39.05.01 · operator-algebras / nuclearity-exactness

Completely Positive Maps and the Stinespring Dilation Theorem

shipped3 tiersLean: none

Anchor (Master): Brown-Ozawa *C*-Algebras and Finite-Dimensional Approximations* (AMS, 2008) Ch. 1; Paulsen *Completely Bounded Maps and Operator Algebras* (Cambridge UP, 2002) Ch. 3-6; Takesaki *Theory of Operator Algebras I* (Springer, 1979) Ch. IV

Intuition Beginner

A state is a single averaging rule that turns each operator into one number. A completely positive map is the next thing up: a rule that turns each operator into another operator, the way a noisy communication line turns one signal into a degraded but still meaningful signal. It should send sensible inputs to sensible outputs, which here means it never turns a square-of-a-star into something with a negative average.

The word "completely" is the surprise. It is not enough to demand that the map alone be sensible. You also have to be allowed to run the map on one system while a second, untouched system sits beside it. Wiring the two together and asking that the joint rule still be sensible is a strictly stronger demand. Some rules pass the single-system test and fail the bystander test; the famous one is "swap rows and columns of a matrix", which looks harmless until a bystander entangled with the system exposes it.

Once a rule passes the bystander test at every size, a clean picture emerges. Every such rule is secretly the same recipe: enlarge the world, let the operators act in the larger world in the honest structure-respecting way, and then squeeze the answer back down. Noise is what honest dynamics looks like after you forget part of the world.

Visual Beginner

A completely positive map factors through a larger space: embed, act honestly, project back.

The dictionary reads: the noisy rule on the left becomes, on the right, the three-step honest process embed-act-project. The isometry $V$ is the embedding, the representation $π$ is the honest dynamics in the bigger space, and $V^{*}$ is the projection that forgets the extra room. Forgetting part of the world is exactly where the noise comes from.

Worked example Beginner

Take the simplest noisy rule on two-by-two complex matrices: average the diagonal and throw away the rest. Concretely, send a matrix $a$ to the matrix that keeps its $(1, 1)$ and $(2, 2)$ entries and zeroes the off-diagonal corners. Call this rule $φ$ . Check it sends sensible to sensible: if $a = b^{*} b$ then the diagonal entries of $a$ are the squared lengths of the columns of $b$ , which are never negative, so $φ (a)$ has non-negative diagonal and is sensible.

Now build the honest picture. Use a larger space with four basis directions instead of two. Embed each input direction $e_{i}$ as the pair $(e_{i}, e_{i})$ spread across two copies, scaled by one over the square root of two. The honest action in the larger space is "apply $a$ to each copy separately". Pulling back, the off-diagonal corners cancel because the two copies are summed with no cross terms, and exactly the diagonal survives.

What this tells us: the rule "keep the diagonal" is not arbitrary damage. It is honest two-copy dynamics seen after forgetting which copy you were in. The lost off-diagonal entries measure exactly the information about "which copy" that the forgetting discarded.

Check your understanding Beginner

Exercise (easy, multiple choice).

What does the word "completely" add to "positive" for a map between algebras of operators?

A. The map must be invertible B. The map must stay sensible even when run alongside an untouched bystander system of any size C. The map must send the identity to the identity D. The map must preserve every eigenvalue

Hint

Plain positivity tests the map alone; the extra word tests the map together with a second system sitting beside it.

Answer

B. It must stay sensible when run alongside an untouched bystander of any size.

Feedback-correct: correct; complete positivity is positivity of the map amplified by an identity on an arbitrarily large bystander, which is strictly stronger than positivity alone. Feedback-wrong: invertibility, unitality, and eigenvalue preservation are separate properties; the word "completely" is specifically about the bystander test.

Formal definition Intermediate+

Let $A$ and $B$ be C*-algebras 39.01.01. For $n \geq 1$ write $M_{n} (A)$ for the C*-algebra of $n \times n$ matrices with entries in $A$ ; its positive cone is inherited from a faithful representation $A \subseteq B (H)$ , so $M_{n} (A) \subseteq M_{n} (B (H)) = B (H^{\oplus n})$ carries the operator order. A linear map $φ : A \to B$ induces the amplification $φ_{n} = φ \otimes id_{n} : M_{n} (A) \to M_{n} (B)$ acting entrywise, $φ_{n} ([a_{ij}]) = [φ (a_{ij})]$ .

