Coexistency on Hilbert space effect algebras and a characterisation of its symmetry transformations
CCoexistency on Hilbert space effect algebras and acharacterisation of its symmetry transformations ∗† Gy¨orgy P´al Geh´er ‡ , Peter ˇSemrl § Abstract
The Hilbert space effect algebra is a fundamental mathematical structure which is used to de-scribe unsharp quantum measurements in Ludwig’s formulation of quantum mechanics. Each effectrepresents a quantum (fuzzy) event. The relation of coexistence plays an important role in thistheory, as it expresses when two quantum events can be measured together by applying a suitableapparatus. This paper’s first goal is to answer a very natural question about this relation, namely,when two effects are coexistent with exactly the same effects? The other main aim is to describe allautomorphisms of the effect algebra with respect to the relation of coexistence. In particular, we willsee that they can differ quite a lot from usual standard automorphisms, which appear for instance inLudwig’s theorem. As a byproduct of our methods we also strengthen a theorem of Moln´ar.
AMS classification: 47B49, 81R15.Keywords: Hilbert space effect algebra, unsharp quantum measurement, coexistency, automorphism.
Throughout this paper H will denote a complex, not necessarily separable, Hilbert space with dimensionat least 2. In the classical mathematical formulation of quantum mechanics such a space is used todescribe experiments at the atomic scale. For instance, the famous Stern–Gerlach experiment (which wasone of the firsts showing the reality of the quantum spin) can be described using the two-dimensionalHilbert space C . In the classical formulation of quantum mechanics, the space of all rank-one projections P ( H ) plays an important role, as its elements represent so-called quantum pure-states (in particular inthe Stern-Gerlach experiment they represent the quantum spin). The so-called transition probabilitybetween two pure states P, Q ∈ P ( H ) is the number tr P Q , where tr denotes the trace. For the physicalmeaning of this quantity we refer the interested reader to e.g. [33]. A very important cornerstone of themathematical foundations of quantum mechanics is
Wigner’s theorem , which states the following.
Wigner’s Theorem.
Given a bijective map φ : P ( H ) → P ( H ) that preserves the transition probability,i.e. tr φ ( P ) φ ( Q ) = tr P Q for all
P, Q ∈ P ( H ) , one can always find either a unitary, or an antiunitaryoperator U : H → H that implements φ , i.e. we have φ ( P ) = U P U ∗ for all P ∈ P ( H ) . For an elementary proof see [11]. As explained thoroughly by Simon in [29], this theorem plays acrucial role (together with Stone’s theorem and some representation theory) in obtaining the generaltime-dependent Schr¨odinger equation that describes quantum systems evolving in time (and which isusually written in the form i (cid:125) ddt | Ψ( t ) (cid:105) = ˆ H | Ψ( t ) (cid:105) , where (cid:125) is the reduced Planck constant, ˆ H is theHamiltonian operator, and | Ψ( t ) (cid:105) is the unit vector that describes the system at time t ).One of the main objectives of quantum mechanics is the study of measurement. In the classicalformulation an observable (such as the position/momentum of a particle, or a component of a particle’s ∗ The first author was supported by the Leverhulme Trust Early Career Fellowship, ECF-2018-125. He was also partlysupported by the Hungarian National Research, Development and Innovation Office – NKFIH (K115383). † The second author was supported by grants N1-0061, J1-8133, and P1-0288 from ARRS, Slovenia. ‡ Department of Mathematics and Statistics, University of Reading, Whiteknights, P.O. Box 220, Reading RG6 6AX,United Kingdom, [email protected] or [email protected] § Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia; Institute ofMathematics, Physics, and Mechanics, Jadranska 19, SI-1000 Ljubljana, Slovenia, [email protected] a r X i v : . [ m a t h . F A ] A ug pin) is represented by a self-adjoint operator. Equivalently, we could say that an observable is representedby a projection-valued measure E : B R → P ( H ) (i.e. the spectral measure of the representing self-adjointoperator), where B R denotes the set of all Borel sets in R and P ( H ) the space of all projections (alsocalled sharp effects) acting on H . If ∆ is a Borel set, then the quantum event that we get a value in ∆corresponds to the projection E (∆). However, this mathematical formulation of observables implicitlyassumes that measurements are perfectly accurate, which is far from being the case in real life. This wasthe crucial thought which led Ludwig to give an alternative axiomatic formulation of quantum mechanicswhich was introduced in his famous books [18] and [19]. This paper is related to
Ludwig’s formulation of quantum mechanics , more precisely, we shall examine oneof the theory’s most important relations, called coexistence (see the definition later). The main differencecompared to the classical formulation is that (due to the fact that no perfectly accurate measurement ispossible in practice) quantum events are not sharp, but fuzzy. Therefore, according to Ludwig, a quantumevent is not necessarily a projection, but rather a self-adjoint operator whose spectrum lies in [0 , effect , and the set of all such operators is called the Hilbert space effect algebra ,or simply the effect algebra , which will be denoted by E ( H ). Clearly, we have P ( H ) ⊂ E ( H ). A fuzzyor unsharp quantum observable corresponds to an effect-valued measure on B R , which is often called anormalised positive operator-valued measure, see e.g. [13] for more details on this. We point out that therole of effects and positive operator-valued measures was already emphasised in the earlier book [8] ofDavies. For some of the subsequent contributions to the theory we refer the reader to the work of Kraus[17] and the recent book of Busch–Lahti–Pellonp¨a¨a–Ylinen [3].Let us point out that, contradicting to its name, E ( H ) is obviously not an actual algebra. There area number of operations and relations on the effect algebra that are relevant in mathematical physics.First of all, the usual partial order ≤ , defined by A ≤ B if and only if (cid:104) Ax, x (cid:105) ≤ (cid:104)
Bx, x (cid:105) for all x ∈ H ,expresses that the occurrence of the quantum event A implies the occurrence of B . We emphasisethat ( E ( H ) , ≤ ) is not a lattice, because usually there is no largest effect C whose occurrence impliesboth A and B (see [1, 25, 31] for more details on this). Note that, as can be easily shown, we have E ( H ) = { A ∈ B ( H ) : A = A ∗ , ≤ A ≤ I } , where B ( H ) denotes the set of all bounded operators on H , A ∗ the adjoint of A , and I the identity operator. Hence sometimes the literature refers to E ( H ) as theoperator interval [0 , I ].Second, the so called ortho-complementation ⊥ is defined by A ⊥ = I − A , and it can be thought ofas the complement event (or negation) of A , i.e. A occurs if and only if A ⊥ does not.We are mostly interested in the relation of coexistence . Ludwig called two effects coexistent if theycan be measured together by applying a suitable apparatus. In the language of mathematics (see [18,Theorem IV.1.2.4]), this translates into the following definition: Definition 1.1.
A, B ∈ E ( H ) are said to be coexistent, in notation A ∼ B , if there are effects E, F, G ∈E ( H ) such that A = E + G, B = F + G and E + F + G ∈ E ( H ) . We point out that in the earlier work [8] Davies examined the simultaneous measurement of unsharpposition and momentum, which is closely related to coexistence. It is apparent from the definition thatcoexistence is a symmetric relation. Although it is not trivial from the above definition, two sharp effects
P, Q ∈ P ( H ) are coexistent if and only if they commute (see Section 2), which corresponds to the classicalformulation. We will denote the set of all effects that are coexistent with A ∈ E ( H ) by A ∼ := { C ∈ E ( H ) : C ∼ A } , and more generally, if M ⊂ E ( H ), then M ∼ := ∩{ A ∼ : A ∈ M} .The relation of order in E ( H ) is fairly well-understood. However, the relation of coexistence is verypoorly understood. In the case of qubit effects (i.e. when dim H = 2) the recent papers of Busch–Schmidt [4], Stano–Reitzner–Heinosaari [30] and Yu–Liu–Li–Oh [36] provide some (rather complicated)characterisations of coexistence. Although there are no similar results in higher dimensions, it was pointedout by Wolf–Perez-Garcia–Fernandez in [35] that the question of coexistence of pairs of effects can bephrased as a so-called semidefinite program, which is a manageable numerical mathematical problem. Wealso mention that Heinosaari–Kiukas–Reitzner in [14] generalised the qubit coexistence characterisationto pairs of effects in arbitrary dimensions that belong to the von Neumann algebra generated by twoprojections. 2o illustrate how poorly the relation of coexistence is understood, we note that the following verynatural question has not been answered before – not even for qubit effects: What does it mean for two effects A and B to be coexistent with exactly the same effects? As our first main result we answer this very natural question. Namely, we will show the following theorem,where F ( H ) and SC ( H ) denote the set off all finite-rank and scalar effects on H , respectively. Theorem 1.1.
For any effects
A, B ∈ E ( H ) the following are equivalent: (i) B ∈ { A, A ⊥ } or A, B ∈ SC ( H ) , (ii) A ∼ = B ∼ .Moreover, if H is separable, then the above statements are also equivalent to (iii) A ∼ ∩ F ( H ) = B ∼ ∩ F ( H ) . Physically speaking, the above theorem says that the (unsharp) quantum events A and B can bemeasured together with exactly the same quantum events if and only if they are the same, or they areeach other’s negation, or both of them are scalar effects. E ( H ) with respect to two relations Automorphisms of mathematical structures related to quantum mechanics are important to study becausethey provide the right tool to understand the time-evolution of certain quantum systems (see e.g. [18,Chapters V-VII] or [29]). In case when this mathematical structure is E ( H ), we call a map φ : E ( H ) →E ( H ) a standard automorphism of the effect algebra if there exists a unitary or antiunitary operator U : H → H that (similarly to Wigner’s theorem) implements φ , i.e. we have φ ( A ) = U AU ∗ ( A ∈ E ( H )) . (1)Obviously, standard automorphisms are automorphisms with respect to the relations of order: A ≤ B ⇐⇒ φ ( A ) ≤ φ ( B ) ( A, B ∈ E ( H )); ( ≤ )of ortho-complementation: φ ( A ⊥ ) = φ ( A ) ⊥ ( A ∈ E ( H )); ( ⊥ )and also of coexistence: A ∼ B ⇐⇒ φ ( A ) ∼ φ ( B ) ( A, B ∈ E ( H )) . ( ∼ )One of the fundamental theorems in the mathematical foundations of quantum mechanics states thatevery ortho-order automorphism is a standard automorphism, which was first stated by Ludwig. Ludwig’s Theorem (1954, Theorem V.5.23 in [18]) . Let H be a Hilbert space with dim H ≥ . Assumethat φ : E ( H ) → E ( H ) is a bijective map satisfying ( ≤ ) and ( ⊥ ) . Then φ is a standard automorphism of E ( H ) . Conversely, every standard automorphism satisfies ( ≤ ) and ( ⊥ ) . We note that Ludwig’s proof was incomplete and that he formulated his theorem under the additionalassumption that dim H ≥
3. The reader can find a rigorous proof of this version for instance in [5]. Letus also point out that the two-dimensional case of Ludwig’s theorem was only proved in 2001 in [22].It is very natural to ask whether the conclusion of Ludwig’s theorem remains true, if one replaceseither ( ≤ ) by ( ∼ ), or ( ⊥ ) by ( ∼ ). Note that in light of Theorem 1.1, in the former case the condition( ⊥ ) becomes almost redundant, except on SC ( H ). However, as scalar effects are exactly those that arecoexistent with every effect (see Section 2), this problem basically reduces to the characterisation ofautomorphisms with respect to coexistence only – which we shall consider later on.In 2001, Moln´ar answered the other question affirmatively under the assumption that dim H ≥ Moln´ar’s Theorem (2001, Theorem 1 in [20]) . Let H be a Hilbert space with dim H ≥ . Assumethat φ : E ( H ) → E ( H ) is a bijective map satisfying ( ≤ ) and ( ∼ ) . Then φ is a standard automorphism of E ( H ) . Conversely, every standard automorphism satisfies ( ≤ ) and ( ∼ ) . In this paper we shall prove the two-dimensional version of Moln´ar’s theorem.3 heorem 1.2.
Assume that φ : E ( C ) → E ( C ) is a bijective map satisfying ( ≤ ) and ( ∼ ) . Then φ is astandard automorphism of E ( C ) . Conversely, every standard automorphism satisfies ( ≤ ) and ( ∼ ) . Note that Moln´ar used the fundamental theorem of projective geometry to prove the aforementionedresult, therefore his proof indeed works only if dim H ≥
3. Here, as an application of Theorem 1.1,we shall give an alternative proof of Moln´ar’s theorem that does not use this dimensionality constraint,hence fill this dimensionality gap in. More precisely, we will reduce Moln´ar’s theorem and Theorem 1.2to Ludwig’s theorem (see the end of Section 2). E ( H ) with respect to only one relation It is certainly a much more difficult problem to describe the general form of automorphisms with respectto only one relation. Of course, here we mean either order preserving ( ≤ ), or coexistence preserving ( ∼ )maps, as it is easy (and not at all interesting) to describe bijective transformations that satisfy ( ⊥ ). Ithas been known for quite some time that automorphisms with respect to the order relation on E ( H ) maydiffer a lot from standard automorphisms, although they are at least always continuous with respect tothe operator norm. We do not state the related result here, but only mention that the answer finally hasbeen given by the second author in [26, Corollary 1.2] (see also [28]).The other main purpose of this paper is to give the characterisation of all automorphisms of E ( H )with respect to the relation of coexistence. As can be seen from our result below, these maps can alsodiffer a lot from standard automorphisms, moreover, unlike in the case of ( ≤ ) they are in general noteven continuous. Theorem 1.3.
