The Set S of States and the Set P of Pure States
LetHbe the Hilbert space of the quantum system, with inner productã,ã, linear in the second argument LetBdenote the set of bounded operators on
Hand letB 1 be its subset of the trace class operators We denote by tr
T the trace of an elementT ∈ B 1 IfA,Bare inB, we writeA≤B, orB≥A, ifB−Ais a positive operator.
A stateT of the system is an element ofB 1 such that T is positive and of trace one We letSbe the set of all states, that is,
The set S is a convex subset of B1, meaning that for any two elements T1 and T2 in S, and any weight w between 0 and 1, the combination wT1 + (1−w)T2 also belongs to S Furthermore, S is σ-convex, which indicates that for any sequence of states (Ti) and a corresponding sequence of weights (wi) where each wi is between 0 and 1, the weighted sum of these states remains within the set S.
If the series ∑∞ i=1 w_i equals 1, then the series ∑∞ i=1 w_i T_i converges in the trace norm to an operator in S, which we denote as w_i T_i The convex structure of S illustrates the physical feasibility of creating new states by mixing existing states with specified weights Specifically, if T is expressed as wT_1 + (1 - w)T_2, it is referred to as a mixture of the states T_1 and T_2 with the weight w.
The convex structure ofSallows one to identify its extreme elements, that is, the elements T ∈ Sfor which the condition T =wT 1+ (1−w)T 2, with
In the context of quantum mechanics, extreme states, denoted as ex(S), are defined as states that cannot be represented as mixtures of other states, occurring only when T equals T1 and T2 within the range of 0 < w < 1 These extreme states are often referred to as pure states, although this concept necessitates further clarification due to the influence of superselection rules.
For anyϕ ∈ H, ϕ = 0, we let P[ϕ] denote the projection on the one- dimensional subspace [ϕ] :={cϕ|c∈C} generated byϕ, that is,
The Set E of Effects and the Set D of Projections
In the context of Hilbert space H, let P represent the set of one-dimensional projections For any projection P in P, it can be expressed as P[ϕ] for some nonzero vector ϕ in P(H), which defines the range of P This set P is crucial within the broader set S Specifically, for any compact self-adjoint operator T in S, it can be decomposed as T = ∑ (from i=0 to ∞) w_i P_i, where (P_i) is a mutually orthogonal sequence in P and w_i are coefficients in the interval [0,1], with the series converging in both the operator norm of B and the trace norm of B_1, since T is trace class The nonzero eigenvalues w_i correspond to the dimensions of their respective eigenspaces Consequently, it follows that the set of extreme states is equivalent to the set of one-dimensional projections, expressed as ex(S) = P.
For this reason we also call the extreme states the vector states The above result also shows that theσ-convex hull ofPis the whole set of states, σ−co (P) =S (1.3)
In other words, vector states exhaust all states in the sense that any state can be expressed as a mixture of at most countably many vector states.
In quantum mechanics, any two or more vector states, denoted as P1 and P2, can be combined through a process known as superposition This concept can be represented by P1 ∨ P2, which signifies the least upper bound of the states P1 and P2 Consequently, any vector state P that exists within this framework is encompassed by this superposition principle.
The expression P ≤ P 1 ∨ P 2 signifies that P is a superposition of vector states P 1 and P 2 Additionally, any vector state P can be represented as a superposition of P 1 and P 2 if and only if P 1 is not orthogonal to P 2.
P, P 1 ≤P ⊥ , that is, if and only if tr
= 0 (we are excluding here the trivial caseP 2=P).
The concept of superposition in vector states is fundamentally illustrated through the linear structure of Hilbert space Specifically, if P1 represents the vector state P[ϕ1] and P2 represents P[ϕ2], then their superpositions are characterized by vector states of the form P = P[c1ϕ1 + c2ϕ2], where c1 and c2 are complex numbers This relationship highlights how any vector state P = P[ϕ] can be expressed in terms of its constituent states P1 and P2.