The map $φ$ is positive if $φ (a^{*} a) \geq 0$ for all $a \in A$ , equivalently $φ (A_{+}) \subseteq B_{+}$ . It is $n$ -positive if $φ_{n}$ is positive on $M_{n} (A)$ , and completely positive (CP) if it is $n$ -positive for every $n \geq 1$ ^{[Paulsen Ch. 3]}. A CP map is automatically bounded, with $∥ φ ∥ = ∥ φ (1) ∥$ in the unital case; it is unital (UCP) if $φ (1) = 1$ . States are exactly the UCP maps into $B = C$ 39.01.03, and $*$ -homomorphisms are CP.

The two structural data of a CP map are recorded by the Choi matrix and the Kraus form. When $A = M_{n} (C)$ with matrix units $e_{ij}$ , the Choi matrix of $φ$ is $$ C_\varphi = [\varphi(e_{ij})]{i,j=1}^n = \varphi_n\big([e{ij}]\big) \in M_n(B), $$ the amplified image of the (positive, rank-one) matrix $[e_{ij}] = (\sum_{i} e_{i 1}) (\sum_{j} e_{1 j})^{*}$ . A Kraus representation is an expression $φ (a) = \sum_{k} V_{k}^{*} a V_{k}$ with $V_{k} : H \to K$ bounded; the sum is finite in finite dimensions and norm- or strongly-convergent in general.

Counterexamples to common slips

Positive does not imply completely positive. Transposition $t : M_{2} (C) \to M_{2} (C)$ , $t (a) = a^{T}$ , is positive (it preserves eigenvalues of self-adjoint matrices) but not $2$ -positive: applied to the positive rank-one matrix $[e_{ij}] \in M_{2} (M_{2} (C))$ it returns the swap operator, which has eigenvalue $- 1$ . Transposition is the canonical positive-but-not-CP map.
$n$ -positive does not imply $(n + 1)$ -positive. For each $n$ there are maps that are $n$ -positive and fail $(n + 1)$ -positivity; complete positivity is genuinely the full tower, not any single level. (On $M_{d}$ , however, $d$ -positivity already forces complete positivity — Choi.)
Unitality and positivity do not give the Schwarz inequality. The inequality $φ (a)^{*} φ (a) \leq φ (a^{*} a)$ can fail for merely positive unital maps; it needs $2$ -positivity (Kadison-Schwarz). Positivity alone controls only self-adjoint elements through $∣ φ (a) ∣ \leq φ (∣ a ∣)$ -type bounds that do not amplify.

Key theorem with proof Intermediate+

Theorem (Stinespring dilation). Let $A$ be a unital C-algebra and $φ : A \to B (H)$ a completely positive map. There exist a Hilbert space $K$ , a unital $*$ -representation $π : A \to B (K)$ , and a bounded operator $V : H \to K$ with $∥ V ∥^{2} = ∥ φ (1) ∥$ such that* $$ \varphi(a) = V^* \pi(a) V \qquad (a \in A). $$ The dilation can be taken minimal, meaning $\overline{π (A) V H} = K$ , and the minimal dilation is unique up to unitary equivalence. ^{[Stinespring; Paulsen Ch. 4]}

Proof. On the algebraic tensor product $A ⊙ H$ define a sesquilinear form determined on elementary tensors by $$ \langle a \otimes \xi, , b \otimes \eta \rangle = \langle \varphi(b^* a)\xi, \eta \rangle_H , $$ extended bilinearly. To see it is positive semidefinite, take a finite sum $u = \sum_{i} a_{i} \otimes ξ_{i}$ . Then $$ \langle u, u \rangle = \sum_{i,j} \langle \varphi(a_j^* a_i)\xi_i, \xi_j \rangle = \big\langle \varphi_n\big([a_j^* a_i]\big), \xi, , \xi \big\rangle, $$ where $ξ = (ξ_{1}, \dots, ξ_{n}) \in H^{\oplus n}$ and $[a_{j}^{*} a_{i}] = [a_{i}]^{*} [a_{i}] \geq 0$ in $M_{n} (A)$ (it is $b^{*} b$ for $b$ the row $[a_{1} \dots a_{n}]$ ). Complete positivity gives $φ_{n} ([a_{j}^{*} a_{i}]) \geq 0$ , so $⟨ u, u ⟩ \geq 0$ . This is the single place the full strength of complete positivity is used.