Let H be a Hilbert space with dim H ≥ , and φ : E ( H ) → E ( H ) be a bijective mapthat satisfies ( ∼ ) . Then there exists a unitary or antiunitary operator U : H → H and a bijective map g : [0 , → [0 , such that we have { φ ( A ) , φ ( A ⊥ ) } = { U AU ∗ , U A ⊥ U ∗ } ( A ∈ E ( H ) \ SC ( H )) (2) and φ ( tI ) = g ( t ) I ( t ∈ [0 , . (3) Conversely, every map of the above form preserves coexistence in both directions.
Observe that in the above theorem if we assume that our automorphism is continuous with respectto the operator norm, then up to unitary-antiunitary equivalence we obtain that φ is either the identitymap, or the ortho-complementation: A (cid:55)→ A ⊥ . Also note that the converse statement of the theoremfollows easily by Theorem 1.1. As we mentioned earlier, the description of all automorphisms with respectto ( ∼ ) and ( ⊥ ) now follows easily, namely, we get the same conclusion as in the above theorem, exceptthat now g further satisfies g (1 − t ) = 1 − g ( t ) for all 0 ≤ t ≤ E ( H ) In order to explain the above automorphism theorems’ physical interpretation, let us go back first toWigner’s theorem. Assume there are two physicists who analyse the same quantum mechanical systemusing the same Hilbert space H , but possibly they might associate different rank-one projections to thesame quantum (pure) state. However, we know that they always agree on the transition probabilities.Then according to Wigner’s theorem, there must be either a unitary, or an antiunitary operator withwhich we can transform from one analysis into the other (like a ”coordinate transformation”).For the interpretation of Ludwig’s theorem, let us say there are two physicists who analyse the samequantum fuzzy measurement, but they might associate different effects to the same quantum fuzzy event.If we at least know that both of them agree on which pairs of effects are ortho-complemented, and whicheffect is larger than the other (i.e. implies the occurrence of the other), then by Ludwig’s theorem theremust exist either a unitary, or an antiunitary operator that gives us the way to transform from oneanalysis into the other.As for the interpretation of our Theorem 1.3, if we only know that our physicists agree on which pairsof effects are coexistent (i.e. which pairs of quantum events can be measured together), then there is amap φ satisfying (2) and (3) that transforms the first physicist’s analysis into the other’s.4 .6 The outline of the paper In the next section we will prove our first main result, Theorem 1.1, and as an application, we proveMoln´ar’s theorem in an alternative way that works for qubit effects as well. This will be followed bySection 3 where we prove our other main result, Theorem 1.3, in the case when dim H = 2. Then inSection 4, using the two-dimensional case, we shall prove the general version of our result. Let us pointout once more that, unless otherwise stated, H is not assumed to be separable. We will close our paperwith some discussion on the qubit case and some open problems in Sections 5–6. We start with some definitions. The symbol P ( H ) will stand for the set of all projections (idempotent andself-adjoint operators) on H , and P ( H ) will denote the set of all rank-one projections. The commutantof an effect A intersected with E ( H ) will be denoted by A c := { C ∈ E ( H ) : CA = AC } , and more generally, for a subset M ⊂ E ( H ) we will use the notation M c := ∩{ A c : A ∈ M} . Also, weset A cc := ( A c ) c and M cc := ( M c ) c .We continue with three known lemmas on the structure of coexistent pairs of effects that can all befound in [27]. The first two have been proved earlier, see [4, 21]. Lemma 2.1.
For any A ∈ E ( H ) and P ∈ P ( H ) the following statements hold: (a) A ∼ = E ( H ) if and only if A ∈ SC ( H ) , (b) P ∼ = P c , (c) A c ⊆ A ∼ . Lemma 2.2.
Let
A, B ∈ E ( H ) so that their matrices are diagonal with respect to some orthogonaldecomposition H = ⊕ i ∈I H i , i.e. A = ⊕ i ∈I A i and B = ⊕ i ∈I B i ∈ E ( ⊕ i ∈I H i ) . Then A ∼ B if and only if A i ∼ B i for all i ∈ I . Lemma 2.3.
Let
A, B ∈ E ( H ) . Then the following are equivalent: (i) A ∼ B , (ii) there exist effects M, N ∈ E ( H ) such that M ≤ A , N ≤ I − A , and M + N = B . We continue with a corollary of Lemma 2.1.
Corollary 2.4.
For any effect A and projection P ∈ A ∼ we have P ∈ A c . In particular, we have A ∼ ∩ P ( H ) = A c ∩ P ( H ) .Proof. Since coexistence is a symmetric relation, we obtain A ∈ P ∼ , which implies AP = P A .The next four statements are easy consequences of Lemma 2.3, we only prove two of them.
Corollary 2.5.
For any effect A we have A ∼ = (cid:0) A ⊥ (cid:1) ∼ . Corollary 2.6.
Let A ∈ E ( H ) such that either / ∈ σ ( A ) , or / ∈ σ ( A ) . Then there exists an ε > suchthat { C ∈ E ( H ) : C ≤ εI } ⊆ A ∼ . We recall the definition of the strength function of A ∈ E ( H ):Λ( A, P ) = max { λ ≥ λP ≤ A } ( P ∈ P ( H )) , see [2] for more details and properties. Corollary 2.7.
Assume that A ∈ E ( H ) , < t ≤ , and P ∈ P ( H ) . Then the following conditions areequivalent: (i) A ∼ tP ; t ≤ Λ( A, P ) + Λ( A ⊥ , P ) . (4) Proof.
By (ii) of Lemma 2.3 we have A ∼ tP if and only if there exist t , t ≥ t = t + t , t P ≤ A and t P ≤ A ⊥ , which is of course equivalent to (4). Corollary 2.8.
Let
A, B ∈ E ( H ) such that A ∼ ⊆ B ∼ . Assume that with respect to the orthogonaldecomposition H = H ⊕ H the two effects have the following block-diagonal matrix forms: A = (cid:20) A A (cid:21) and B = (cid:20) B B (cid:21) ∈ E ( H ⊕ H ) . Then we also have A ∼ ⊆ B ∼ and A ∼ ⊆ B ∼ . (5) In particular, if A ∼ = B ∼ , then A ∼ = B ∼ and A ∼ = B ∼ .Proof. Let P be the orthogonal projection onto H . By Lemma 2.2 we observe that (cid:40) (cid:20) C D (cid:21) ∈ E ( H ⊕ H ) : C ∼ A , D ∼ A (cid:41) = P c ∩ A ∼ ⊆ P c ∩ B ∼ = (cid:40) (cid:20) E F (cid:21) ∈ E ( H ⊕ H ) : E ∼ B , F ∼ B (cid:41) , which immediately implies (5).Next, we recall the Busch–Gudder theorem about the explicit form of the strength function, whichwe shall use frequently here. We also adopt their notation, so whenever it is important to emphasisethat the range of a rank-one projection P is C · x with some x ∈ H such that (cid:107) x (cid:107) = 1, we write P x instead. Furthermore, the symbol A − / denotes the algebraic inverse of the bijective restriction A / | (Im A ) − : (Im A ) − → Im ( A / ), where · − stands for the closure of a set. In particular, for all x ∈ Im ( A / ) the vector A − / x is the unique element in (Im A ) − which A / maps to x . Busch–Gudder Theorem (1999, Theorem 4 in [2]) . For every effect A ∈ E ( H ) and unit vector x ∈ H we have Λ( A, P x ) = (cid:26) (cid:107) A − / x (cid:107) − , if x ∈ Im ( A / ) , , otherwise. (6)We proceed with proving some new results which will be crucial in the proofs of our main theorems.The first lemma is probably well-known, but as we did not find it in the literature, we state and prove ithere. Recall that WOT and SOT stand for the weak- and strong operator topologies, respectively. Lemma 2.9.
For any effect A ∈ E ( H ) , the set A ∼ is convex and WOT-compact, hence it is also SOT-and norm-closed. Moreover, if H is separable, then the subset A ∼ ∩ F ( H ) is SOT-dense, hence alsoWOT-dense, in A ∼ .Proof. Let t ∈ [0 ,
1] and B , B ∈ A ∼ . By Lemma 2.3 there are M , M , N , N ∈ E ( H ) such that M + N = B , M + N = B , M ≤ A, N ≤ I − A and M ≤ A, N ≤ I − A . Hence setting M = tM + (1 − t ) M ∈ E ( H ) and N = tN + (1 − t ) N ∈ E ( H ) gives M + N = tB + (1 − t ) B and M ≤ A, N ≤ I − A , thus tB + (1 − t ) B ∼ A , so A ∼ is indeed convex.Next, we prove that A ∼ is WOT-compact. Clearly, E ( H ) is WOT-compact, as it is a bounded WOT-closed subset of B ( H ) (see [7, Proposition IX.5.5]), therefore it is enough to show that A ∼ is WOT-closed.Let { B ν } ν ⊆ A ∼ be an arbitrary net that WOT-converges to B , we shall show that B ∼ A holds. Forevery ν we can find two effects M ν and N ν such that M ν + N ν = B ν , M ν ≤ A and N ν ≤ I − A . ByWOT-compactness of E ( H ), there exists a subnet { B ξ } ξ such that M ξ → M in WOT with some effect M . Again, by WOT-compactness of E ( H ), there exists a subnet { B η } η of the subnet { B ξ } ξ such that N η → N in WOT with some effect N . Obviously we also have B η → B and M η → M in WOT. Thereforewe have M + N = B and by definition of WOT convergence we also obtain M ≤ A , N ≤ I − A , henceindeed B ∼ A . Closedness with respect to the other topologies is straightforward.Concerning our last statement for separable spaces, first we point out that for every effect C thereexists a net of finite rank effects { C ν } ν such that C ν ≤ C holds for all ν and C ν → C in SOT. Denote6y E C the projection-valued spectral measure of C , and set C n = (cid:80) nj =0 jn E C (cid:0)(cid:2) jn , j +1 n (cid:1)(cid:1) for every n ∈ N .Clearly, each C n has finite spectrum, satisfies C n ≤ C , and (cid:107) C n − C (cid:107) → n → ∞ . For eachspectral projection E C (cid:0)(cid:2) jn , j +1 n (cid:1)(cid:1) we can take a sequence of finite-rank projections { P j,nk } ∞ k =1 such that P j,nk ≤ E C (cid:0)(cid:2) jn , j +1 n (cid:1)(cid:1) for all k and P j,nk → E C (cid:0)(cid:2) jn , j +1 n (cid:1)(cid:1) in SOT as k → ∞ . Define C n,k := (cid:80) nj =0 jn P j,nk .It is apparent that C n,k ≤ C n for all n and k , and that for each n we have C n,k → C n in SOT as k → ∞ .Therefore the SOT-closure of { C n,k : n, k ∈ N } contains each C n , hence also C , thus we can construct anet { C ν } ν with the required properties.Now, let B ∈ A ∼ be arbitrary, and consider two other effects M, N ∈ E ( H ) that satisfy the conditionsof Lemma 2.3 (ii). Set C := M ⊕ N ∈ E ( H ⊕ H ), and denote by E M and E N the projection-valued spectralmeasures of M and N , respectively. Clearly, E C (cid:0)(cid:2) jn , j +1 n (cid:1)(cid:1) = E M (cid:0)(cid:2) jn , j +1 n (cid:1)(cid:1) ⊕ E N (cid:0)(cid:2) jn , j +1 n (cid:1)(cid:1) for each j and n . In the above construction we can choose finite-rank projections of the form P j,nk = Q j,nk ⊕ R j,nk ∈P ( H ⊕ H ) where Q j,nk , R j,nk ∈ P ( H ), Q j,nk ≤ E M (cid:0)(cid:2) jn , j +1 n (cid:1)(cid:1) and R j,nk ≤ E N (cid:0)(cid:2) jn , j +1 n (cid:1)(cid:1) holds for all k, n .Then each element C ν of the convergent net is an orthogonal sum of the form M ν ⊕ N ν ∈ E ( H ⊕ H )). Itis apparent that M ν , N ν ∈ F ( H ), M ν ≤ M and N ν ≤ N for all ν , and that M ν → M , N ν → N holdsin SOT. Therefore M ν + N ν ∈ F ( H ) ∩ A ∼ and M ν + N ν converges to M + N = B in SOT, the proof iscomplete.We proceed to investigate when do we have the equation A ∼ = B ∼ for two effects A and B , which willtake several steps. We will denote the set of all rank-one effects by F ( H ) := { tP : P ∈ P ( H ) , < t ≤ } . Lemma 2.10.
