Choi's theorem on completely positive maps

Choi's theorem on completely positive maps In mathematics, Choi's theorem on completely positive maps is a result that classifies completely positive maps between finite-dimensional (matrix) C*-algebras. An infinite-dimensional algebraic generalization of Choi's theorem is known as Belavkin's "Radon–Nikodym" theorem for completely positive maps.

Contents 1 Statement 2 Proof 2.1 (i) implies (ii) 2.2 (iii) implies (i) 2.3 (ii) implies (iii) 3 Consequences 3.1 Kraus operators 3.2 Completely copositive maps 3.3 Hermitian-preserving maps 4 See also 5 References Statement Choi's theorem. Let {displaystyle Phi :mathbb {C} ^{ntimes n}to mathbb {C} ^{mtimes m}} be a linear map. The following are equivalent: (i) Φ is n-positive (i.e. {displaystyle Phi (A)in mathbb {C} ^{mtimes m}} is positive whenever {displaystyle Ain mathbb {C} ^{ntimes n}} is positive). (ii) The matrix with operator entries {displaystyle C_{Phi }=left(operatorname {id} _{n}otimes Phi right)left(sum _{ij}E_{ij}otimes E_{ij}right)=sum _{ij}E_{ij}otimes Phi (E_{ij})in mathbb {C} ^{nmtimes nm}} is positive, where {displaystyle E_{ij}in mathbb {C} ^{ntimes n}} is the matrix with 1 in the ij-th entry and 0s elsewhere. (The matrix CΦ is sometimes called the Choi matrix of Φ.) (iii) Φ is completely positive. Proof (i) implies (ii) We observe that if {displaystyle E=sum _{ij}E_{ij}otimes E_{ij},} then E=E* and E2=nE, so E=n−1EE* which is positive. Therefore CΦ =(In ⊗ Φ)(E) is positive by the n-positivity of Φ.

(iii) implies (i) This holds trivially.

(ii) implies (iii) This mainly involves chasing the different ways of looking at Cnm×nm: {displaystyle mathbb {C} ^{nmtimes nm}cong mathbb {C} ^{nm}otimes (mathbb {C} ^{nm})^{*}cong mathbb {C} ^{n}otimes mathbb {C} ^{m}otimes (mathbb {C} ^{n}otimes mathbb {C} ^{m})^{*}cong mathbb {C} ^{n}otimes (mathbb {C} ^{n})^{*}otimes mathbb {C} ^{m}otimes (mathbb {C} ^{m})^{*}cong mathbb {C} ^{ntimes n}otimes mathbb {C} ^{mtimes m}.} Let the eigenvector decomposition of CΦ be {displaystyle C_{Phi }=sum _{i=1}^{nm}lambda _{i}v_{i}v_{i}^{*},} where the vectors {displaystyle v_{i}} lie in Cnm . By assumption, each eigenvalue {displaystyle lambda _{i}} is non-negative so we can absorb the eigenvalues in the eigenvectors and redefine {displaystyle v_{i}} so that {displaystyle ;C_{Phi }=sum _{i=1}^{nm}v_{i}v_{i}^{*}.} The vector space Cnm can be viewed as the direct sum {displaystyle textstyle oplus _{i=1}^{n}mathbb {C} ^{m}} compatibly with the above identification {displaystyle textstyle mathbb {C} ^{nm}cong mathbb {C} ^{n}otimes mathbb {C} ^{m}} and the standard basis of Cn.

If Pk ∈ Cm × nm is projection onto the k-th copy of Cm, then Pk* ∈ Cnm×m is the inclusion of Cm as the k-th summand of the direct sum and {displaystyle ;Phi (E_{kl})=P_{k}cdot C_{Phi }cdot P_{l}^{*}=sum _{i=1}^{nm}P_{k}v_{i}(P_{l}v_{i})^{*}.} Now if the operators Vi ∈ Cm×n are defined on the k-th standard basis vector ek of Cn by {displaystyle ;V_{i}e_{k}=P_{k}v_{i},} then {displaystyle Phi (E_{kl})=sum _{i=1}^{nm}P_{k}v_{i}(P_{l}v_{i})^{*}=sum _{i=1}^{nm}V_{i}e_{k}e_{l}^{*}V_{i}^{*}=sum _{i=1}^{nm}V_{i}E_{kl}V_{i}^{*}.} Extending by linearity gives us {displaystyle Phi (A)=sum _{i=1}^{nm}V_{i}AV_{i}^{*}} for any A ∈ Cn×n. Any map of this form is manifestly completely positive: the map {displaystyle Ato V_{i}AV_{i}^{*}} is completely positive, and the sum (across {displaystyle i} ) of completely positive operators is again completely positive. Thus {displaystyle Phi } is completely positive, the desired result.

