and complete in the trace norm): preparations are the states $D (H)$ ; countable probability-weighted sums of states converge absolutely in the trace norm to a state, finite convex combinations of states are states, and consecutive finite blocks of a weighted family may be grouped with the grouped weights producing the same sum.

[F3]

Spectral expansion (Density operators and scalar spectral probability clause (a), Purity, spectral weights and the criterion for a pure state): every density operator $ρ$ admits an expansion $ρ = \sum_{n} λ_{n} T_{u_{n}, u_{n}}$ converging in trace norm, with $λ_{n} > 0$ , $\sum_{n} λ_{n} = 1$ , and $(u_{n})$ an orthonormal family of eigenvectors; the $λ_{n}$ are the positive eigenvalues of $ρ$ counted with multiplicity. The purity criterion gives that $ρ$ is pure exactly when the multiplicity list has one positive eigenvalue equal to $1$ .

[F4]

Pure states are extreme (The extreme points of the state space are exactly the pure states): the extreme points of $D (H)$ are exactly the pure states.

[F5]

Pure states and rays (Rays, unit vectors and rank-one Hilbert projections, The state space: density operators, pure and mixed states, and purity): $T_{u, u}$ with $u \in S (H)$ are exactly the pure states, and $T_{u_{1}, u_{1}} \neq T_{u_{2}, u_{2}}$ for orthonormal unit vectors $u_{1}, u_{2}$ .

[F6]

Extremality condition (The extreme points of the state space are exactly the pure states): a state $x \in D (H)$ is extreme when $x = (1 - t) y + t z$ with $y, z \in D (H)$ and real $0 < t < 1$ implies $y = z = x$ .

Proof

1.1givenF1F2algebra

[(a) mixtures and affinity] By [F2] the sum $T = \sum_{n} w_{n} ρ_{n}$ converges absolutely in trace norm to a state, so it is a preparation in the sense of [F1]; the identity $\sum_{n} w_{n} ρ_{n} = \sum_{n} w_{n} ρ_{n} + 0 \cdot ρ^{'}$ for an extra preparation $ρ^{'}$ and the identity $w ρ = w_{1} ρ + w_{2} ρ$ for $w_{1} + w_{2} = w$ are identities in the real vector space of self-adjoint trace-class operators, so the value of the mixture is unchanged; and the grouping statement for consecutive blocks is clause (c) of [F2].

1.2givenF2F3F5algebra

[(b) the spectral decomposition is a mixture of pure preparations] By [F3] the preparation $ρ$ has an expansion $ρ = \sum_{n} λ_{n} T_{u_{n}, u_{n}}$ with positive weights $λ_{n}$ , $\sum_{n} λ_{n} = 1$ and $(u_{n})$ orthonormal; by [F5] each $T_{u_{n}, u_{n}}$ is a pure state, and pairwise orthogonality of the $u_{n}$ makes the pure preparations pairwise distinct; the convergence in trace norm is the convergence of a statistical mixture as in [F2]. Thus $ρ$ is exhibited as the mixture of the pure preparations $T_{u_{n}, u_{n}}$ with the weights $λ_{n}$ , which by [F3] are its positive eigenvalues counted with multiplicity.

1.3givenF2F4

[(c) extremality] By [F4] the extreme points of $D (H)$ are exactly the pure states; since preparations are the states of $D (H)$ by [F2], a preparation is pure exactly when its density operator is extreme.

2.1givenstep 1.3F1F2F3F5F6algebra

[(c) the nontrivial splitting of a mixed preparation] By [F3], a mixed $ρ$ has at least two positive eigenvalues counted with multiplicity. Choose two with $λ_{1} \geq λ_{2} > 0$ and orthonormal eigenvectors $u_{1}, u_{2}$ , and put $ε = λ_{2} / 2$ . In the spectral expansion of [F3], choose the orthonormal eigenvectors to include $u_{1}, u_{2}$ (within either selected eigenspace, replace its finite orthonormal basis by one containing the specified vectors). The operators $σ_{\pm} = ρ \pm ε (T_{u_{1}, u_{1}} - T_{u_{2}, u_{2}})$ therefore have the same trace-norm convergent expansion with the two selected weights replaced by $λ_{1} \pm ε$ and $λ_{2} \mp ε$ . All weights remain positive, since $λ_{1} - ε \geq λ_{2} / 2 > 0$ and $λ_{2} - ε = λ_{2} / 2 > 0$ , and their sum remains one. By [F2] and [F5] each $σ_{\pm}$ is a density operator and hence a preparation. Their average is $ρ$ , while $σ_{+} - σ_{-} = 2 ε (T_{u_{1}, u_{1}} - T_{u_{2}, u_{2}}) \neq 0$ by [F5]. Thus [F1] reads the identity as a nontrivial statistical mixture with weights $1 / 2$ , so $ρ$ is not extreme by [F6], consistent with step 1.3.

