Coexistency on Hilbert space effect algebras and a characterisation of its symmetry transformations

The Hilbert space effect algebra is a fundamental mathematical structure which is used to describe unsharp quantum measurements in Ludwig's formulation of quantum mechanics. Each effect represents a quantum (fuzzy) event. The relation of coexistence plays an important role in this theory, as it expresses when two quantum events can be measured together by applying a suitable apparatus. 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 all automorphisms of the effect algebra with respect to the relation of coexistence. In particular, we will see that they can differ quite a lot from usual standard automorphisms, which appear for instance in Ludwig's theorem. As a byproduct of our methods we also strengthen a theorem of Molnar.


On the classical mathematical formulation of quantum mechanics
Throughout this paper H will denote a complex, not necessarily separable, Hilbert space with dimension at least 2. In the classical mathematical formulation of quantum mechanics such a space is used to describe experiments at the atomic scale. For instance, the famous Stern-Gerlach experiment (which was one of the firsts showing the reality of the quantum spin) can be described using the two-dimensional Hilbert space C 2 . In the classical formulation of quantum mechanics, the space of all rank-one projections P 1 (H) plays an important role, as its elements represent so-called quantum pure-states (in particular in the Stern-Gerlach experiment they represent the quantum spin). The so-called transition probability between two pure states P, Q ∈ P 1 (H) is the number trP Q, where tr denotes the trace. For the physical meaning of this quantity we refer the interested reader to e.g. [33]. A very important cornerstone of the mathematical foundations of quantum mechanics is Wigner's theorem, which states the following.
Wigner's Theorem. Given a bijective map φ : P 1 (H) → P 1 (H) that preserves the transition probability, i.e. trφ(P )φ(Q) = trP Q for all P, Q ∈ P 1 (H), one can always find either a unitary, or an antiunitary operator U : H → H that implements φ, i.e. we have φ(P ) = U P U * for all P ∈ P 1 (H).
For an elementary proof see [11]. As explained thoroughly by Simon in [29], this theorem plays a crucial role (together with Stone's theorem and some representation theory) in obtaining the general time-dependent Schrödinger equation that describes quantum systems evolving in time (and which is usually written in the form i d dt |Ψ(t) =Ĥ|Ψ(t) , where is the reduced Planck constant,Ĥ is the Hamiltonian operator, and |Ψ(t) 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 classical formulation an observable (such as the position/momentum of a particle, or a component of a particle's To illustrate how poorly the relation of coexistence is understood, we note that the following very natural 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: Moreover, if H is separable, then the above statements are also equivalent to Physically speaking, the above theorem says that the (unsharp) quantum events A and B can be measured together with exactly the same quantum events if and only if they are the same, or they are each other's negation, or both of them are scalar effects.

Automorphisms of E(H) with respect to two relations
Automorphisms of mathematical structures related to quantum mechanics are important to study because they provide the right tool to understand the time-evolution of certain quantum systems (see e.g. [18, 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 Obviously, standard automorphisms are automorphisms with respect to the relations of order: One of the fundamental theorems in the mathematical foundations of quantum mechanics states that every ortho-order automorphism is a standard automorphism, which was first stated by Ludwig.
We note that Ludwig's proof was incomplete and that he formulated his theorem under the additional assumption that dim H ≥ 3. The reader can find a rigorous proof of this version for instance in [5]. Let us 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 replaces either (≤) 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 are coexistent with every effect (see Section 2), this problem basically reduces to the characterisation of automorphisms with respect to coexistence only -which we shall consider later on.
In 2001, Molnár answered the other question affirmatively under the assumption that dim H ≥ 3.
In this paper we shall prove the two-dimensional version of Molnár's theorem.
Note that Molnár used the fundamental theorem of projective geometry to prove the aforementioned result, 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 Molnár's theorem that does not use this dimensionality constraint, hence fill this dimensionality gap in. More precisely, we will reduce Molnár's theorem and Theorem 1.2 to Ludwig's theorem (see the end of Section 2).

Automorphisms of E(H) with respect to only one relation
It is certainly a much more difficult problem to describe the general form of automorphisms with respect to 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 (⊥). It has been known for quite some time that automorphisms with respect to the order relation on E(H) may differ a lot from standard automorphisms, although they are at least always continuous with respect to the operator norm. We do not state the related result here, but only mention that the answer finally has been 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 also differ a lot from standard automorphisms, moreover, unlike in the case of (≤) they are in general not even continuous. and 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 respect to the operator norm, then up to unitary-antiunitary equivalence we obtain that φ is either the identity map, or the ortho-complementation: A → A ⊥ . Also note that the converse statement of the theorem follows easily by Theorem 1.1. As we mentioned earlier, the description of all automorphisms with respect to (∼) and (⊥) now follows easily, namely, we get the same conclusion as in the above theorem, except that now g further satisfies g(1 − t) = 1 − g(t) for all 0 ≤ t ≤ 1.