P 1 ≤P ⊥ , thenϕ, ϕ 1 = 0, andP is a superposition ofP 1 and, for instance,
1.2 The Set E of Effects and the Set D of Projections
Any stateT ∈ Sinduces an expectation functionalE→tr
T E on the setB of bounded operators The requirement that the numbers tr
T E represent probabilities implies that the operatorEis positive and bounded by the unit operator: O ≤ E ≤ I Such operators are called effects and the number tr
T E is the probability for the effectE in the stateT Let
E:={E∈ B |O≤E≤I} (1.4) denote the set of all effects.
As a subset of B, E is an ordered set with O and I serving as its order bounds Notably, the order on E is closely tied to the fundamental principles of quantum mechanics, specifically the basic probabilities In this context, for any elements E and F within E, E is less than or equal to F if and only if the trace of F minus E is greater than or equal to O.
T F for all T ∈ S The map E E → E ⊥ := I−E ∈ E is a kind of complementation, since it reverses the order (if E ≤ F, then
F ⊥ ≤ E ⊥ ) and, when applied twice, it yields the identity ((E ⊥ ) ⊥ = E).
The de Morgan laws are upheld in the context of the lattice E, where the greatest lower bound (E ∧ F) of elements E and F exists, ensuring that the least upper bound of their complements (E ⊥ and F ⊥) also exists within E This relationship is expressed as (E ∧ F) ⊥ = E ⊥ ∨ F ⊥ However, the implication E → E ⊥ does not constitute an orthocomplementation, as the greatest lower bound of E and its complement E ⊥ may not exist, and even if it does, it may not equate to the null effect.
The set of projections D is an important subset of E For any E ∈ E,
In the context of projections, it is established that EE ⊥ = E ⊥ E, indicating that the effect EE ⊥ is inherent to both E and E ⊥ Consequently, projections can be defined as effects E for which the set of lower bounds, l.b.{E, E ⊥ }, exclusively includes the null effect.
In addition to its order structure, the setE of effects is a convex subset of the set of bounded operators B: for anyE, F ∈ Eand 0≤w≤1,wE+
(1−w)F ∈ E This structure reflects the physical possibility of combining measurements into new measurements by mixing them An effect E ∈ E is an extreme effect if the condition E =wE 1+ (1−w)E 2, with E 1 , E 2 ∈ E,
In the context of quantum mechanics, when the value of w is between 0 and 1, it leads to the conclusion that the effects E1 and E2 are equivalent, represented as E = E1 = E2 Extreme effects are characterized by pure measurements, which cannot be derived from a combination of other measurements Utilizing the spectral theorem, it can be demonstrated that the set of extreme effects, denoted as ex(E), is equal to the set of projections, expressed mathematically as ex(E) = D.
The algebraic structure of B provides E with the characteristics of a partial algebra, where the sum E + F of any two elements E, F in E is considered an effect, provided that this operation is bounded by the unit operator Additionally, for each element E in E, there exists a unique counterpart E in E such that E + E equals the identity operator I, indicating that E is equal to its orthogonal complement E ⊥ This framework is intrinsically linked to the physical feasibility of the effects represented by E.
In the context of the measurement of F, where E + F ≤ I, we can establish an order on the set E For any elements E and F within E, we denote E ≤ F if there exists a G in E such that E + G = F This ordering aligns with the concept of a positive operator Additionally, it is important to note that for any D1, D2 in D, the sum D1 + D2 qualifies as an effect if and only if it is a projection, thereby endowing D with a partial algebra structure through this restriction.
Observables
the partially defined sum ofE The order defined onD by this partial sum is obviously the standard one.
There is a significant distinction between the structures of partial algebras in D and E, particularly regarding their relationship with ortho-ordered sets For elements D1 and D2 in D, the condition D1 + D2 belongs to D if and only if D1 is less than or equal to D2 orthogonally, resulting in D1 + D2 equating to D1 ∨ D2 This indicates a mutual dependency between the partial algebra structure of D and its order structure Conversely, in the set of effects E, there exist elements E and F such that E ≤ F orthogonally, and E + F is still in E, yet E + F does not necessarily equal E ∨ F This discrepancy arises because E ∨ F may not exist at all.
E =αD 1 , F =βD 2, with 0< α < β