3.1step 1.1step 1.2step 1.3step 2.1∎

[Conclusion] Steps 1.1-1.2 prove (a) and (b); steps 1.3 and 2.1 prove (c).

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Scope audit of the state-geometry page: what is mathematics, what is adopted, and what is deferred

Physical scope

A scope audit of the state-geometry pair: it separates the mathematical layer (rays, normalization, rank-one projections, convexity, purity and extremality of the state space) from the readings of the state postulate adopted on the framework page, records that this page adopts no new physical assumption and no empirical claim, and lists the deliberate deferrals: observable consequences of phase, the probability law, measurement models, superselection and identical-particle sectors, and the photon interpretation of two-level polarization vectors. The remark proves nothing and asserts no observation.

Scope of the state-geometry page

This remark audits the status of the items of the pair. It records which statements are conditional mathematics, which are readings of an adopted model assumption, and which questions are deliberately left to other pages. It proves nothing and asserts no empirical result.

Mathematical layer, independent of any physical assumption. The items State vectors, unit vectors, rays, normalization and phases of a Hilbert space, Normalization, ray equivalence and the phase orbits of the unit sphere, Rays, unit vectors and rank-one Hilbert projections, The state space: density operators, pure and mixed states, and purity, The state space is convex, closed under countable mixtures, and complete in the trace norm, Purity, spectral weights and the criterion for a pure state and The extreme points of the state space are exactly the pure states are theorems and definitions of mathematics. They use only published or previously authored mathematical items, assume the stated choice hypotheses of those suppliers, and would be true of any complex Hilbert space regardless of any physical interpretation. In particular, the convexity of the state space, the criterion that purity is trace of the square, and the identification of the extreme points with the pure states carry no physical premise.

The adopted state postulate is homed on the framework page. The model assumption that a system is assigned a nonzero separable complex Hilbert space, that pure preparations are its unit rays and that general preparations are its density operators is clause-level content of State and composition postulate P1: Hilbert spaces, preparations and composite systems, authored on the framework page of this category. The present page does not restate it as a second postulate; The state space of the framework: preparations, pure rays and density operators only fixes the readings of its clauses in the geometry vocabulary, and the two physical theorems Pure preparations are well defined by their rays, and phase factors do not change their states and Statistical mixtures, spectral decompositions and the extremal role of pure preparations are conditional consequences of it together with the mathematical layer. No new physical assumption is adopted anywhere on this page.

No observable consequence of phase is claimed here. The two physical theorems use the redundancy of the unit representative and of phase factors only at the level of the assigned density operator. That a global phase has no consequence for any outcome probability is a statement about the framework's probability law and is not asserted on this page; the probability law, the moment conditions and the role of the phase in interferences belong to the observable and Born-probability items of this category.

Ensembles are not determined by states. The state postulate states, and the companion page exemplifies, that distinct weighted ensembles of pure preparations can determine the same density operator. This is a feature of the model's state assignment and not a claim that the model is incomplete, that nature is classical, or that some ensemble is the physically realized one. The mixture weights are classical preparation weights; they are not amplitudes, and no coherence between the alternatives of a mixture is asserted.

Deliberate deferrals and exclusions. The page adopts no superselection restriction, no identical-particle symmetry sector, no external field or radiation content and no measurement model; those are separate assumptions of the framework adopted on the pages that name them. The two-level vector space of the fixed-mode examples is treated purely as a two-dimensional Hilbert space; the identification of its vectors with photon polarization states, and any statement about photon number, energy or momentum, is deferred to the radiation assumptions of the category and is not made here.

Empirical status. No item of this page reports, predicts or compares a measured outcome. The worked checks of the companion page are finite computations verifying the definitions and the mathematical lemmas; they are not experiments, they use no data, and agreement of the model with any dataset is not established by them.

5 · Examples, experiments and counterexamples

None yet.

Qm State Geometry

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

State vectors, unit vectors, rays, normalization and phases of a Hilbert space

Definition

Normalization, ray equivalence and the phase orbits of the unit sphere

Statement

Facts & Assumptions

Proof

Rays, unit vectors and rank-one Hilbert projections

Statement

Facts & Assumptions

Proof

The state space: density operators, pure and mixed states, and purity

Definition

The state space is convex, closed under countable mixtures, and complete in the trace norm

Statement

Facts & Assumptions

Proof

Purity, spectral weights and the criterion for a pure state

Statement

Facts & Assumptions

Proof

The extreme points of the state space are exactly the pure states

Statement

Facts & Assumptions

Proof

The state space of the framework: preparations, pure rays and density operators

Definition

Pure preparations are well defined by their rays, and phase factors do not change their states

Statement

Facts & Assumptions

Proof

Statistical mixtures, spectral decompositions and the extremal role of pure preparations

Statement

Facts & Assumptions

Proof

Scope audit of the state-geometry page: what is mathematics, what is adopted, and what is deferred

Scope of the state-geometry page

5 · Examples, experiments and counterexamples

Sources