Let H = H ⊕ H be an orthogonal decomposition and assume that A, B ∈ E ( H ) have thefollowing matrix decompositions: A = (cid:20) λ I λ I (cid:21) and B = (cid:20) µ I µ I (cid:21) ∈ E ( H ⊕ H ) (7) where λ , λ , µ , µ ∈ [0 , , and I and I denote the identity operators on H and H , respectively. Thenthe following are equivalent: (i) Λ( A, P ) + Λ( A ⊥ , P ) = Λ( B, P ) + Λ( B ⊥ , P ) holds for all P ∈ P ( H ) , (ii) A ∼ ∩ F ( H ) = B ∼ ∩ F ( H ) , (iii) either λ = λ and µ = µ , or λ = µ and λ = µ , or λ + µ = λ + µ = 1 .Proof. The directions (iii)= ⇒ (ii) ⇐⇒ (i) are trivial by Lemma 2.1 (a) and Corollaries 2.5, 2.7, so weshall only consider the direction (i)= ⇒ (iii). First, a straightforward calculation using the Busch–Guddertheorem gives the following for every x ∈ H , x ∈ H , (cid:107) x (cid:107) = (cid:107) x (cid:107) = 1 and 0 ≤ α ≤ π :Λ ( A, P cos αx +sin αx ) = 1 (cid:16) λ (cid:17) · cos α + (cid:16) λ (cid:17) · sin α , (8)where we use the interpretations = ∞ , ∞ = 0, ∞ · ∞ + ∞ = ∞ , and ∞ + a = ∞ , ∞ · a = ∞ ( a > λ = 0 or λ = 0. Clearly, (8) dependsonly on α , but not on the specific choices of x and x . We define the following two functions T A : (cid:2) , π (cid:3) → [0 , , T A ( α ) = Λ ( A, P cos αx +sin αx ) + Λ (cid:0) A ⊥ , P cos αx +sin αx (cid:1) and T B : (cid:2) , π (cid:3) → [0 , , T B ( α ) = Λ ( B, P cos αx +sin αx ) + Λ (cid:0) B ⊥ , P cos αx +sin αx (cid:1) , which are the same by our assumptions. By (8), for all 0 ≤ α ≤ π we have T A ( α ) = 1 (cid:16) λ (cid:17) · cos α + (cid:16) λ (cid:17) · sin α + 1 (cid:16) − λ (cid:17) · cos α + (cid:16) − λ (cid:17) · sin α = 1 (cid:16) µ (cid:17) · cos α + (cid:16) µ (cid:17) · sin α + 1 (cid:16) − µ (cid:17) · cos α + (cid:16) − µ (cid:17) · sin α = T B ( α ) . (9)Next, we observe the following implications: • if λ = λ , then T A ( α ) is the constant 1 function,7 if λ = 0 and λ = 1, then T A ( α ) is the characteristic function χ { ,π/ } ( α ), • if λ = 0 and 0 < λ <
1, then T A ( α ) is continuous on (cid:2) , π (cid:1) , but has a jump at π , namelylim α → π − T A ( α ) = 1 − λ and T A ( π ) = 1, • if λ = 1 and 0 < λ <
1, then T A ( α ) is continuous on (cid:2) , π (cid:1) , but has a jump at π , namelylim α → π − T A ( α ) = λ and T A ( π ) = 1, • if λ , λ ∈ (0 , T A ( α ) is continuous on (cid:2) , π (cid:3) , • if λ (cid:54) = λ , then we have T A (0) = T A ( π ) = 1 and T A ( α ) < < α < π .All of the above statements are rather straightforward computations using the formula (9), let us onlyshow the last one here. Clearly, T A (0) = T A ( π ) = 1 is obvious. As for the other assertion, if λ , λ ∈ (0 , λ cos α + λ sin α + 1 − λ cos α + − λ sin α< ( λ cos α + λ sin α ) + ((1 − λ ) cos α + (1 − λ ) sin α ) = 1 (0 < α < π ) . If λ = 0 < λ <
1, then we calculate in the following way:1 (cid:0) (cid:1) cos α + λ sin α + 1cos α + − λ sin α = 11 − sin α + − λ sin α < < α < π ) . The remaining cases are very similar.The above observations together with Corollary 2.5 and (9) readily imply the following: • A ∈ SC ( H ) if and only if B ∈ SC ( H ), • A ∈ P ( H ) \ SC ( H ) if and only if B ∈ P ( H ) \ SC ( H ), in which case B ∈ { A, A ⊥ } , • there exists a P ∈ P ( H ) \SC ( H ) and a t ∈ (0 ,
1) with A ∈ { tP, I − tP } if and only if B ∈ { tP, I − tP } , • λ , λ ∈ (0 ,
1) and λ (cid:54) = λ if and only if µ , µ ∈ (0 ,
1) and µ (cid:54) = µ .So what remained is to show that in the last case we further have B ∈ { A, A ⊥ } , which is what we shalldo below.Let us introduce the following functions: T A : [0 , → [0 , , T A ( s ) := T A (arcsin √ s ) = λ λ λ s + λ (1 − s ) + (1 − λ )(1 − λ )(1 − λ ) s + (1 − λ )(1 − s )and T B : [0 , → [0 , , T B ( s ) := T B (arcsin √ s ) = µ µ µ s + µ (1 − s ) + (1 − µ )(1 − µ )(1 − µ ) s + (1 − µ )(1 − s ) . Our aim is to prove that T A ( s ) = T B ( s ) ( s ∈ [0 , λ = µ and λ = µ , or λ + µ = λ + µ = 1. The derivative of T A is d T A ds ( s ) = ( λ − λ ) (cid:18) − λ λ ( λ s + λ (1 − s )) + (1 − λ )(1 − λ )((1 − λ ) s + (1 − λ )(1 − s )) (cid:19) , from which we calculate d T A ds (0) = − ( λ − λ ) (1 − λ ) λ and d T A ds (1) = ( λ − λ ) (1 − λ ) λ . Therefore, if we managed to show that the function F : (0 , → R , F ( x, y ) = (cid:16) ( x − y ) (1 − x ) x , ( x − y ) (1 − y ) y (cid:17)
8s injective on the set ∆ := { ( x, y ) ∈ R : 0 < y < x < } , then we are done (note that F ( x, y ) = F (1 − x, − y )). For this assume that with some c, d > x − y ) (1 − x ) x = 1 c and ( x − y ) (1 − y ) y = 1 d , or equivalently, (1 − x ) x = c ( x − y ) and (1 − y ) y = d ( x − y ) . If we substitute u = x − y and v = x + y , then we get( u + v ) − ( u + v ) = − cu and ( v − u ) − ( v − u ) = − du . Now, considering the sum and difference of these two equations and manipulate them a bit gives v − v = − (2 c + 2 d + 1) u and v = ( d − c ) u + . From these latter equations we conclude x − y = 2 u = (cid:113) d − c ) +2 c +2 d +1 and x + y = 2 v = 2( d − c ) u + 1 , which clearly implies that F is globally injective on ∆, and the proof is complete.We have an interesting consequence in finite dimensions. Corollary 2.11.
Assume that ≤ dim H < ∞ and A, B ∈ E ( H ) . Then the following are equivalent: (i) Λ( A, P ) + Λ( A ⊥ , P ) = Λ( B, P ) + Λ( B ⊥ , P ) for all P ∈ P ( H ) , (ii) A ∼ ∩ F ( H ) = B ∼ ∩ F ( H ) , (iii) either A, B ∈ SC ( H ) , or A = B , or A = B ⊥ .Proof. The directions (i) ⇐⇒ (ii) ⇐ =(iii) are trivial, so we shall only prove the (ii)= ⇒ (iii) direction. First,let us consider the two-dimensional case. As we saw in the proof of Lemma 2.10, we have A ∼ ∩ F ( H ) = F ( H ) if and only if A is a scalar effect (see the first set of bullet points there). Therefore, without lossof generality we may assume that none of A and B are scalar effects. Notice that by Lemma 2.1, A and B commute with exactly the same rank-one projections, hence A and B possess the forms in (7) withsome one-dimensional subspaces H and H , and an easy application of Lemma 2.10 gives (iii).As for the general case, since again A and B commute with exactly the same rank-one projections,we can jointly diagonalise them with respect to some orthonormal basis { e j } nj =1 , where n = dim H : A = λ . . . λ . . . . . . λ n −
00 0 . . . λ n and B = µ . . . µ . . . . . . µ n −
00 0 . . . µ n . Of course, for any two distinct i, j ∈ { , . . . , n } we have the following equation for the strength functions:Λ( A, P ) + Λ( A ⊥ , P ) = Λ( B, P ) + Λ( B ⊥ , P ) ( P ∈ P ( C · e i + C · e j )) , which instantly implies (cid:20) µ i µ j (cid:21) ∼ ∩ F ( C · e i + C · e j ) = (cid:20) λ i λ j (cid:21) ∼ ∩ F ( C · e i + C · e j ) . By the two-dimensional case this means that we have one of the following cases: • λ i = λ j and µ i = µ j , • λ i (cid:54) = λ j and either µ i = λ i and µ j = λ j , or µ i = 1 − λ i and µ j = 1 − λ j .From here it is easy to conclude (iii). 9he commutant of an operator T ∈ B ( H ) will be denoted by T (cid:48) := { S ∈ B ( H ) : ST = T S } , and moregenerally, if M ⊆ B ( H ), then we set M (cid:48) := ∩{ T (cid:48) : T ∈ M} . We shall use the notations T (cid:48)(cid:48) := ( T (cid:48) ) (cid:48) and M (cid:48)(cid:48) := ( M (cid:48) ) (cid:48) for the double commutants. Lemma 2.12.
For any
A, B ∈ E ( H ) the following three assertions hold: (a) If A ∼ ⊆ B ∼ , then B ∈ A (cid:48)(cid:48) . (b) If dim H ≤ ℵ and A ∼ ⊆ B ∼ , then there exists a Borel function f : [0 , → [0 , such that B = f ( A ) . (c) If B is a convex combination of A, A ⊥ , and I , then A ∼ ⊆ B ∼ .Proof. (a): Assume that C ∈ A (cid:48) . Our goal is to show B ∈ C (cid:48) . We express C in the following way: C = C (cid:60) + iC (cid:61) , C (cid:60) = C + C ∗ , C (cid:61) = C − C ∗ i where C (cid:60) and C (cid:61) are self-adjoint (they are usually called the real and imaginary parts of C ). Since A isself-adjoint, C ∗ ∈ A (cid:48) , hence C (cid:60) , C (cid:61) ∈ A (cid:48) . Let E (cid:60) and E (cid:61) denote the projection-valued spectral measuresof C (cid:60) and C (cid:61) , respectively. By the spectral theorem ([7, Theorem IX.2.2]), Lemma 2.1 and Corollary2.4, we have E (cid:60) (∆) , E (cid:61) (∆) ∈ A c ⊆ A ∼ ⊆ B ∼ , therefore also E (cid:60) (∆) , E (cid:61) (∆) ∈ B (cid:48) for all ∆ ∈ B R , whichgives C ∈ B (cid:48) . (b): This is an easy consequence of [7, Proposition IX.8.1 and Lemma IX.8.7]. (c): If A ∼ C , then also A ⊥ , I ∼ C . Hence by the convexity of C ∼ we obtain B ∼ C .Now, we are in the position to prove our first main result. Proof of Theorem 1.1. If H is separable, then the equivalence (ii) ⇐⇒ (iii) is straightforward by Lemma2.9. For general H the direction (i)= ⇒ (ii) is obvious, therefore we shall only prove (ii)= ⇒ (i), first in theseparable, and then in the general case. By Lemma 2.1, we may assume throughout the rest of the proofthat A and B are non-scalar effects. We will denote the spectral subspace of a self-adjoint operator T associated to a Borel set ∆ ⊆ R by H T (∆). (ii) = ⇒ (i) in the separable case: We split this part into two steps.