Let $N = {u : ⟨ u, u ⟩ = 0}$ , a subspace by Cauchy-Schwarz, and let $K$ be the Hilbert-space completion of $(A ⊙ H) / N$ , with $[a \otimes ξ]$ the image of $a \otimes ξ$ . Define $π (a)$ by left multiplication on the first leg, $π (a) [b \otimes ξ] = [ab \otimes ξ]$ . It is well-defined and bounded: for fixed $a$ , $$ \langle \pi(a)u, \pi(a)u\rangle = \big\langle \varphi_n\big([b_j^* a^* a, b_i]\big)\xi, \xi\big\rangle \le |a|^2 \big\langle \varphi_n([b_j^* b_i])\xi, \xi\big\rangle = |a|^2 \langle u, u\rangle, $$ using $a^{*} a \leq ∥ a ∥^{2} 1$ amplified to $M_{n} (A)$ and CP positivity of $φ_{n}$ . So $π (a)$ extends to $K$ with $∥ π (a) ∥ \leq ∥ a ∥$ ; the map $π$ is linear, multiplicative ( $π (a) π (a^{'}) [b \otimes ξ] = [a a^{'} b \otimes ξ]$ ), unital, and $*$ -preserving by the symmetry of the form, hence a representation.

Define $V : H \to K$ by $V ξ = [1 \otimes ξ]$ . It is bounded since $∥ V ξ ∥^{2} = ⟨ φ (1) ξ, ξ ⟩ \leq ∥ φ (1) ∥ ∥ ξ ∥^{2}$ , so $∥ V ∥^{2} \leq ∥ φ (1) ∥$ . Then $$ \langle V^\pi(a)V\xi, \eta\rangle = \langle \pi(a)[1\otimes \xi], [1\otimes\eta]\rangle = \langle [a \otimes \xi], [1\otimes\eta]\rangle = \langle \varphi(a)\xi, \eta\rangle, $$ so $\varphi(a) = V^\pi(a)V $; t ak in g$ a = 1 $g i v es$ \varphi(1) = V^*V $an d h e n ce$ |V|^2 = |\varphi(1)| $. B y co n s t r u c t i o n$ \pi(A)VH = {[a \otimes \xi]}$ spans a dense subspace, so this dilation is minimal.

Uniqueness: if $(π^{'}, K^{'}, V^{'})$ is another minimal dilation, set $U (π (a) V ξ) = π^{'} (a) V^{'} ξ$ . This is isometric because $$ \langle \pi'(a)V'\xi, \pi'(b)V'\eta\rangle = \langle V'^*\pi'(b^*a)V'\xi, \eta\rangle = \langle \varphi(b^*a)\xi, \eta\rangle = \langle \pi(a)V\xi, \pi(b)V\eta\rangle, $$ so $U$ is well-defined on the dense set $π (A) V H$ , extends to a unitary $K \to K^{'}$ , intertwines $U π (a) = π^{'} (a) U$ , and satisfies $U V = V^{'}$ . $□$

Bridge. Stinespring dilation builds toward the entire structure theory of operator systems and nuclearity, and it appears again in 39.05.02 where Arveson's extension theorem extends a CP map off an operator system to the ambient C*-algebra by dilating, extending the representation, and compressing back. The foundational reason it holds is exactly the GNS construction 39.01.03 amplified to matrices: a state is the scalar case $H = C$ , where the form $φ (b^{*} a)$ is the GNS form and $V$ is the cyclic vector, so this is exactly the GNS theorem with the target $C$ replaced by $B (H)$ and positivity replaced by complete positivity. The dilation generalises GNS in one direction and is dual to the Kraus picture in another — collapsing $π$ into a multiplicity space recovers $φ (a) = \sum_{k} V_{k}^{*} a V_{k}$ , the central insight that CP maps are precisely operator-sum channels. Putting these together, every completely positive map is an honest representation seen through a single compression, and the bridge is that this one compression is what separates reversible $*$ -homomorphisms from irreversible quantum channels.

Exercises Intermediate+

Exercise 6 (hard, short-answer).

Prove Choi's theorem: a linear map $φ : M_{n} (C) \to B$ is completely positive if and only if its Choi matrix $C_{φ} = [φ (e_{ij})] \in M_{n} (B)$ is positive.

Hint

One direction is amplification applied to the positive matrix $[e_{ij}]$ . For the converse, diagonalise $C_{φ} = \sum_{k} η_{k} η_{k}^{*}$ and read off Kraus operators $V_{k}$ from the blocks of $η_{k}$ .

Answer

( $\Rightarrow$ ) If $φ$ is CP then $φ_{n}$ is positive, and $[e_{ij}] = (\sum_{i} e_{i 1}) (\sum_{j} e_{1 j})^{*} \geq 0$ in $M_{n} (M_{n} (C))$ , so $C_{φ} = φ_{n} ([e_{ij})]) \geq 0$ .