Quantum mechanical interpretation of automorphisms of E(H)
In order to explain the above automorphism theorems' physical interpretation, let us go back first to Wigner's theorem. Assume there are two physicists who analyse the same quantum mechanical system using the same Hilbert space H, but possibly they might associate different rank-one projections to the same 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 with which 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 same quantum 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 which effect is larger than the other (i.e. implies the occurrence of the other), then by Ludwig's theorem there must exist either a unitary, or an antiunitary operator that gives us the way to transform from one analysis into the other.
As for the interpretation of our Theorem 1.3, if we only know that our physicists agree on which pairs of effects are coexistent (i.e. which pairs of quantum events can be measured together), then there is a map φ satisfying (2) and (3) that transforms the first physicist's analysis into the other's.

The outline of the paper
In the next section we will prove our first main result, Theorem 1.1, and as an application, we prove Molnár's theorem in an alternative way that works for qubit effects as well. This will be followed by Section 3 where we prove our other main result, Theorem 1.3, in the case when dim H = 2. Then in Section 4, using the two-dimensional case, we shall prove the general version of our result. Let us point out once more that, unless otherwise stated, H is not assumed to be separable. We will close our paper with some discussion on the qubit case and some open problems in Sections 5-6.
2 Proofs of Theorems 1.1, 1.2, and Molnár's theorem We start with some definitions. The symbol P(H) will stand for the set of all projections (idempotent and self-adjoint operators) on H, and P 1 (H) will denote the set of all rank-one projections. The commutant of an effect A intersected with E(H) will be denoted by and more generally, for a subset M ⊂ E(H) we will use the notation M c := ∩{A c : A ∈ M}. Also, we set 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 be found 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: 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 2.7. Assume that A ∈ E(H), 0 < t ≤ 1, and P ∈ P 1 (H). Then the following conditions are equivalent: Proof. By (ii) of Lemma 2.3 we have A ∼ tP if and only if there exist t 1 , t 2 ≥ 0 such that t = t 1 + t 2 , t 1 P ≤ A and t 2 P ≤ A ⊥ , which is of course equivalent to (4).
Assume that with respect to the orthogonal decomposition H = H 1 ⊕ H 2 the two effects have the following block-diagonal matrix forms: Then we also have In particular, if Proof. Let P 1 be the orthogonal projection onto H 1 . By Lemma 2.2 we observe that which immediately implies (5).
Next, we recall the Busch-Gudder theorem about the explicit form of the strength function, which we shall use frequently here. We also adopt their notation, so whenever it is important to emphasise that the range of a rank-one projection P is C · x with some x ∈ H such that x = 1, we write P x instead. Furthermore, the symbol A −1/2 denotes the algebraic inverse of the bijective restriction A 1/2 | (Im A) − : (Im A) − → Im (A 1/2 ), where · − stands for the closure of a set. In particular, for all x ∈ Im (A 1/2 ) the vector A −1/2 x is the unique element in (Im A) − which A 1/2 maps to x.
Busch-Gudder Theorem (1999, Theorem 4 in [2]). For every effect A ∈ E(H) and unit vector x ∈ H we have 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 it here. 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 SOTand norm-closed. Moreover, if H is separable, then the subset Next, we prove that A ∼ is WOT-compact. Clearly, E(H) is WOT-compact, as it is a bounded WOTclosed 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. For every 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. Concerning our last statement for separable spaces, first we point out that for every effect C there exists a net of finite rank effects {C ν } ν such that C ν ≤ C holds for all ν and C ν → C in SOT. Denote by E C the projection-valued spectral measure of C, and set C n = n j=0 j n E C j n , j+1 n for every n ∈ N. Clearly, each C n has finite spectrum, satisfies C n ≤ C, and C n − C → 0 as n → ∞. For each spectral projection E C j n , j+1 n we can take a sequence of finite-rank projections {P j,n k } ∞ k=1 such that P j,n k ≤ E C j n , j+1 n for all k and P j,n k → E C j n , j+1 n in SOT as k → ∞. Define C n,k := n j=0 j n P j,n k . 