STEP 1:
Here, we establish two estimations, (11) and (12), for the strength functions of A and B oncertain subspaces of H . Let λ , λ ∈ σ ( A ) , λ (cid:54) = λ and 0 < ε < | λ − λ | . Then the spectral subspaces H = H A (( λ − ε, λ + ε )) and H = H A (( λ − ε, λ + ε )) are non-trivial and orthogonal. Set H to bethe orthogonal complement of H ⊕ H , then the matrix of A written in the orthogonal decomposition H = H ⊕ H ⊕ H is diagonal: A = A A
00 0 A ∈ B ( H ⊕ H ⊕ H ) . Note that H might be a trivial subspace. Since by Corollary 2.4 A and B commute with exactly thesame projections, the matrix of B in H = H ⊕ H ⊕ H is also diagonal: B = B B
00 0 B ∈ B ( H ⊕ H ⊕ H ) . At this point, let us emphasise that of course H j , A j and B j ( j = 1 , ,
3) all depend on λ , λ and ε , butin order to keep our notation as simple as possible, we will stick with these symbols. However, if at anypoint it becomes important to point out this dependence, we shall use for instance B ( λ ,λ ,ε ) j instead of B j . Similar conventions apply later on.Observe that by Corollary 2.8 we have (cid:20) A A (cid:21) ∼ = (cid:20) B B (cid:21) ∼ . Now, we pick two arbitrary points µ ∈ σ ( B ) and µ ∈ σ ( B ). Then obviously, the following twosubspaces are non-zero subspaces of H and H , respectively: (cid:98) H := ( H ) B (cid:0) ( µ − ε, µ + ε ) (cid:1) , (cid:98) H := ( H ) B (cid:0) ( µ − ε, µ + ε ) (cid:1) . H j = H j (cid:9) (cid:98) H j ( j = 1 , B = (cid:20) (cid:98) B
00 ˇ B (cid:21) ∈ B ( (cid:98) H ⊕ ˇ H ) and B = (cid:20) (cid:98) B
00 ˇ B (cid:21) ∈ B ( (cid:98) H ⊕ ˇ H )and A = (cid:20) (cid:98) A
00 ˇ A (cid:21) ∈ B ( (cid:98) H ⊕ ˇ H ) and A = (cid:20) (cid:98) A
00 ˇ A (cid:21) ∈ B ( (cid:98) H ⊕ ˇ H ) . Note that ˇ H or ˇ H might be trivial subspaces. Again by Corollary 2.8, we have (cid:34) (cid:98) A (cid:98) A (cid:35) ∼ = (cid:34) (cid:98) B (cid:98) B (cid:35) ∼ . Let us point out that by construction σ ( (cid:98) A j ) ⊆ [ λ j − ε, λ j + ε ] and σ ( (cid:98) B j ) ⊆ [ µ j − ε, µ j + ε ]. Corollary 2.7gives the following identity for the strength functions, where (cid:98) I j denotes the identity on (cid:98) H j ( j = 1 , (cid:32)(cid:34) (cid:98) A (cid:98) A (cid:35) , P (cid:33) + Λ (cid:32)(cid:34)(cid:98) I − (cid:98) A (cid:98) I − (cid:98) A (cid:35) , P (cid:33) = Λ (cid:32)(cid:34) (cid:98) B (cid:98) B (cid:35) , P (cid:33) + Λ (cid:32)(cid:34)(cid:98) I − (cid:98) B (cid:98) I − (cid:98) B (cid:35) , P (cid:33) (cid:16) ∀ P ∈ P (cid:16) (cid:98) H ⊕ (cid:98) H (cid:17)(cid:17) . (10)Define Θ : R → [0 , , Θ( t ) = t < t if 0 ≤ t ≤
11 if 1 < t , and notice that we have the following two estimations for all rank-one projections P :Λ (cid:32)(cid:34) Θ( λ − ε ) (cid:98) I
00 Θ( λ − ε ) (cid:98) I (cid:35) , P (cid:33) + Λ (cid:32)(cid:34) Θ(1 − λ − ε ) (cid:98) I
00 Θ(1 − λ − ε ) (cid:98) I (cid:35) , P (cid:33) ≤ the expression in (10) ≤ Λ (cid:32)(cid:34) Θ( λ + ε ) (cid:98) I
00 Θ( λ + ε ) (cid:98) I (cid:35) , P (cid:33) + Λ (cid:32)(cid:34) Θ(1 − λ + ε ) (cid:98) I
00 Θ(1 − λ + ε ) (cid:98) I (cid:35) , P (cid:33) (11)and Λ (cid:32)(cid:34) Θ( µ − ε ) (cid:98) I
00 Θ( µ − ε ) (cid:98) I (cid:35) , P (cid:33) + Λ (cid:32)(cid:34) Θ(1 − µ − ε ) (cid:98) I
00 Θ(1 − µ − ε ) (cid:98) I (cid:35) , P (cid:33) ≤ the expression in (10) ≤ Λ (cid:32)(cid:34) Θ( µ + ε ) (cid:98) I
00 Θ( µ + ε ) (cid:98) I (cid:35) , P (cid:33) + Λ (cid:32)(cid:34) Θ(1 − µ + ε ) (cid:98) I
00 Θ(1 − µ + ε ) (cid:98) I (cid:35) , P (cid:33) . (12)Note that the above estimations hold for any arbitrarily small ε and for all suitable choices of µ and µ (which of course depend on ε ). STEP 2:
Here we show that B ∈ { A, A ⊥ } . Let us define the following set that depends only on λ j : C j = C ( λ j ) j := (cid:92) (cid:110) σ (cid:16) B ( λ ,λ ,ε ) j (cid:17) : 0 < ε < | λ − λ | (cid:111) = (cid:92) (cid:8) σ (cid:0) B | H A (( λ j − ε,λ j + ε )) (cid:1) : 0 < ε (cid:9) ( j = 1 , . Notice that as this set is an intersection of monotonically decreasing (as ε (cid:38) µ ∈ C and µ ∈ C , then (11) and (12)hold for all ε > C = { λ } and C = { λ } , or C = { − λ } and C = { − λ } hold. Fix two arbitrary elements µ ∈ C and µ ∈ C , and assume that neither λ = µ and λ = µ ,11or λ + µ = λ + µ = 1 hold. From here our aim is to get a contradiction. As we showed in the proofof Lemma 2.10, there exists an α ∈ (cid:0) , π (cid:1) such that we have1 (cid:16) λ (cid:17) · cos α + (cid:16) λ (cid:17) · sin α + 1 (cid:16) − λ (cid:17) · cos α + (cid:16) − λ (cid:17) · sin α (cid:54) = 1 (cid:16) µ (cid:17) · cos α + (cid:16) µ (cid:17) · sin α + 1 (cid:16) − µ (cid:17) · cos α + (cid:16) − µ (cid:17) · sin α where we interpret both sides as in (8). Notice that both summands on both sides depend continuouslyon λ , λ , µ and µ . Therefore there exists an ε > P = P cos α (cid:98) x +sin α (cid:98) x , with (cid:98) x ∈ (cid:98) H , (cid:98) x ∈ (cid:98) H , (cid:107) (cid:98) x (cid:107) = (cid:107) (cid:98) x (cid:107) = 1, such that the closed intervals bounded by theright- and left-hand sides of (11), and those of (12) are disjoint – which is a contradiction.Observe that as we can do the above for any two disjoint elements of the spectrum σ ( A ), we canconclude that one of the following possibilities occur: { λ } = (cid:92) (cid:8) σ (cid:0) B | H A (( λ − ε,λ + ε )) (cid:1) : ε > (cid:9) ( λ ∈ σ ( A )) (13)or { − λ } = (cid:92) (cid:8) σ (cid:0) B | H A (( λ − ε,λ + ε )) (cid:1) : ε > (cid:9) ( λ ∈ σ ( A )) . (14)From here, we show that (13) implies A = B , and (14) implies B = A ⊥ . As the latter can be reducedto the case (13), by considering B ⊥ instead of B , we may assume without loss of generality that (13)holds. By Lemma 2.12 and [7, Theorem IX.8.10], there exists a function f ∈ L ∞ ( µ ), where µ is a scalar-valued spectral measure of A , such that B = f ( A ). Moreover, we have B = A if and only if f ( λ ) = λµ -a.e, so we only have to prove the latter equation. Let us fix an arbitrarily small number δ >
0. By thespectral mapping theorem ([7, Theorem IX.8.11]) and (13) we notice that for every λ ∈ σ ( A ) there existsan 0 < ε λ < δ such that µ − essran (cid:0) f | ( λ − ε λ ,λ + ε λ ) (cid:1) = σ (cid:0) B | H A (( λ − ε λ ,λ + ε λ )) (cid:1) ⊆ ( λ − δ, λ + δ ) , (15)where µ − essran denotes the essential range of a function with respect to µ (see [7, Example IX.2.6]).Now, for every λ ∈ σ ( A ) we fix such an ε λ . Clearly, the intervals { ( λ − ε λ , λ + ε λ ) : λ ∈ σ ( A ) } coverthe whole spectrum σ ( A ), which is a compact set. Therefore we can find finitely many of them, let’s say λ , . . . , λ n so that σ ( A ) ⊆ n (cid:91) j =1 (cid:0) λ j − ε λ j , λ j + ε λ j (cid:1) . Finally, we define the function h ( λ ) = λ j , where | λ − λ i | ≥ ε λ i for all 1 ≤ i < j and | λ − λ j | < ε λ j . By definition we have (cid:107) h − id σ ( A ) (cid:107) ∞ ≤ δ where the ∞ -norm is taken with respect to µ and id σ ( A ) ( λ ) = λ ( λ ∈ σ ( A )). But notice that by (15) we also have (cid:107) h − f (cid:107) ∞ ≤ δ , and hence (cid:107) f − id σ ( A ) (cid:107) ∞ ≤ δ . As thisinequality holds for all positive δ , we actually get that f ( λ ) = λ for µ -a.e. λ . (ii) = ⇒ (i) in the non-separable case: It is well-known that there exists an orthogonal decomposition H = ⊕ i ∈I H i such that each H i is a non-trivial, separable, invariant subspace of A , see for instance [7,Proposition IX.4.4]. Since A and B commute with exactly the same projections, both are diagonal withrespect to the decomposition H = ⊕ i ∈I H i : A = ⊕ i ∈I A i and B = ⊕ i ∈I B i ∈ E ( ⊕ i ∈I H i ) . By Corollary 2.8 we have A ∼ i = B ∼ i for all i ∈ I , therefore the separable case implieseither A i = B i , or B i = A ⊥ i , or A i , B i ∈ SC ( H i ) ( i ∈ I ) . Without loss of generality we may assume from now on that there exists an i ∈ I so that A i is nota scalar effect. (In case all of them are scalar, we simply combine two subspaces H i and H i so that σ ( A i ) (cid:54) = σ ( A i )). This implies either A i = B i , or B i = A ⊥ i . By considering B ⊥ instead of B ifnecessary, we may assume from now on that A i = B i holds.Finally, let i ∈ I \ { i } be arbitrary, and let us consider the orthogonal decomposition H = ⊕ i ∈I\{ i ,i } H i ⊕ K where K = H i ⊕ H i . Similarly as above, we obtain either A i ⊕ A i = B i ⊕ B i , or B i ⊕ B i = A ⊥ i ⊕ A ⊥ i , but since A i = B i , we must have A i = B i . As this holds for arbitrary i , theproof is complete. 12ow, we are in the position to give an alternative proof of Moln´ar’s theorem which also extends tothe two-dimensional case. Proof of Theorem 1.2 and Moln´ar’s theorem.
By (a) of Lemma 2.1 and ( ∼ ) we obtain φ ( SC ( H )) = SC ( H ), moreover, the property ( ≤ ) implies the existence of a strictly increasing bijection g : [0 , → [0 , φ ( λI ) = g ( λ ) I for every λ ∈ [0 , φ ( A ⊥ ) = φ ( A ) ⊥ ( A ∈ E ( H ) \ SC ( H )) . We only have to show that the same holds for scalar operators, because then the theorem is reduced toLudwig’s theorem. For any effect A and any set of effects S let us define the following sets A ≤ := { B ∈E ( H ) : A ≤ B } , A ≥ := { B ∈ E ( H ) : A ≥ B } and S ⊥ := { B ⊥ : B ∈ S} . Observe that for any s, t ∈ [0 , (cid:0) ( sI ) ≤ ∩ ( tI ) ≥ \ SC ( H ) (cid:1) ⊥ = ( sI ) ≤ ∩ ( tI ) ≥ \ SC ( H ) (cid:54) = ∅ (16)if and only if t = 1 − s and s < . Thus for all s < we obtain ∅ (cid:54) = (cid:0) ( g ( s ) I ) ≤ ∩ ( g (1 − s ) I ) ≥ \ SC ( H ) (cid:1) ⊥ = φ (cid:16)(cid:0) ( sI ) ≤ ∩ ((1 − s ) I ) ≥ \ SC ( H ) (cid:1) ⊥ (cid:17) = φ (cid:0) ( sI ) ≤ ∩ ((1 − s ) I ) ≥ \ SC ( H ) (cid:1) = ( g ( s ) I ) ≤ ∩ ( g (1 − s ) I ) ≥ \ SC ( H ) , which by (16) implies g (1 − s ) = 1 − g ( s ) and g ( s ) < , therefore we indeed have ( ⊥ ) for every effect. In this section we prove our other main theorem for qubit effects. In order to do that we need to prove afew preparatory lemmas. We start with a characterisation of rank-one projections in terms of coexistence.