( $\Leftarrow$ ) Suppose $B = B (H)$ and $C_{φ} \geq 0$ in $M_{n} (B (H)) = B (H^{\oplus n})$ . Write $C_{φ} = \sum_{k} η_{k} η_{k}^{*}$ with $η_{k} \in H^{\oplus n}$ , i.e. $η_{k} = (η_{k}^{(1)}, \dots, η_{k}^{(n)})$ , so that $φ (e_{ij}) = \sum_{k} η_{k}^{(i)} (η_{k}^{(j)})^{*}$ . Define $V_{k} : H \to H^{\oplus n}$ wait — instead define operators $V_{k} : C^{n} \to H$ by $V_{k} e_{j} = η_{k}^{(j)}$ . Then for $a = [a_{ij}] = \sum_{ij} a_{ij} e_{ij}$ , $$ \sum_k V_k^* a V_k \ \text{evaluated correctly gives} \ \varphi(a) = \sum_k V_k a V_k^, $$ with $V_{k}$ the $n$ -to- $H$ blocks: $\varphi(e_{ij}) = V_k e_i (V_k e_j)^ $s u mm e d o v er$ k $ma t c h es t h e d ia g o na l i s a t i o n . E a c h s u mman d$ a \mapsto V_k a V_k^* $i s C P (i t i s$ \pi(a) = a $co m p r esse d), an d a f ini t es u m o f C P ma p s i s C P . H e n ce$ \varphi $i s C P, w i t h K r a u so p er a t or sr e a d o f f t h ee i g e n v ec t or so f$ C_\varphi$.

Rubric: full credit for the amplification direction and the diagonalisation-to-Kraus converse; minor index/adjoint bookkeeping not penalised.

Exercise 7 (hard, short-answer).

Show that for a unital completely positive map $φ : A \to B$ the multiplicative domain $A_{φ} = {a \in A : φ (a^{*} a) = φ (a)^{*} φ (a) and φ (a a^{*}) = φ (a) φ (a)^{*}}$ is a C*-subalgebra on which $φ$ is a $*$ -homomorphism, and that $φ (ab) = φ (a) φ (b)$ whenever $a \in A_{φ}$ .

Hint

Use the Stinespring dilation $φ = V^{*} π (\cdot) V$ with $V$ an isometry. Equality in Kadison-Schwarz $φ (a^{*} a) = φ (a)^{*} φ (a)$ means $∥ π (a) V ξ - V φ (a) ξ ∥ = 0$ , i.e. $π (a) V = V φ (a)$ .

Answer

By Stinespring, $φ (a) = V^{*} π (a) V$ with $V^{*} V = 1$ (unital case, $V$ isometry). Compute $$ \varphi(a^a) - \varphi(a)^\varphi(a) = V^\pi(a)^\pi(a)V - V^\pi(a)^VV^\pi(a)V = V^\pi(a)^(1 - VV^)\pi(a)V \ge 0, $$ since $1 - V V^{*}$ is the projection onto $(V H)^{⊥}$ . Thus equality holds iff $(1 - V V^{*}) π (a) V = 0$ , i.e. $π (a) V = V V^{*} π (a) V = V φ (a)$ . So $a \in A_{φ}$ (left condition) iff $π (a) V = V φ (a)$ ; the right condition gives $π (a^{*}) V = V φ (a^{*})$ . For $a \in A_{φ}$ and any $b$ , $$ \varphi(ab) = V^\pi(a)\pi(b)V = V^V\varphi(a)V^... $$ more directly $\varphi(ab) = V^\pi(ab)V = V^\pi(a)\pi(b)V = \varphi(a)V^\pi(b)V = \varphi(a)\varphi(b) $, u s in g$ V^\pi(a) = \varphi(a)V^ $(a d j o in t o f$ \pi(a^)V = V\varphi(a^) $) . S y mm e t r i c a l l y$ \varphi(ba) = \varphi(b)\varphi(a) $f or$ a \in A_\varphi $. C l os u r eo f$ A_\varphi $u n d er t h eo p er a t i o n s f o l l o w s b ec a u se$ a \mapsto \pi(a)V - V\varphi(a) $i s l in e a r an d t h eco n d i t i o n s a r e t h e v ani s hin g o f a co n t in u o u se x p r ess i o n; t h e m u l t i pl i c a t i v i t y j u s t s h o w nmak es$ A_\varphi $c l ose d u n d er p r o d u c t s an d a d j o in t s, h e n ce a C * - s u ba l g e b r a, w i t h$ \varphi|{A\varphi} $a$ *$-homomorphism.

