Open Systems

Abstract

Let H be a separable complex Hilbert space corresponding to a physical system and let T(H) he the Banach space of trace class operators on H. The positive cone, i.e., the set of all positive operators from T(H), will be denoted by V ⁺(H). The set of all states (density operators) can be embedded in T(H). More precisely,

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaiaacI % cacaWGibGaaiykaiabg2da9maacmaabaGaeqyWdiNaeyicI4SaamOv % amaaCaaaleqabaGaey4kaScaaOGaaiikaiaadIeacaGGPaGaaiOoai % aadshacaWGYbGaeqyWdiNaeyypa0JaaGymaaGaay5Eaiaaw2haaaaa % !49B6!></EquationSource><EquationSource Format="TEX"><![CDATA[$$S(H) = \left\{ {\rho \in {V^ + }(H):tr\rho = 1} \right\}$$

(7.1)

cf. [197], [198].