Lemma 3.1.
For any A ∈ E ( C ) the following are equivalent: (i) there are no effects B ∈ E ( C ) such that B ∼ (cid:40) A ∼ , (ii) A ∈ P ( C ) .Proof. The case when A ∈ SC ( C ) is trivial, therefore we may assume otherwise throughout the proof. (i) = ⇒ (ii): Suppose that
A / ∈ P ( C ), then by Corollary 2.6 there exists an ε > { C ∈E ( C ) : C ≤ εI } ⊆ A ∼ . Let B ∈ P ( C ) ∩ A c , then we have B ∼ = B c = A c ⊆ A ∼ . But it is very easy tofind a C ∈ E ( C ) such that C ≤ εI and C / ∈ B c , therefore we conclude B ∼ (cid:40) A ∼ . (ii) = ⇒ (i): If A ∈ P ( C ), B ∈ E ( C ) and B ∼ (cid:40) A ∼ , then also B c (cid:40) A c , which is impossible.Note that the above statement does not hold in higher dimensions, see the final section of this paperfor more details. We continue with a characterisation of rank-one and ortho-rank-one qubit effects interms of coexistence. Lemma 3.2.
Let A ∈ E ( C ) \ SC ( C ) . Then the following are equivalent: (i) A or A ⊥ ∈ F ( C ) \ P ( C ) , (ii) There exists at least one B ∈ E ( C ) such that B ∼ (cid:40) A ∼ , and for every such pair of effects B , B we have either B ∼ ⊆ B ∼ , or B ∼ ⊆ B ∼ .Moreover, if (i) holds, i.e. A or A ⊥ = tP with P ∈ P ( C ) and < t < , then we have B ∼ ⊆ A ∼ if andonly if B or B ⊥ = sP with some t ≤ s ≤ .Proof. First, notice that by Theorem 1.1 and Lemma 2.12 (c) we have( sP ) ∼ ⊆ ( tP ) ∼ ⇐⇒ t ≤ s ( P ∈ P ( C ) , t, s ∈ (0 , . (i) = ⇒ (ii): If we have B ∼ ⊆ ( tP ) ∼ with some rank-one projection P , t ∈ (0 ,
1] and qubit effect B , thenby Lemma 2.12 (b) we obtain P ∈ B c and B / ∈ SC ( C ). Furthermore, since B ∼ ∩F ( C ) ⊆ ( tP ) ∼ ∩F ( C ),by Corollary 2.7 we obtain T B ( α ) ≤ T tP ( α ) (0 ≤ α ≤ π ) , T tP ( α ) at either α = 0, or α = π , implies the discontinuity of T B ( α ) at the same α . Whence we conclude either B = sP ,or B = I − sP with some t ≤ s ≤ (ii) = ⇒ (i): By Lemma 3.1, (ii) cannot hold for elements of P ( C ), so we only have to check that if A, A ⊥ / ∈ F ( C ) ∪ SC ( C ), then (ii) fails. Suppose that the spectral decomposition of A is λ P + λ P ⊥ where 1 > λ > λ >
0. Then by Lemma 2.12 (c) we find that ( λ P ) ∼ ⊆ A ∼ and (cid:0) (1 − λ ) P ⊥ (cid:1) ∼ ⊆ A ∼ (see Figure 1), but by the previous part neither ( λ P ) ∼ ⊆ (cid:0) (1 − λ ) P ⊥ (cid:1) ∼ , nor (cid:0) (1 − λ ) P ⊥ (cid:1) ∼ ⊆ ( λ P ) ∼ holds. 0 P P ⊥ IA A ⊥ λ P (1 − λ ) P ⊥ Figure 1: The figure shows all effects commuting with A ∈ E ( C ) \ SC ( C ), whose spectral decompositionis A = λ P + λ P ⊥ with 1 > λ > λ > tP ) ∼ ∩ F ( C ) see Section 5. Before we proceed with the proof of Theorem 1.3for qubit effects, we need a few more lemmas about rank-one projections acting on C . Lemma 3.3.
For all
P, Q ∈ P ( C ) we have (cid:107) P − Q (cid:107) = − det( P − Q ) = 1 − tr P Q = 1 − (cid:107) P ⊥ − Q (cid:107) . (17) Proof.
Since tr( P − Q ) = 0, the eigenvalues of the self-adjoint operator P − Q are λ and − λ with some λ ≥
0. Hence we have (cid:107) P − Q (cid:107) = − det( P − Q ). Applying a unitary similarity if necessary, we mayassume without loss of generality that (1 , ∈ Im P . Obviously, there exist 0 ≤ ϑ ≤ π and 0 ≤ µ ≤ π such that (cos ϑ, e iµ sin ϑ ) ∈ Im Q . Thus the matrix forms of P and Q in the standard basis are P = P (1 , = (cid:20) (cid:21) · (cid:20) (cid:21) ∗ = (cid:20) (cid:21) (18)and Q = P (cos ϑ,e iµ sin ϑ ) = (cid:20) cos ϑe iµ sin ϑ (cid:21) · (cid:20) cos ϑe iµ sin ϑ (cid:21) ∗ = (cid:20) cos ϑ e − iµ cos ϑ sin ϑe iµ cos ϑ sin ϑ sin ϑ (cid:21) , (19)where we used the notation of the Busch–Gudder theorem. Now, an easy calculation gives us det( P − Q ) = − sin ϑ and tr P Q = cos ϑ . Hence the second equation in (17) is proved, and the third one follows fromtr P ⊥ Q = 1 − tr P Q .For P ∈ P ( C ) and s ∈ [0 , M P,s := (cid:8) Q ∈ P ( C ) : (cid:107) P − Q (cid:107) = s (cid:9) . Next, we examine this set.
Lemma 3.4.
For all P ∈ P ( C ) the following statements are equivalent: (i) s = sin π , (ii) there exists an R ∈ M P,s such that R ⊥ ∈ M P,s , (iii) for all R ∈ M P,s we have also R ⊥ ∈ M P,s . roof. One could use the Bloch representation (see Section 5), however, let us give here a purely linearalgebraic proof. Note that for any R , R ∈ P ( C ) we have (cid:107) R − R (cid:107) = 1 if and only if R = R ⊥ .Without loss of generality we may assume that P has the matrix form of (18). Then for any 0 ≤ ϑ ≤ π and R , R ∈ M P, sin ϑ we have R = (cid:20) cos ϑ e − iµ cos ϑ sin ϑe iµ cos ϑ sin ϑ sin ϑ (cid:21) and R = (cid:20) cos ϑ e − iµ cos ϑ sin ϑe iµ cos ϑ sin ϑ sin ϑ (cid:21) with some µ , µ ∈ R . Hence, we get (cid:107) R − R (cid:107) = (cid:112) − tr R R = (cid:113) sin ϑ cos ϑ (2 − e i ( µ − µ ) − e i ( µ − µ ) )= | e iµ − e iµ | cos ϑ sin ϑ = | e iµ − e iµ | sin(2 ϑ ) . Notice that the right-hand side is always less than or equal to 1. Moreover, for any µ ∈ R there exist a µ ∈ R such that (cid:107) R − R (cid:107) = 1 if and only if ϑ = π . This completes the proof. Lemma 3.5.
Let
P, Q ∈ P ( C ) and s, t ∈ (0 , . Then the following are equivalent: (i) tP ∼ sQ (ii) either Q = P , or Q = P ⊥ , or s ≤ − t (cid:107) P ⊥ − Q (cid:107) + (cid:107) P − Q (cid:107) . Proof.
The case when Q ∈ { P, P ⊥ } is trivial, so from now on we assume otherwise. Recall that tworank-one effects with different images are coexistent if and only if their sum is an effect, see [20, Lemma2]. Therefore, (i) is equivalent to I − tP − sQ ≥
0. Since tr( I − tP − sQ ) = 2 − t − s >
0, the latteris further equivalent to det( I − tP − sQ ) ≥
0. Without loss of generality we may assume that P and Q have the matrix forms written in (18) and (19) with 0 < ϑ < π . Then a calculation givesdet( I − tP − sQ ) = s ( t −
1) sin ϑ − s cos ϑ + 1 − t = 1 − t − s + ts (cid:107) P − Q (cid:107) . From the latter we get that det( I − tP − sQ ) ≥ s ≤ − t − t (cid:107) P − Q (cid:107) , which, by (17) is equivalent to (ii).Note that we have0 < − t (cid:107) P ⊥ − Q (cid:107) + (cid:107) P − Q (cid:107) < t ∈ (0 , , P, Q ∈ P ( C ) , Q / ∈ { P, P ⊥ } ) . We need one more lemma.
Lemma 3.6.
Let
P, Q ∈ P ( C ) . Then there exists a projection R ∈ P ( C ) such that (cid:107) P − R (cid:107) = (cid:107) Q − R (cid:107) = sin π . Proof.
Again, one could use the Bloch representation, however, let us give here a purely linear algebraicproof. We may assume without loss of generality that P and Q are of the form (18) and (19). Then forany z ∈ C , | z | = 1 the rank-one projection R = 1 √ (cid:20) z (cid:21) · (cid:18) √ (cid:20) z (cid:21)(cid:19) ∗ = 12 (cid:20) zz (cid:21) satisfies (cid:107) P − R (cid:107) = sin π . In order to complete the proof we only have to find a z with | z | = 1 such thattr RQ = , which is an easy calculation. Namely, we find that z = ie iµ is a suitable choice.Now, we are in the position to prove our second main result in the low-dimensional case.15 roof of Theorem 1.3 in two dimensions. The proof is divided into the following three steps:1 we show some basic properties of φ , in particular, that it preserves commutativity in both directions,2 we show that φ maps pairs of rank-one projections with distance sin π into pairs of rank-oneprojections with the same distance,3 we finish the proof by examining how φ acts on rank-one projections and rank-one effects. STEP 1:
First of all, the properties of φ imply φ ( A ) ∼ = φ ( A ∼ ) ( A ∈ E ( C )) , and B ∼ ⊆ A ∼ ⇐⇒ φ ( B ) ∼ ⊆ φ ( A ) ∼ ( A, B ∈ E ( C )) . Hence, it is straightforward from Lemma 2.1 that there exists a bijection g : [0 , → [0 ,
1] such that φ ( tI ) = g ( t ) I ( t ∈ [0 , . (20)Also, by Lemma 3.1 we easily infer φ ( P ( C )) = P ( C ) , thus, in particular, we get φ ( P c ) = φ ( P ∼ ) = φ ( P ) ∼ = φ ( P ) c ( P ∈ P ( C )) . By Theorem 1.1 we also obtain φ ( A ⊥ ) = φ ( A ) ⊥ ( A ∈ E ( C ) \ SC ( C )) . Now, we observe that φ preserves commutativity in both directions. Indeed we have the following forevery A, B ∈ E ( C ) \ SC ( C ): AB = BA ⇐⇒ A ∼ ∩ P ( C ) = B ∼ ∩ P ( C ) = { P, P ⊥ } for some P ∈ P ( C ) ⇐⇒ φ ( A ) ∼ ∩ P ( C ) = φ ( B ) ∼ ∩ P ( C ) = { Q, Q ⊥ } for some Q ∈ P ( C ) ⇐⇒ φ ( A ) φ ( B ) = φ ( B ) φ ( A ) . Note that we easily get the same conclusion using (20) if any of the two effects is a scalar effect.Next, notice that Lemma 3.2 implies A or A ⊥ ∈ F ( C ) \ P ( C ) ⇐⇒ φ ( A ) or φ ( A ) ⊥ ∈ F ( C ) \ P ( C ) . Therefore, by interchanging the φ -images of tP and I − tP for some 0 < t < P ∈ P ( C ), we mayassume without loss of generality that φ (cid:0) F ( C ) \ P ( C ) (cid:1) = F ( C ) \ P ( C ) . Hence we obtain the following for all rank-one projections P : φ (cid:0) { tP, tP ⊥ : 0 < t ≤ } (cid:1) = φ (cid:0) P c ∩ F ( C ) (cid:1) = φ ( P ) c ∩ F ( C ) = { tφ ( P ) , tφ ( P ) ⊥ : 0 < t ≤ } . Thus, again by interchanging the φ -images of P and P ⊥ for some P ∈ P ( C ), and using Lemma 3.2,we may assume without loss of generality that for every P ∈ P ( C ) there exists a strictly increasingbijective map f P : (0 , → (0 ,