Rubric: full credit for the $(1 - V V^{*})$ identity, the intertwining $π (a) V = V φ (a)$ , and the bimodule multiplicativity $φ (ab) = φ (a) φ (b)$ .

Lean formalization Intermediate+

lean_status: none — Mathlib carries the C*-algebra layer (CStarAlgebra, positivity through cfc, matrices over a C*-algebra) but neither the matrix amplifications $φ \otimes id_{n}$ as a named construction, the order on $M_{n} (A)$ defining $n$ -positivity, the GNS construction (see 39.01.03), nor the Stinespring dilation, Choi criterion, Kraus form, Kadison-Schwarz inequality, and multiplicative domain. The intended statement reads schematically:

import Mathlib.Analysis.CStarAlgebra.Basic
import Mathlib.Analysis.CStarAlgebra.Matrix

variable {A : Type*} [CStarAlgebra A] {H : Type*}
  [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]

/-- Stinespring: a unital completely positive map dilates to a
*-representation compressed by an isometry. -/
theorem stinespring_dilation
    (φ : A →ₗ[ℂ] (H →L[ℂ] H))
    (hcp : ∀ n (x : Matrix (Fin n) (Fin n) A), 0 ≤ x →
            0 ≤ (x.map φ))            -- complete positivity
    (hunital : φ 1 = 1) :
    ∃ (K : Type*) (_ : NormedAddCommGroup K) (_ : InnerProductSpace ℂ K)
      (π : A →⋆ₐ[ℂ] (K →L[ℂ] K)) (V : H →L[ℂ] K),
      (∀ a, φ a = V.adjoint ∘L (π a) ∘L V) ∧ Dense (⋃ a, Set.range (π a ∘L V)) :=
  sorry  -- GNS-style construction on A ⊗ H, quotient by the null space, compress

Advanced results Master

The dilation theorem organises a cluster of equivalent and derived structures.

Choi's theorem and the operator-sum form. For $φ : M_{n} (C) \to M_{m} (C)$ the following are equivalent: $φ$ is completely positive; $φ$ is $n$ -positive; the Choi matrix $C_{φ} = [φ (e_{ij})] \in M_{n} (M_{m} (C))$ is positive; $φ$ admits a Kraus representation $φ (a) = \sum_{k = 1}^{nm} V_{k}^{*} a V_{k}$ with at most $nm$ terms. The equivalence is the matricial shadow of Stinespring: a finite-dimensional minimal dilation has $dim K \leq nm$ , the representation $π$ on $K$ decomposes as $a \mapsto a \otimes 1$ on a multiplicity space, and compressing by $V$ produces the Kraus operators as the columns of $V$ . The number of Kraus operators in a minimal representation equals the rank of the Choi matrix, the Choi rank, an invariant of the channel.

Kadison-Schwarz, Choi inequalities, and the Russo-Dye theorem. A unital $2$ -positive map satisfies $φ (a)^{*} φ (a) \leq φ (a^{*} a)$ , and a unital CP map satisfies the stronger Choi inequality $φ (a)^{*} φ (a) \leq φ (a^{*} a)$ together with $φ (a^{*} a) φ (a)^{*} φ (a) \leq ∥ φ ∥ φ (a^{*} a a^{*} a)$ -type refinements; for normal $a$ in the multiplicative domain both become equalities. The Russo-Dye theorem identifies the norm of a unital positive map as $∥ φ ∥ = ∥ φ (1) ∥ = 1$ , so UCP maps are unital contractions, placing the state space ( $m = 1$ ) as the base case of a tower of matrix-state spaces $S_{n} (A) = {UCP A \to M_{n}}$ whose union is the matrix state space, the complete invariant of the operator-system structure.

The multiplicative domain and rigidity. For UCP $φ$ , the multiplicative domain $A_{φ}$ is the largest C*-subalgebra on which $φ$ restricts to a $*$ -homomorphism; via Stinespring it is ${a : π (a) V = V φ (a)}$ , the elements whose dilation commutes with the compression. When $φ$ is a conditional expectation onto a subalgebra $B \subseteq A$ , the multiplicative domain contains $B$ and the bimodule property $φ (ba b^{'}) = b φ (a) b^{'}$ for $b, b^{'} \in B$ holds — the defining feature of expectations, recovered here as the multiplicative-domain bimodule identity.

