保持凸组合映射及选择性量子测量的几何特征

Convex combination preserving maps and geometric features for selective quantum measurements

令H和K为复Hilbert空间,2≤dim H≤∞,S(H)为H上所有量子态(即迹为1的正迹类算子)构成的凸子集.本文讨论S(H)到S(K)∪{0}保持凸组合的映射的刻画问题以及与量子测量之间的联系,并证明映射Φ:S(H)→S(K)保持严格凸组合并将纯态映为纯态的充分必要条件是下列结论之一成立:(1)存在K上纯态σ_(0)使得Φ(ρ)=σ_(0)对于所有ρ∈S(H)都成立;(2)Φ是单射选择性量子测量映射或单射选择性量子测量映射与转置映射的复合,即存在单射有界线性算子M:H→K使得Φ(ρ)=(MρM^(*))/(Tr(MρM^(*)))对于所有ρ∈S(H)成立,或Φ(ρ)=(Mρ^(t)M^(*))/(Tr(Mρ^(t)M^(*)))对于所有ρ∈S(H)成立;(3)存在σ_(0),σ_(1)∈S(K),其中σ_(0)是纯态,使得Φ(Pur(H))={σ_(0)},Φ(S(H))■[σ_(0),σ_(1)),进而存在映射h:S(H)→[0,1)使得,对于任意ρ_(1),ρ_(2)∈S(H)及任意t∈(0,1),都有h((1-t)ρ_(1)+tρ_(2)))=(1-s)h(ρ_(1))+sh(ρ_(2))对某个s∈(0,1)成立,并且Φ(ρ)=(1-h(ρ))σ_(0)+hρ)σ_(1)对于所有ρ∈S(H)都成立.特别地,情形(3)在H为有限维时不出现.最后,对于单量子比特系统即dim H=2的情形,给出从S(H)到S(K)∪{0}的保持凸组合并将纯态映为纯态或0的一般映射的刻画.以上结果揭示了选择性量子测量的几何特征.

Let H and K be the complex Hilbert spaces and S(H)the convex set of all quantum states(i.e.,the positive operators with trace one)on H.In this paper,the convex combination preserving maps from S(H)into S(K)∪{0}and their connection to quantum measurements are discussed.It is shown that a mapΦ:S(H)→S(K)with 2≤dim H≤∞preserves pure states and strict convex combinations if and only if one of the following holds:(1)there exists a pure stateσ_(0)on K such thatΦ(ρ)=σ_(0)for allρ∈S(H);(2)Φis an injective selective quantum measurement map up to the transpose,i.e.,there exists an injective bounded linear operator M:H→K such thatΦ(ρ)=(MρM^(*))/(Tr(MρM^(*)))for allρ∈S(H)orΦ(ρ)=(Mρ^(t)M^(*))/(Tr(Mρ^(t)M^(*)))for allρ∈S(H);(3)there existσ_(0),σ_(1)∈S(K)withσ_(0)a pure state,such thatΦ(Pur(H))={σ_(0)},Φ(S(H))■[σ_(0),σ_(1)),and there exists a functional h:S(H)→[0,1)such that,for anyρ_(1),ρ_(2)∈S(H)and any t∈(0,1),h((1-t)ρ_(1)+tρ_(2))=(1-s)h(ρ_(1))+sh(ρ_(2))for some s∈(0,1),andΦ(ρ)=(1-h(ρ))σ_(0)+h(ρ)σ_(1)for allρ∈S(H).Particularly,Case(3)cannot occur in the finite-dimensional cases.In addition,for the qubit system,i.e.,dim H=2,the maps sending pure states to pure states or zero and preserving convex combinations are completely characterized.These results reveal geometric features for selective quantum measurements.