1] such that φ ( tP ) = f P ( t ) φ ( P ) (0 < t ≤ , P ∈ P ( C )) . (21) STEP 2:
We define the following set for any qubit effect of the form tP , 0 < t < , P ∈ P ( C ): (cid:96) tP := (cid:40) − t (cid:107) P ⊥ − Q (cid:107) + (cid:107) P − Q (cid:107) Q : Q ∈ P ( C ) \ { P, P ⊥ } (cid:41) . (22)16For a visualisation of (cid:96) tP see Section 5.) Using Lemma 3.5 we see that (cid:96) tP = (cid:0) ( tP ) ∼ \ ∪{ ( sP ) ∼ : t < s < } (cid:1) ∩ F ( C ) (0 < t < , P ∈ P ( C )) . By the properties of φ we obtain φ ( (cid:96) tP ) = (cid:96) φ ( tP ) = (cid:96) f P ( t ) φ ( P ) (0 < t < , P ∈ P ( C )) . (23)Next, using the set introduced in (22), we prove the following property of φ : (cid:107) P − Q (cid:107) = sin π ⇐⇒ (cid:107) φ ( P ) − φ ( Q ) (cid:107) = sin π ( P, Q ∈ P ( C )) . (24)By a straightforward calculation we get that (cid:96) tP ∩ (cid:96) rP ⊥ = (cid:26) − t − t · s ( t, r ) Q : Q ∈ P ( C ) , (cid:107) P − Q (cid:107) = s ( t, r ) (cid:27) ( t, r ∈ (0 , , P ∈ P ( C ))where s ( t, r ) := (cid:115) t − tt − t + r − r . Note that s ( t, r ) = sin π holds if and only if t = r . By Lemma 3.4, this is further equivalent to thefollowing: ∀ A ∈ (cid:96) tP ∩ (cid:96) rP ⊥ , ∃ A ∈ (cid:96) tP ∩ (cid:96) rP ⊥ , A (cid:54) = A : ( A ) ∼ ∩ P ( C ) = ( A ) ∼ ∩ P ( C ) . Notice that by (23) this is equivalent to the following: ∀ B ∈ (cid:96) f P ( t ) φ ( P ) ∩ (cid:96) f P ⊥ ( r ) φ ( P ) ⊥ , ∃ B ∈ (cid:96) f P ( t ) φ ( P ) ∩ (cid:96) f P ⊥ ( r ) φ ( P ) ⊥ , B (cid:54) = B :( B ) ∼ ∩ P ( C ) = ( B ) ∼ ∩ P ( C ) , which is further equivalent to f P ( t ) = f P ⊥ ( r ).Hence we can conclude a few important properties of φ . First, we have f P ( t ) = f P ⊥ ( t ) (0 < t ≤ , P ∈ P ( C )) . Second, since for every 0 < t < P ∈ P ( C ) we have (cid:110) f Q (cid:16) − t − t/ (cid:17) φ ( Q ) : Q ∈ P ( C ) , (cid:107) P − Q (cid:107) = sin π (cid:111) = φ ( (cid:96) tP ∩ (cid:96) tP ⊥ )= (cid:96) f P ( t ) φ ( P ) ∩ (cid:96) f P ( t ) φ ( P ) ⊥ = (cid:110) − f P ( t )1 − f P ( t ) / R : R ∈ P ( C ) , (cid:107) φ ( P ) − R (cid:107) = sin π (cid:111) , therefore using (21) gives (24).Furthermore, we also obtain f Q (cid:16) − t − t/ (cid:17) = − f P ( t )1 − f P ( t ) / (cid:0) < t < , P, Q ∈ P ( C ) , (cid:107) P − Q (cid:107) = sin π (cid:1) . (25)By Lemma 3.6, for all Q , Q ∈ P ( C ) there exists a rank-one projection P such that (cid:107) Q − P (cid:107) = (cid:107) Q − P (cid:107) = sin π . Therefore, applying (25) and noticing that t (cid:55)→ − t − t/ is a strictly decreasing bijection of (0 ,
1) gives that f Q ( t ) = f Q ( t ) ( t ∈ (0 , , Q , Q ∈ P ( C )) . Thus we conclude that there exists a strictly increasing bijection f : (0 , → (0 ,
1] such that φ ( tP ) = f ( t ) φ ( P ) (0 < t ≤ , P ∈ P ( C )) . (26)We also observe that (25) implies f (cid:16) − t − t/ (cid:17) = − f ( t )1 − f ( t ) / , (27)17herefore we notice that f (cid:16) − √ (cid:17) = 2 − √ , (28)which is a consequence of the fact that the unique solution of the equation t = − t − t/ , 0 < t <
1, is t = 2 − √ STEP 3:
Next, applying [12, Theorem 2.3] gives that there exists a unitary or antiunitary operator U : C → C such that we have U ∗ φ ( P ) U ∈ { P, P ⊥ } ( P ∈ P ( C )) . Since either both U ∗ φ ( · ) U and φ ( · ) satisfy our assumptions simultaneously, or none of them does, thereforewithout loss of generality we may assume that we have φ ( P ) ∈ { P, P ⊥ } ( P ∈ P ( C )) . We now claim that either φ ( P ) = P ( P ∈ P ( C )) , or φ ( P ) = P ⊥ ( P ∈ P ( C )) (29)Let us assume otherwise, then there exist two rank-one projections P and Q such that (cid:107) P − Q (cid:107) < sin π , φ ( P ) = P and φ ( Q ) = Q ⊥ . Note that (cid:107) P − Q ⊥ (cid:107) = (cid:112) − (cid:107) P − Q (cid:107) > sin π > (cid:107) P − Q (cid:107) . By (23) and(28) we have (cid:110) √ − − (2 −√ (cid:107) P − R (cid:107) R : R ∈ P ( C ) \ { P, P ⊥ } (cid:111) = (cid:96) (2 −√ P = φ (cid:16) (cid:96) (2 −√ P (cid:17) = (cid:110) f (cid:16) √ − − (2 −√ (cid:107) P − R (cid:107) (cid:17) φ ( R ) : R ∈ P ( C ) \ { P, P ⊥ } (cid:111) . (30)Therefore putting first R = Q and then R = Q ⊥ gives φ (cid:16) √ − − (2 −√ (cid:107) P − Q (cid:107) Q (cid:17) = f (cid:16) √ − − (2 −√ (cid:107) P − Q (cid:107) (cid:17) φ ( Q )= f (cid:16) √ − − (2 −√ (cid:107) P − Q (cid:107) (cid:17) Q ⊥ = √ − − (2 −√ (cid:107) P − Q ⊥ (cid:107) Q ⊥ and φ (cid:16) √ − − (2 −√ (cid:107) P − Q ⊥ (cid:107) Q ⊥ (cid:17) = f (cid:16) √ − − (2 −√ (cid:107) P − Q ⊥ (cid:107) (cid:17) φ ( Q ⊥ )= f (cid:16) √ − − (2 −√ (cid:107) P − Q ⊥ (cid:107) (cid:17) Q = √ − − (2 −√ (cid:107) P − Q (cid:107) Q. But this implies that f interchanges two different numbers which contradicts to its strict increasingness– proving our claim (29).Note that for every 0 ≤ ϑ ≤ π and 0 ≤ µ < π we have( P (cos ϑ,e iµ sin ϑ ) ) ⊥ = (cid:20) sin ϑ − e − iµ cos ϑ sin ϑ − e iµ cos ϑ sin ϑ cos ϑ (cid:21) = (cid:20) − (cid:21) ( P (cos ϑ,e iµ sin ϑ ) ) t (cid:20) − (cid:21) ∗ where · t stands for the transposition, and we used the notation of the Busch–Gudder theorem. It iswell-known, and can be verified by an easy computation, that we have A t = KAK ∗ for every qubit effect A , where K is the coordinate-wise conjugation antiunitary operator: K ( z , z ) = ( z , z ) ( z , z ∈ C ).Therefore from now on we may assume without loss of generality that we have φ ( P ) = P ( P ∈ P ( C )) , (31)i.e. φ fixes all rank-one projections.Finally, observe that (30) and (31) implies f (cid:16) √ − − (2 −√ τ (cid:17) = √ − − (2 −√ τ (0 < τ < , thus we obtain φ ( tP ) = tP for all P ∈ P ( C ) and √ − < t <
1. But this further implies (cid:110) − t − t (cid:107) P − Q (cid:107) Q : Q ∈ P ( C ) \ { P, P ⊥ } (cid:111) = (cid:96) tP = φ ( (cid:96) tP ) = (cid:110) f (cid:16) − t − t (cid:107) P − Q (cid:107) (cid:17) Q : Q ∈ P ( C ) \ { P, P ⊥ } (cid:111) for all √ − < t <
1, from which we conclude φ ( tP ) = tP (0 < t < , (32)i.e. φ fixes all rank-one effects. From here we only need to apply Corollary 2.11 and transform back toour original φ to complete the proof. 18 Proof of Theorem 1.3 in the general case
Here we prove the general case of our main theorem, utilising the above proved low-dimensional case. Westart with two lemmas.
Lemma 4.1.
Let P ∈ P ( H ) \ SC ( H ) and A ∈ E ( H ) \ { P, P ⊥ } . Then there exists a rank-one effect R ∈ F ( H ) such that R ∼ A but R (cid:54)∼ P .Proof. Assume that A ∈ E ( H ) such that A ∼ ∩ F ( H ) ⊆ P ∼ = P c holds. We have to show that theneither A = P , or A = P ⊥ . Clearly, A is not a scalar effect. By Corollary 2.7 we obtain thatΛ( A, Q ) + Λ( A ⊥ , Q ) ≤ Λ( P, Q ) + Λ( P ⊥ , Q ) ( Q ∈ P ( H )) . Notice that the setsupp (cid:0) Λ( P, · ) + Λ( P ⊥ , · ) (cid:1) := (cid:8) Q ∈ P ( H ) : Λ( P, Q ) + Λ( P ⊥ , Q ) > (cid:9) has two connected components (with respect to the operator norm topology), namely { Q ∈ P ( H ) : Im Q ⊂ Im P } and { Q ∈ P ( H ) : Im Q ⊂ Ker P } . (33)However, by the Busch–Gudder theorem we obtain that { Q ∈ P ( H ) : Im Q ⊂ Im A ∪ Im ( I − A ) } ⊆ (cid:110) Q ∈ P ( H ) : Im Q ⊂ Im A / ∪ Im ( I − A ) / (cid:111) ⊆ supp (cid:0) Λ( A, · ) + Λ( A ⊥ , · ) (cid:1) ⊆ supp (cid:0) Λ( P, · ) + Λ( P ⊥ , · ) (cid:1) . Since supp (cid:0) Λ( P, · ) + Λ( P ⊥ , · ) (cid:1) is a closed set, we obtain (cid:8) Q ∈ P ( H ) : Im Q ⊂ (Im A ) − ∪ (Im ( I − A )) − (cid:9) = (cid:8) Q ∈ P ( H ) : Im Q ⊂ (Ker A ) ⊥ ∪ (Ker( I − A )) ⊥ (cid:9) ⊆ supp (cid:0) Λ( P, · ) + Λ( P ⊥ , · ) (cid:1) . (34)Notice that the left-hand side of (34) is connected if and only if A is not a projection, in which case itmust be a subset of one of the components of the right-hand side. However, this is impossible because theleft-hand side contains a maximal set of pairwise orthogonal rank-one projections. Therefore A ∈ P ( H ),and in particular supp (cid:0) Λ( A, · ) + Λ( A ⊥ , · ) (cid:1) has two connected components. From here using (33) for both A and P we easily complete the proof.We introduce a new relation on E ( H ) \ SC ( H ). For A, B ∈ E ( H ) \ SC ( H ) we write A ≺ B if andonly if for every C ∈ A ∼ \ SC ( H ) there exists a D ∈ B ∼ \ SC ( H ) such that C ∼ ⊆ D ∼ . Clearly, for everynon-scalar effect B we have B ≺ B and B ⊥ ≺ B . In particular ≺ is a reflexive relation, but it is notantisymmetric. It is also straightforward from the definition that ≺ is a transitive relation, i.e. A ≺ B and B ≺ C imply A ≺ C .We proceed with characterising non-trivial projections in terms of the relation of coexistence. Lemma 4.2.