Arveson's extension theorem. A CP map $φ : S \to B (H)$ defined on an operator system $S \subseteq A$ (a self-adjoint unital subspace) extends to a CP map $\tilde{φ} : A \to B (H)$ on the whole algebra. The proof dilates $φ$ on $S$ , extends the resulting positive map by a Hahn-Banach argument for ordered operator spaces, and recompresses; it is the CP analogue of the Hahn-Banach extension of states and the engine behind injectivity of $B (H)$ in the category of operator systems. This previews 39.05.02 and underlies the definition of nuclear and exact C*-algebras through CP approximation.

Synthesis. The Stinespring dilation is the foundational reason the matricial criteria all agree: Choi's positivity of $[φ (e_{ij})]$ , the Kraus operator-sum, and the dilation $V^{*} π (\cdot) V$ are three faces of one construction, and this is exactly the GNS theorem 39.01.03 with scalar target promoted to $B (H)$ and positivity promoted to complete positivity. The Kadison-Schwarz inequality is dual to the multiplicative-domain rigidity — the defect $V^{*} π (a)^{*} (1 - V V^{*}) π (a) V$ measures exactly how far $a$ is from the subalgebra where $φ$ is multiplicative, so equality detects the multiplicative domain and the central insight is that loss of multiplicativity is loss of the dilation's commutation with the compression. Putting these together, Arveson's extension theorem generalises Hahn-Banach from states to CP maps, and the bridge to nuclearity is that a C*-algebra is nuclear precisely when the identity factors approximately through matrix algebras by UCP maps — so the entire chapter 39.05.02 is the study of which algebras the Stinespring/CP machinery can approximate by finite-dimensional pieces.

Full proof set Master

Proposition (complete positivity of compressions and $*$ -homomorphisms). For any bounded $V : H \to K$ and any $*$ -representation $π : A \to B (K)$ , the map $φ (a) = V^{*} π (a) V$ is completely positive. Indeed $φ_{n} ([a_{ij}]) = (1_{n} \otimes V)^{*} π_{n} ([a_{ij}]) (1_{n} \otimes V)$ where $π_{n} = π \otimes id_{n}$ is a $*$ -representation of $M_{n} (A)$ , so it carries positive elements to positive elements, and conjugation by $1_{n} \otimes V$ preserves positivity. In particular every $*$ -homomorphism ( $V$ an isometry, no compression) is CP, and Stinespring asserts the converse: these are all the CP maps.

Proposition (Stinespring existence, restated with the key positivity step). The form $⟨ a \otimes ξ, b \otimes η ⟩ = ⟨ φ (b^{*} a) ξ, η ⟩$ on $A ⊙ H$ is positive semidefinite iff $φ$ is completely positive. For $u = \sum_{i} a_{i} \otimes ξ_{i}$ , $⟨ u, u ⟩ = ⟨ φ_{n} ([a_{j}^{*} a_{i}]) ξ, ξ ⟩$ with $[a_{j}^{*} a_{i}] = [a_{i}]^{*} [a_{i}] \geq 0$ ; positivity of all such expressions for every $n$ is precisely $n$ -positivity of $φ$ for every $n$ , i.e. complete positivity. The completion, the representation by left multiplication, and the operator $V ξ = [1 \otimes ξ]$ then satisfy $φ = V^{*} π (\cdot) V$ as computed in the Key theorem.

Proposition (minimal dilation and Choi rank). Among Stinespring dilations of $φ$ , the minimal one (with $\overline{π (A) V H} = K$ ) is unique up to a unitary intertwining $π, V$ , and any dilation contains the minimal one as the restriction to the reducing subspace $\overline{π (A) V H}$ . In finite dimensions, decomposing the minimal $π$ into irreducibles $a \mapsto a \otimes 1$ on a multiplicity space identifies $dim K$ with the rank of the Choi matrix; the columns of $V$ relative to a basis of the multiplicity space are the Kraus operators, and their number equals the Choi rank, the minimal number of Kraus terms.

Proposition (Kadison-Schwarz via the Stinespring defect). For unital CP $φ = V^{*} π (\cdot) V$ with $V$ an isometry, $$ \varphi(a^a) - \varphi(a)^\varphi(a) = V^\pi(a)^(1 - VV^)\pi(a)V \ge 0, $$ because $1 - VV^ $i s a p r o j ec t i o n . T h u s$ \varphi(a)^\varphi(a) \le \varphi(a^a) $, an d e q u a l i t y h o l d s i f f$ (1 - VV^)\pi(a)V = 0 $, i . e .$ \pi(a)V = V\varphi(a) $. T h ese t o f s u c h$ a $, in t er sec t e d w i t hi t s a d j o in t co u n t er p a r t, i s t h e m u l t i pl i c a t i v e d o main; o ni t$ \varphi(ab) = V^\pi(a)\pi(b)V = \varphi(a)V^\pi(b)V = \varphi(a)\varphi(b) $, so$ \varphi $i s a$ $- h o m o m or p hi s m t h er e . H e n ceK a d i so n - S c h w a r z n ee d so n l y$ 2 $- p os i t i v i t y (t h e p r oo f o f t h e in e q u a l i t y u se d o n l y$ \varphi_2 $o n t h e ma t r i x$ \begin{pmatrix} a^a & a^\ a & 1\end{pmatrix}$), while the full equality analysis uses the dilation.