The above is essentially Choi's original proof. Alternative proofs have also been known.

Consequences Kraus operators In the context of quantum information theory, the operators {Vi} are called the Kraus operators (after Karl Kraus) of Φ. Notice, given a completely positive Φ, its Kraus operators need not be unique. For example, any "square root" factorization of the Choi matrix CΦ = B∗B gives a set of Kraus operators.

Let {displaystyle B^{*}=[b_{1},ldots ,b_{nm}],} where bi*'s are the row vectors of B, then {displaystyle C_{Phi }=sum _{i=1}^{nm}b_{i}b_{i}^{*}.} The corresponding Kraus operators can be obtained by exactly the same argument from the proof.

When the Kraus operators are obtained from the eigenvector decomposition of the Choi matrix, because the eigenvectors form an orthogonal set, the corresponding Kraus operators are also orthogonal in the Hilbert–Schmidt inner product. This is not true in general for Kraus operators obtained from square root factorizations. (Positive semidefinite matrices do not generally have a unique square-root factorizations.) If two sets of Kraus operators {Ai}1nm and {Bi}1nm represent the same completely positive map Φ, then there exists a unitary operator matrix {displaystyle {U_{ij}}_{ij}in mathbb {C} ^{nm^{2}times nm^{2}}quad {text{such that}}quad A_{i}=sum _{j=1}U_{ij}B_{j}.} This can be viewed as a special case of the result relating two minimal Stinespring representations.

Alternatively, there is an isometry scalar matrix {uij}ij ∈ Cnm × nm such that {displaystyle A_{i}=sum _{j=1}u_{ij}B_{j}.} This follows from the fact that for two square matrices M and N, M M* = N N* if and only if M = N U for some unitary U.

Completely copositive maps It follows immediately from Choi's theorem that Φ is completely copositive if and only if it is of the form {displaystyle Phi (A)=sum _{i}V_{i}A^{T}V_{i}^{*}.} Hermitian-preserving maps Choi's technique can be used to obtain a similar result for a more general class of maps. Φ is said to be Hermitian-preserving if A is Hermitian implies Φ(A) is also Hermitian. One can show Φ is Hermitian-preserving if and only if it is of the form {displaystyle Phi (A)=sum _{i=1}^{nm}lambda _{i}V_{i}AV_{i}^{*}} where λi are real numbers, the eigenvalues of CΦ, and each Vi corresponds to an eigenvector of CΦ. Unlike the completely positive case, CΦ may fail to be positive. Since Hermitian matrices do not admit factorizations of the form B*B in general, the Kraus representation is no longer possible for a given Φ.

See also Stinespring factorization theorem Quantum operation Holevo's theorem References M.-D. Choi, Completely Positive Linear Maps on Complex Matrices, Linear Algebra and its Applications, 10, 285–290 (1975). V. P. Belavkin, P. Staszewski, Radon-Nikodym Theorem for Completely Positive Maps, Reports on Mathematical Physics, v.24, No 1, 49–55 (1986). J. de Pillis, Linear Transformations Which Preserve Hermitian and Positive Semidefinite Operators, Pacific Journal of Mathematics, 23, 129–137 (1967). hide vte Functional analysis (topics – glossary) Spaces BanachBesovFréchetHilbertHölderNuclearOrliczSchwartzSobolevtopological vector Properties barrelledcompletedual (algebraic/topological)locally convexreflexiveseparable Theorems Hahn–BanachRiesz representationclosed graphuniform boundedness principleKakutani fixed-pointKrein–Milmanmin–maxGelfand–NaimarkBanach–Alaoglu Operators adjointboundedcompactHilbert–Schmidtnormalnucleartrace classtransposeunboundedunitary Algebras Banach algebraC*-algebraspectrum of a C*-algebraoperator algebragroup algebra of a locally compact groupvon Neumann algebra Open problems invariant subspace problemMahler's conjecture Applications Hardy spacespectral theory of ordinary differential equationsheat kernelindex theoremcalculus of variationsfunctional calculusintegral operatorJones polynomialtopological quantum field theorynoncommutative geometryRiemann hypothesisdistribution (or generalized functions) Advanced topics approximation propertybalanced setChoquet theoryweak topologyBanach–Mazur distanceTomita–Takesaki theory Categories: Linear algebraOperator theoryTheorems in functional analysis