Assume that A ∈ E ( H ) \ SC ( H ) . Then the following two statements are equivalent: (i) A ∈ P ( H ) , (ii) { B ∈ E ( H ) \ SC ( H ) : B ≺ A } = 2 .Proof. (i) = ⇒ (ii): Suppose that B ∈ E ( H ) \ SC ( H ), B (cid:54) = A , B (cid:54) = A ⊥ and B ≺ A . We need to show thatthis assumption leads to a contradiction. By Lemma 4.1 there exists a rank one effect tQ , with some Q ∈ P ( H ) and t ∈ (0 , tQ ∼ B but tQ (cid:54)∼ A . From B ≺ A we know that there exists anon-scalar effect D such that ( tQ ) ∼ ⊆ D ∼ and D ∼ A. By Lemma 2.12 (a) we have D ∈ ( tQ ) (cid:48)(cid:48) ∩ E ( H ) = Q (cid:48)(cid:48) ∩ E ( H ) = (cid:8) sQ + rQ ⊥ ∈ E ( H ) : s, r ∈ [0 , (cid:9) , where the latter equation is easy to see (even in non-separable Hilbert spaces). Since we also have D ∈ A c ,we obtain Q ∈ A c , hence the contradiction tQ ∈ A c = A ∼ .19 ii) = ⇒ (i): Here we use contraposition, so let us assume that A ∈ ( E ( H ) \ P ( H )) \ SC ( H ). We shallconstruct a non-trivial projection P (which is obviously different from both A and A ⊥ ) such that P ≺ A .First, notice that there exists an 0 < ε < such that H A (( ε, − ε ]) / ∈ {{ } , H } . Indeed, otherwise anelementary examination of the spectrum gives that σ ( A ) ⊆ { ε , − ε } holds with some 0 < ε < . As A is non-scalar, we actually get σ ( A ) = { ε , − ε } , which implies that H A (( ε , − ε ]) is a non-trivialsubspace.Let us now consider the orthogonal decomposition H = H ⊕ H ⊕ H where H = H A ([0 , ε ]) , H = H A (( ε, − ε ]) and H = H A ((1 − ε, . With respect to this orthogonal decomposition we have A = A A
00 0 A ∈ E ( H ⊕ H ⊕ H ) . Since coexistence is invariant under taking the ortho-complements, we may assume without loss of gen-erality that H (cid:54) = { } . Let us set P = I I
00 0 0 / ∈ SC ( H ⊕ H ⊕ H ) . Our goal is to show that P ≺ A . Let C be an arbitrary non-scalar effect coexistent with P . Then, since C and P commute, the matrix form of C is C = C C C ∗ C
00 0 C ∈ E ( H ⊕ H ⊕ H ) . Consider the effect D := ε · C and notice that ε · C C C ∗ C
00 0 0 ≤ I − A and ε · C ≤ A. Clearly, by Lemmas 2.3 and 2.12 we have D ∼ A and C ∼ ⊆ D ∼ , which completes the proof.Next, we characterise commutativity preservers on P ( H ). We note that the following theorem hasbeen proved before implicitly in [23] for separable spaces, and was stated explicitly in [24, Theorem 2.8].In order to prove the theorem for general spaces, one only has to use the ideas of [23], however, we decidedto include the proof for the sake of completeness and clarity. Theorem 4.3.
Let H be a Hilbert space of dimension at least three and φ : P ( H ) → P ( H ) be a bijectivemapping that preserves commutativity in both directions, i.e. P Q = QP ⇐⇒ φ ( P ) φ ( Q ) = φ ( Q ) φ ( P ) ( P, Q ∈ P ( H )) . (35) Then there exists a unitary or antiunitary operator U : H → H such that φ ( P ) ∈ { U P U ∗ , U P ⊥ U ∗ } ( P ∈ P ( H )) . Proof.
For an arbitrary set
M ⊆ P ( H ) let us use the following notations: M c := M c ∩ P ( H ) and M cc := ( M c ) c . By the properties of φ we immediately get φ ( M c ) = φ ( M ) c and φ ( M cc ) = φ ( M ) cc forall subset M .Next, let P and Q be two arbitrary commuting projections. Then (for instance by the Halmos’s twoprojections theorem) we have P = I I and Q = I I
00 0 0 0 ∈ B ( H ⊕ H ⊕ H ⊕ H )20here H = Im P ∩ Im Q , H = Im P ∩ Ker Q , H = Ker P ∩ Im Q , H = Ker P ∩ Ker Q and H = H ⊕ H ⊕ H ⊕ H . Note that some of these subspaces might be trivial. We observe that { P, Q } cc = ( { P, Q } c ) c = R R R
00 0 0 R : R j ∈ P ( H j ) , j = 1 , , , c = λ I λ I λ I
00 0 0 λ I : λ j ∈ { , } , j = 1 , , , . Hence we conclude that { P, Q } cc = 2 { j : H j (cid:54) = { }} . In particular, { P, Q } cc = 2 if and only if P, Q ∈{ , I } , and { P, Q } cc = 4 if and only if either P / ∈ { , I } and Q ∈ { I, , P, P ⊥ } , or Q / ∈ { , I } and P ∈ { I, , Q, Q ⊥ } .Now, we easily conclude the following characterisation of rank-one and co-rank-one projections: P or P ⊥ ∈ P ( H ) ⇐⇒ { P, Q } cc ∈ { , } holds for all Q ∈ P c . This implies that φ ( { P : P or P ⊥ ∈ P ( H ) } ) = { P : P or P ⊥ ∈ P ( H ) } . Note that we also have φ ( P ⊥ ) = φ ( P ) ⊥ for every P ∈ P ( H ), as P c = Q c holds exactly when P = Q or P + Q = I . Since changing the images of some pairs of ortho-complemented projections to theirorto-complementations does not change the property (35), we may assume without loss of generality that φ ( P ( H )) = P ( H ). It is easy to see that two rank-one projections commute if and only if either theycoincide, or they are orthogonal to each other. Thus, as dim H ≥
3, Uhlhorn’s theorem [32] gives thatthere exist a unitary or antiunitary operator U : H → H such that φ ( P ) = U P U ∗ ( P ∈ P ( H )) . Finally, note that for every projection Q ∈ P ( H ) we have Q c ∩ P ( H ) = { P ∈ P ( H ) : Im P ⊂ Im Q ∪ Ker Q } , from which we easily complete the proof.Before we prove Theorem 1.3 in the general case, we need one more technical lemma for non-separableHilbert spaces. We will use the notation E fs ( H ) for the set of all effects whose spectrum has finitely manyelements. Lemma 4.4.
For all A ∈ E fs ( H ) we have A cc = A (cid:48)(cid:48) ∩ E ( H ) = { p ( A ) ∈ E ( H ) : p is a polynomial } . Proof.
We only have to observe the following for all A ∈ E ( H ) with σ ( A ) = n ∈ N , where E , . . . E n are the spectral projections and H j = Im E j ( j = 1 , , . . . n ): A cc = n (cid:92) j =1 E cj c = n (cid:77) j =1 B j : B j ∈ E ( H j ) for all j c = n (cid:88) j =1 µ j E j : µ j ∈ [0 ,
1] for all j = n (cid:77) j =1 T j : T j ∈ B ( H j ) for all j (cid:48) ∩ E ( H ) = n (cid:92) j =1 E (cid:48) j (cid:48) ∩ E ( H ) = A (cid:48)(cid:48) ∩ E ( H ) . Now, we are in the position to prove our second main theorem in the general case.
Proof of Theorem 1.3 for spaces of dimension at least three.
The proof will be divided into the followingsteps:1 we show that φ maps E fs ( H ) onto itself, 21 we prove that φ has the form (2) on E fs ( H ) \ SC ( H ),3 we show that φ has the form (2) on E ( H ) \ SC ( H ). STEP 1:
First, similarly as in the previous section, we easily get the existence of a bijective function g : [0 , → [0 ,
1] such that φ ( tI ) = g ( t ) I ( t ∈ [0 , . Of course, the properties of φ imply φ ( A ) ∼ = φ ( A ∼ ) for all A ∈ E ( H ), and also B ∼ ⊆ A ∼ ⇐⇒ φ ( B ) ∼ ⊆ φ ( A ) ∼ ( A, B ∈ E ( H )) . (36)From the latter it follows that B ≺ A ⇐⇒ φ ( B ) ≺ φ ( A ) ( A, B ∈ E ( H ) \ SC ( H )) . (37)Hence by Lemma 4.2 we obtain φ ( P ( H ) \ { , I } ) = P ( H ) \ { , I } , and therefore Lemma 2.1 (b) implies that the restriction φ | P ( H ) \{ ,I } preserves commutativity in bothdirections. Applying Theorem 4.3 then gives that up to unitary–antiunitary equivalence and element-wiseortho-complementation, we have φ ( P ) = P ( P ∈ P ( H ) \ { , I } ) . (38)From now on we may assume without loss of generality that this is the case.Next, by the spectral theorem [7, Theorem IX.2.2] we have A c = (cid:92) ∆ ∈B [0 , E A (∆) c = (cid:92) ∆ ∈B [0 , E A (∆) ∼ ( A ∈ E ( H )) . Therefore we obtain φ ( A c ) = (cid:92) ∆ ∈B [0 , φ ( E A (∆)) ∼ = (cid:92) ∆ ∈B [0 , E A (∆) ∼ = A c ( A ∈ E ( H )) , and thus also φ ( A cc ) = φ (cid:32) (cid:92) B ∈ A c B c (cid:33) = (cid:92) B ∈ A c φ ( B c ) = (cid:92) B ∈ A c B c = A cc ( A ∈ E ( H )) . In particular, we have φ ( A ) ∈ A cc ( A ∈ E ( H )) . Hence for all A ∈ E fs ( H ) there exists a polynomial p A such that p A ( σ ( A )) ⊂ [0 ,
1] and φ ( A ) = p A ( A ) ( A ∈ E fs ( H )) . As a similar statement holds for φ − , we immediately get φ ( E fs ( H )) = E fs ( H ). Also, notice that σ ( φ ( A )) = σ ( p A ( A )) ≤ σ ( A ) and σ ( φ − ( A )) ≤ σ ( A ) hold for all A ∈ E fs ( H ). Whence weobtain σ ( φ ( A )) = σ ( A ) ( A ∈ E fs ( H )) . (39)In particular, the restriction p A | σ ( A ) is injective. STEP 2:
Now, let M be an arbitrary two-dimensional subspace of H and let P M ∈ P ( H ) be theorthogonal projection onto M . Consider two arbitrary effects A, B ∈ ( P M ) ∼ ∩ E fs ( H ) which thereforehave the following matrix representations: A = (cid:20) A M A M ⊥ (cid:21) and B = (cid:20) B M B M ⊥ (cid:21) ∈ E fs ( M ⊕ M ⊥ ) . Obviously, φ ( A ) = p A ( A ) = (cid:20) p A ( A M ) 00 p A ( A M ⊥ ) (cid:21) and φ ( B ) = p B ( B ) = (cid:20) p B ( B M ) 00 p B ( B M ⊥ ) (cid:21) . p A acts injectively on σ ( A ), therefore A M ∈ SC ( M ) ⇐⇒ p A ( A M ) ∈ SC ( M ) , and of course, similarly for B . We observe that by Lemma 2.2 the following two equations hold: A ∼ (cid:92) (cid:20) I
00 0 (cid:21) ∼ (cid:92) (cid:92) P ∈P ( M ⊥ ) (cid:20) P (cid:21) ∼ = (cid:26)(cid:20) D λI (cid:21) : D ∼ A M , λ ∈ [0 , (cid:27) (40)and φ ( A ) ∼ (cid:92) (cid:20) I
00 0 (cid:21) ∼ (cid:92) (cid:92) P ∈P ( M ⊥ ) (cid:20) P (cid:21) ∼ = (cid:26)(cid:20) D λI (cid:21) : D ∼ p A ( A M ) , λ ∈ [0 , (cid:27) . (41)It is important to observe that by (38) the set in (41) is the φ -image of (40). Thus we obtain the followingequivalence if A M / ∈ SC ( M ): B M ∈ { A M , A ⊥ M } ⇐⇒ A ∼ M = B ∼ M ⇐⇒ (cid:26)(cid:20) D λI (cid:21) : D ∼ A M , λ ∈ [0 , (cid:27) = (cid:26)(cid:20) E µI (cid:21) : E ∼ B M , µ ∈ [0 , (cid:27) ⇐⇒ (cid:26)(cid:20) D λI (cid:21) : D ∼ p A ( A M ) , λ ∈ [0 , (cid:27) = (cid:26)(cid:20) E µI (cid:21) : E ∼ p B ( B M ) , µ ∈ [0 , (cid:27) ⇐⇒ ( p A ( A M )) ∼ = ( p B ( B M )) ∼ ⇐⇒ p B ( B M ) ∈ { p A ( A M ) , I − p A ( A M ) } . (42)Now, we are in the position to use the previously proved two-dimensional version. Let E ( M ) := (cid:8)(cid:8) D, D ⊥ (cid:9) : D ∈ E ( M ) \ SC ( M ) (cid:9) ∪ {SC ( M ) } , and let us say that two elements of E ( M ) are coexistent, in notation ≈ , if either one of them is SC ( M ),or the two elements are { D, D ⊥ } and { E, E ⊥ } with D ∼ E . Clearly, the bijective restriction φ | ( P M ) ∼ ∩E fs ( H ) : ( P M ) ∼ ∩ E fs ( H ) → ( P M ) ∼ ∩ E fs ( H )induces a well-defined bijection on E ( M ) by SC ( M ) (cid:55)→ SC ( M ) , { A M , A ⊥ M } (cid:55)→ { p A ( A M ) , p A ( A M ) ⊥ } ( A M / ∈ SC ( M )) . Notice that this map also preserves the relation ≈ in both directions. Indeed, for all A, B ∈ ( P M ) ∼ ∩E fs ( H ), A M , B M / ∈ SC ( H ) we have { A M , A ⊥ M } ≈ { B M , B ⊥ M } ⇐⇒ ˆ A := (cid:20) A M