Proposition (Choi's criterion, $M_{n} \to M_{m}$ ). A linear $φ : M_{n} (C) \to M_{m} (C)$ is CP iff $C_{φ} = [φ (e_{ij})] \geq 0$ in $M_{nm} (C)$ . Necessity is amplification on $[e_{ij}] \geq 0$ . Sufficiency: write $C_{φ} = \sum_{k} ∣ η_{k} ⟩ ⟨ η_{k} ∣$ with $η_{k} \in C^{n} \otimes C^{m}$ , define $V_{k} \in M_{m \times n} (C)$ by the partial-isometry reshaping $η_{k} \leftrightarrow V_{k}$ (the Choi-Jamiołkowski correspondence), and verify $φ (a) = \sum_{k} V_{k} a V_{k}^{*}$ ; each term is CP and the finite sum is CP. The Kraus operators $V_{k}$ are determined up to a unitary mixing, and the minimal count is the rank of $C_{φ}$ .

Proposition (Arveson extension). Let $S \subseteq A$ be an operator system and $φ : S \to B (H)$ completely positive. Then $φ$ extends to a CP $\tilde{φ} : A \to B (H)$ . Sketch: reduce to $H$ finite-dimensional by a weak-* limit / pointwise-bounded argument; identify CP maps $S \to M_{m}$ with positive functionals on $M_{m} (S)$ via the Choi-type duality; extend the positive functional from $M_{m} (S)$ to $M_{m} (A)$ by the order-theoretic Hahn-Banach theorem (a positive functional on a subsystem containing an order unit extends preserving positivity); translate back to a CP extension. The injectivity of $B (H)$ as an operator system is the categorical content, and the theorem is the CP lift of the Hahn-Banach extension of states used in Gelfand-Naimark 39.01.03.

Connections Master

States, the GNS construction, and Gelfand-Naimark 39.01.03 — Stinespring is the GNS theorem with the scalar target $C$ replaced by $B (H)$ : a state is the special case $H = C$ , the form $φ (b^{*} a)$ is the GNS form, the cyclic vector becomes the operator $V$ , and minimality / uniqueness specialise to the GNS minimality. This unit is the matrix-valued completion of that scalar construction.
C-algebras: axioms, spectrum, and the continuous functional calculus 39.01.01* — the positivity of $M_{n} (A)$ that defines $n$ -positivity is inherited from the order on $A$ given by the functional calculus, and the amplified inequality $a^{*} a \leq ∥ a ∥^{2} 1$ in $M_{n} (A)$ is what bounds the dilating representation $π$ , so the whole construction runs on the C*-order developed there.
Arveson extension and operator systems 39.05.02 — the CP extension theorem, the injectivity of $B (H)$ , and the operator-system viewpoint are the immediate sequel; Stinespring supplies the dilation that the extension argument extends and recompresses, making this unit the structural prerequisite for the operator-system theory.
Density matrix, pure and mixed states 12.17.01 — the finite-dimensional quantum-channel theory there (Kraus operators, Stinespring as system-plus-environment dilation, Choi-Jamiołkowski isomorphism) is exactly the $M_{n} \to M_{m}$ specialisation of this C*-structure theory, with the partial trace as the canonical compression $V^{*} (\cdot) V$ .
Nuclear and exact C-algebras 39.05.03* — nuclearity is defined by approximating the identity map by UCP maps factoring through matrix algebras $A \to M_{n} \to A$ , so the CP machinery of this unit is the language in which the whole nuclearity-exactness chapter is phrased.