00 0 (cid:21) ∼ ˆ B := (cid:20) B M
00 0 (cid:21) ⇐⇒ (cid:20) p ˆ A ( A M ) 00 p ˆ A (0) I (cid:21) ∼ (cid:20) p ˆ B ( B M ) 00 p ˆ B (0) I (cid:21) ⇐⇒ { p ˆ A ( A M ) , p ˆ A ( A M ) ⊥ } ≈ { p ˆ B ( B M ) , p ˆ B ( B M ) ⊥ }⇐⇒ { p A ( A M ) , p A ( A M ) ⊥ } ≈ { p B ( B M ) , p B ( B M ) ⊥ } . Therefore, using the two-dimensional version of Theorem 1.3, we obtain a unitary or antiunitary operator U M : M → M such that p A ( A M ) ∈ { U M ( A M ) U ∗ M , U M ( A M ) ⊥ U ∗ M } ( A ∈ ( P M ) ∼ ∩ E fs ( H ) , A M / ∈ SC ( M ))and p A ( A M ) ∈ SC ( M ) ( A ∈ ( P M ) ∼ ∩ E fs ( H ) , A M ∈ SC ( M )) . Observe that this implies the following: for any pair of orthogonal unit vectors x, y ∈ M we musthave either U M ( C · x ) = C · x and U M ( C · y ) = C · y , or U M ( C · x ) = C · y and U M ( C · y ) = C · x . As U M is continuous, we have either the first case for all orthogonal pairs C · x, C · y , or the second for every23uch pair. But a similar statement holds for all two-dimensional subspaces, therefore it is easy to showthat the second possibility cannot occur. Consequently, we have U M ( C · x ) = C · x for all unit vectors x ∈ M , from which it follows that U M is a scalar multiple of the identity operator. Thus we obtain thefollowing for every two-dimensional subspace M : p A ( A M ) ∈ { A M , A ⊥ M } ( A ∈ ( P M ) ∼ ∩ E fs ( H ) , A M / ∈ SC ( M )) . From here it is rather straightforward to obtain φ ( A ) = p A ( A ) ∈ { A, A ⊥ } ( A ∈ E fs ( H ) \ SC ( M )) . (43) STEP 3:
Observe that (43) holds for every A ∈ F ( H ), therefore an application of Theorem 1.1and Corollary 2.5 completes the proof in the separable case. As for the general case, let us consider anarbitrary effect A ∈ E ( H ) \ E fs ( H ) and an orthogonal decomposition H = ⊕ i ∈I H i such that each H i isa separable invariant subspace of A . By (38) and Lemma 2.1 (b), each H i is an invariant subspace alsofor φ ( A ), in particular, we have A = ⊕ i ∈I A i , and φ ( A ) = ⊕ i ∈I A i ∈ E ( ⊕ i ∈I H i ) . Without loss of generality we may assume from now on that there exists an i ∈ I so that A i is not ascalar effect.Now, let i ∈ I , F ∈ F ( H ) and Im F ⊆ H i be arbitrary. Then by (43) we have A i ∼ P i F | H i ⇐⇒ A ∼ F ⇐⇒ φ ( A ) ∼ F ⇐⇒ A i ∼ P i F | H i . In particular, A ∼ i ∩ F ( H i ) = A ∼ i ∩ F ( H i ), therefore by Theorem 1.1 we get that for all i we have either A i , A i ∈ SC ( H ), or A i = A i , or A i = A ⊥ i . By considering A ⊥ instead of A if necessary, we may assumethat we have A i = A i . Finally, for any i ∈ I \ { i } let us consider the orthogonal decomposition H = ⊕ i ∈I\{ i ,i } H i ⊕ ( H i ⊕ H i ). Similarly as above, we then get A i ⊕ A i = A i ⊕ A i , and the proofis complete. Here we visualise the set A ∼ ∩F ( C ) for a general rank-one qubit effect A . First, let us introduce Bloch’srepresentation. Consider the following vector space isomorphism between the space of all 2 × B sa ( C ) and R , see also [4]: ρ : B sa ( C ) → R , ρ ( A ) = ρ ( x σ + x σ + x σ + x σ ) = ( x , x , x , x ) , where σ = (cid:20) (cid:21) , σ = (cid:20) (cid:21) , σ = (cid:20) − ii (cid:21) , σ = (cid:20) − (cid:21) are the Pauli matrices. Clearly, we have ρ (0) = (0 , , , ρ ( I ) = (1 , , , ρ | P ( C ) which maps P ( C ) onto a sphere of the three-dimensionalaffine subspace { (1 / , x , x , x ) : x j ∈ R , j = 1 , , } with centre at (1 / , , ,
0) and radius 1 /
2. Indeed,as the general form of a rank-one projection in C is P (cos ϑ,e iµ sin ϑ ) = (cid:20) cos ϑe iµ sin ϑ (cid:21) (cid:20) cos ϑe iµ sin ϑ (cid:21) ∗ = (cid:20) cos ϑ e − iµ cos ϑ sin ϑe iµ cos ϑ sin ϑ sin ϑ (cid:21) where 0 ≤ ϑ ≤ π and 0 ≤ µ < π , a not too hard calculation gives that ρ ( P (cos ϑ,e iµ sin ϑ ) ) = · (1 , cos µ sin 2 ϑ, sin µ sin 2 ϑ, cos 2 ϑ ) . (44)Recall the remarkable angle doubling property of the Bloch representation, namely, we have (cid:107) P − Q (cid:107) =sin θ if and only if the angle between the vectors ρ ( P ) − e and ρ ( Q ) − e is exactly 2 θ .Next, we call a positive (semi-definite) element of B sa ( C ) a density matrix if its trace is 1, or inother words, if it is a convex combination of some rank-one projections. Therefore ρ maps the set of all2 × { (1 / , x , x , x ) : x j ∈ R , j = 1 , , } with centre at (1 / , , ,
0) and radius 1 /
2. Hence, we see that the cone of all positive (semi-definite) 2 × , , ,
0) and the aforementioned24igure 2: Illustration of ρ ( E ( C )) ∩ S µ . The circle is ρ ( P ( C )) ∩ S µ .ball. Thus ρ maps E ( C ) onto the intersection of this cone and its reflection through the point ρ ( I ) =( , , , ρ ( P (cos ϑ,e iµ sin ϑ ) ) = · ( e + sin 2 ϑ · e µ + cos 2 ϑ · e ) , (45)where e := (1 , , , , e µ := (0 , cos µ, sin µ, , e := (0 , , , R . Let S µ be the three-dimensional subspace spanned by e , e µ , e . Thenthe set ρ ( E ( C )) ∩ S µ can be visualised as a double cone of R , by regarding e , e µ , e as the standardbasis of R , see Figure 2. Note that ρ ( P ( C )) ∩ S µ is the circle where the boundaries of the two conesmeet.We continue with visualising the set ( tP (1 , ) ∼ for an arbitrary 0 < t <
1. Note that then visualising( tP ) ∼ for a general rank-one projection P is very similar, we simply have to apply a unitary similarity(which by well-known properties of the Bloch representation, acts as a rotation on the sphere ρ ( P ( C ))).Equation (9) gives the following:Λ (cid:0) tP (1 , , P (cos ϑ,e iµ sin ϑ ) (cid:1) + Λ (cid:0) I − tP (1 , , P (cos ϑ,e iµ sin ϑ ) (cid:1) = 1 t cos ϑ + (cid:0) (cid:1) sin ϑ + 1 − t cos ϑ + sin ϑ = (cid:40) − t cos ϑ +sin ϑ if ϑ >
01 if ϑ = 0 . (46)Now, let us consider the vector u = (2 − t ) · e + t · e , which is orthogonal to ρ (cid:16) (1 − t ) P (1 , − P ⊥ (1 , (cid:17) = − [ t · e + ( t − · e ] . From here a bit tedious computation gives (cid:42) u, − t cos ϑ + sin ϑ · ρ (cid:0) P (cos ϑ,e iµ sin ϑ ) (cid:1) − ρ (cid:16) P ⊥ (1 , (cid:17)(cid:43) = 0 (0 ≤ ϑ ≤ π ) . ρ (cid:0) ( tP (1 , ) ∼ ∩ F ( C ) (cid:1) is the union of the linesegment { ρ (cid:0) sP (1 , (cid:1) : 0 < s ≤ } = { s e + s e : 0 < s ≤ } and of the area on the boundary of ρ ( E ( C ))which is either on, or below the affine hyperplane whose normal vector is u and which contains ρ ( P ⊥ (1 , ),see Figure 3. We note that using the notation of (22), the ellipse on the boundary is exactly the set (cid:16) ρ ( (cid:96) tP (1 , ) ∪ (cid:8) (1 − t ) · ρ ( P (1 , ) , ρ ( P ⊥ (1 , ) (cid:9)(cid:17) ∩ S µ . Therefore ρ ( (cid:96) tP (1 , ) is a punctured ellipsoid.Figure 3: Illustration of ρ (cid:0) ( tP (1 , ) ∼ (cid:1) ∩ ρ (cid:0) F ( C ) (cid:1) ∩ S µ (thick ellipse, thick line segment and the shadedarea). The dotted circle is ρ ( P ( C )) ∩ S µ .If one illustrates the set ρ (( A ) ∼ ) ∩ ρ (cid:0) F ( C ) (cid:1) ∩ S µ with A, A ⊥ / ∈ SC ( C ) ∪ F ( C ) in the way asabove, then one gets a set on the boundary of the cone which is bounded by a continuous closed curvecontaining the ρ -images of the spectral projections. First, we prove the analogue of Lemma 3.1 for finite dimensional spaces of dimension at least three.
Lemma 6.1.
Let H be a Hilbert space with ≤ dim H < ∞ and A ∈ E ( H ) . Then the following areequivalent: (i) 0 , ∈ σ ( A ) , (ii) there exists no effect B ∈ E ( H ) such that B ∼ (cid:40) A ∼ .Proof. If dim H = 2, then (i) ⇐⇒ (ii) was proved in Lemma 3.1, so from now on we will assume2 < dim H < ∞ . Also, as the case when A ∈ SC ( H ) is trivial, we assume otherwise throughout theproof. (i) = ⇒ (ii): Suppose that 0 , ∈ σ ( A ) and consider an arbitrary effect B with B ∼ ⊆ A ∼ . By Lemma2.12, A and B commute. If 0 = λ ≤ λ ≤ · · · ≤ λ n − ≤ λ n = 1 are the eigenvalues of A , then thematrices of A and B written in an orthonormal basis of joint eigenvectors are the following: A = . . . λ . . . . . . λ n −
00 0 . . . and B = µ . . . µ . . . . . . µ n −
00 0 . . . µ n with some µ , . . . µ n ∈ [0 , ≤ i < j ≤ n we have (cid:20) µ i µ j (cid:21) ∼ ⊆ (cid:20) λ i λ j (cid:21) ∼ . (47)26n particular, choosing i = 1 , j = n implies either µ = 0 and µ n = 1, or µ = 1 and µ n = 0. Assumethe first case. If we set i = 1, then Lemma 3.2 and (47) imply µ j ≥ λ j for all j = 2 , . . . , n −
1. But onthe other hand, setting j = n implies µ i ≤ λ i for all i = 2 , . . . , n −
1. Therefore we conclude B = A .Similarly, assuming the second case implies B = A ⊥ . (ii) = ⇒ (i): Assume (i) does not hold, then there exists a positive number ε such that σ ( A ) ⊆ [0 , − ε ]or σ ( A ⊥ ) ⊆ [0 , − ε ]. Suppose the first possibility holds, then − ε A / ∈ {
A, A ⊥ } and (cid:16) − ε A (cid:17) ∼ ⊆ A ∼ .The second case is very similar.We only proved the above lemma and Corollary 2.11 in the finite dimensional case. The following twoquestions would be interesting to examine: Question 6.2.
Does the statement of Corollary 2.11 remain true for general infinite dimensional Hilbertspaces?
Question 6.3.
Does the statement of Lemma 6.1 hold if dim H ≥ ℵ ? Finally, our first main theorem characterises completely when A ∼ = B ∼ happens for two effects A and B . However, we gave only some partial results about when A ∼ ⊆ B ∼ occurs, e.g. Lemma 2.12. Question 6.4.
How can we characterise the relation A ∼ ⊆ B ∼ for effects A, B ? We believe that a complete answer to this latter question would represent a substantial step towardsthe better understanding of coexistence.
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