Historical & philosophical context Master

The dilation theorem is due to W. Forrest Stinespring, who in 1955 proved that every completely positive map from a C*-algebra into the bounded operators on a Hilbert space has the form $V^{*} π (\cdot) V$ for a $*$ -representation $π$ and a bounded $V$ ^{[Stinespring 1955]}. The result generalised the GNS construction (Gelfand-Naimark 1943, Segal 1947), recovering states as the scalar case, and isolated complete positivity — positivity stable under tensoring with matrix algebras of every size — as the correct morphism class for operator algebras. Man-Duen Choi sharpened the finite-dimensional picture in 1975, proving that complete positivity of a map on $M_{n}$ is equivalent to positivity of the single matrix $[φ (e_{ij})]$ and giving the operator-sum representation ^{[Choi 1975]}; the same operator-sum form had appeared in Karl Kraus's 1971 axiomatisation of quantum operations. William Arveson, in his 1969 study of subalgebras of C*-algebras, established the extension theorem for CP maps on operator systems and launched the noncommutative theory of operator systems and completely bounded maps ^{[Arveson 1969]}.

The physical content is the theory of open quantum systems: a completely positive trace-preserving map is the most general physically allowed evolution of a quantum state, and the Stinespring dilation is the statement that any such evolution arises from unitary dynamics on a larger system-plus-environment followed by forgetting the environment. Complete positivity rather than mere positivity is forced precisely because a quantum system can be entangled with an untouched ancilla; the transpose map, positive but not completely positive, is the mathematical core of the Peres-Horodecki entanglement criterion. Vern Paulsen's monograph organised the subject around completely bounded maps, and the CP-approximation viewpoint became the foundation of the Brown-Ozawa development of nuclearity and exactness through finite-dimensional approximation.

Bibliography Master

Stinespring, W. F., "Positive functions on C*-algebras", Proceedings of the American Mathematical Society 6 (1955), 211-216.
Choi, M.-D., "Completely positive linear maps on complex matrices", Linear Algebra and its Applications 10 (1975), 285-290.
Kraus, K., "General state changes in quantum theory", Annals of Physics 64 (1971), 311-335.
Arveson, W. B., "Subalgebras of C*-algebras", Acta Mathematica 123 (1969), 141-224.
Paulsen, V., Completely Bounded Maps and Operator Algebras, Cambridge Studies in Advanced Mathematics 78, Cambridge University Press, 2002.
Brown, N. P. and Ozawa, N., C-Algebras and Finite-Dimensional Approximations*, Graduate Studies in Mathematics 88, American Mathematical Society, 2008. Ch. 1.
Takesaki, M., Theory of Operator Algebras I, Springer, 1979. Ch. IV.

Operator-algebras spine, opening unit of the nuclearity-exactness chapter. Produced as the completely-positive-maps / Stinespring anchor: positivity versus complete positivity and the transpose counterexample, the dilation theorem with its GNS-style proof and minimal-dilation uniqueness, the Choi matrix criterion and Kraus operator-sum form, the Kadison-Schwarz inequality and the multiplicative domain, and Arveson's extension theorem previewing the operator-system theory of 39.05.02.

Prerequisites

39.01.03
39.01.01

Tier anchors

beginner: Strogatz-style 'a quantum process is an averaging rule that stays sensible when you watch it alongside an untouched bystander'; the noise-channel picture from quantum information
intermediate: Paulsen *Completely Bounded Maps and Operator Algebras* (Cambridge UP, 2002) Ch. 3-4; Brown-Ozawa *C*-Algebras and Finite-Dimensional Approximations* (AMS, 2008) Ch. 1
master: Brown-Ozawa *C*-Algebras and Finite-Dimensional Approximations* (AMS, 2008) Ch. 1; Paulsen *Completely Bounded Maps and Operator Algebras* (Cambridge UP, 2002) Ch. 3-6; Takesaki *Theory of Operator Algebras I* (Springer, 1979) Ch. IV

References

Stinespring, W. F. — Positive functions on C*-algebras · Proc. Amer. Math. Soc. 6 (1955), 211-216 — the dilation theorem
Paulsen, V. — Completely Bounded Maps and Operator Algebras · Cambridge Studies in Advanced Mathematics 78 (2002), Ch. 3-4 completely positive maps, Stinespring, Arveson extension
Brown, N. P. and Ozawa, N. — C*-Algebras and Finite-Dimensional Approximations · Graduate Studies in Mathematics 88, AMS (2008), Ch. 1 completely positive maps, Stinespring, Arveson, multiplicative domains
Choi, M.-D. — Completely positive linear maps on complex matrices · Linear Algebra Appl. 10 (1975), 285-290 — the Choi matrix criterion and Kraus form
Arveson, W. B. — Subalgebras of C*-algebras · Acta Math. 123 (1969), 141-224 — the extension theorem for operator systems

Estimated time

beginner: 18m
intermediate: 52m
master: 92m