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 a net {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 conditions of n holds for all k, n. Then each element C ν of the convergent net is an orthogonal sum of the form We proceed to investigate when do we have the equation A ∼ = B ∼ for two effects A and B, which will take several steps. We will denote the set of all rank-one effects by Lemma 2.10. Let H = H 1 ⊕ H 2 be an orthogonal decomposition and assume that A, B ∈ E(H) have the following matrix decompositions: where λ 1 , λ 2 , µ 1 , µ 2 ∈ [0, 1], and I 1 and I 2 denote the identity operators on H 1 and H 2 , respectively. Then the following are equivalent: (iii) either λ 1 = λ 2 and µ 1 = µ 2 , or λ 1 = µ 1 and λ 2 = µ 2 , or λ 1 + µ 1 = λ 2 + µ 2 = 1.
Proof. The directions (iii)=⇒(ii) ⇐⇒ (i) are trivial by Lemma 2.1 (a) and Corollaries 2.5, 2.7, so we shall only consider the direction (i)=⇒(iii). First, a straightforward calculation using the Busch-Gudder theorem gives the following for every where we use the interpretations 1 0 = ∞, 1 ∞ = 0, ∞ · 0 = 0, ∞ + ∞ = ∞, and ∞ + a = ∞, ∞ · a = ∞ (a > 0), in order to make the formula valid also for the case when λ 1 = 0 or λ 2 = 0. Clearly, (8) depends only on α, but not on the specific choices of x 1 and x 2 . We define the following two functions which are the same by our assumptions. By (8), for all 0 ≤ α ≤ π 2 we have Next, we observe the following implications: • if λ 1 = 0 and λ 2 = 1, then T A (α) is the characteristic function χ {0,π/2} (α), • if λ 1 = 0 and 0 < λ 2 < 1, then T A (α) is continuous on 0, π 2 , but has a jump at π 2 , namely lim α→ π 2 − T A (α) = 1 − λ 2 and T A ( π 2 ) = 1, • if λ 1 = 1 and 0 < λ 2 < 1, then T A (α) is continuous on 0, π 2 , but has a jump at π 2 , namely lim α→ π 2 − T A (α) = λ 2 and T A ( π 2 ) = 1, All of the above statements are rather straightforward computations using the formula (9), let us only show the last one here. Clearly, T A (0) = T A ( π 2 ) = 1 is obvious. As for the other assertion, if λ 1 , λ 2 ∈ (0, 1), then we can use the strict version of the weighted harmonic-arithmetic mean inequality: If λ 1 = 0 < λ 2 < 1, then we calculate in the following way: The remaining cases are very similar. The above observations together with Corollary 2.5 and (9) readily imply the following: • 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 1 , λ 2 ∈ (0, 1) and λ 1 = λ 2 if and only if µ 1 , µ 2 ∈ (0, 1) and So what remained is to show that in the last case we further have B ∈ {A, A ⊥ }, which is what we shall do below. Let us introduce the following functions: Our aim is to prove that Therefore, if we managed to show that the function . For this assume that with some c, d > 0 we have If we substitute u = x−y 2 and v = x+y 2 , then we get Now, considering the sum and difference of these two equations and manipulate them a bit gives From these latter equations we conclude 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 2 ≤ dim H < ∞ and A, B ∈ E(H). Then the following are equivalent: 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 1 (H) = F 1 (H) if and only if A is a scalar effect (see the first set of bullet points there). Therefore, without loss of 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) with some one-dimensional subspaces H 1 and H 2 , 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 } n j=1 , where n = dim H: Of course, for any two distinct i, j ∈ {1, . . . , n} we have the following equation for the strength functions: which instantly implies 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 = 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). Lemma 2.12. For any A, B ∈ E(H) the following three assertions hold: Proof. (a): Assume that C ∈ A . Our goal is to show B ∈ C . We express C in the following way: where C and C are self-adjoint (they are usually called the real and imaginary parts of C). Since A is self-adjoint, C * ∈ A , hence C , C ∈ A . Let E and E denote the projection-valued spectral measures of C and C , respectively.  Proof of Theorem 1.1. If H is separable, then the equivalence (ii) ⇐⇒ (iii) is straightforward by Lemma 2.9. For general H the direction (i)=⇒(ii) is obvious, therefore we shall only prove (ii)=⇒(i), first in the separable, and then in the general case. By Lemma 2.1, we may assume throughout the rest of the proof that 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 on certain subspaces of H. Let λ 1 , λ 2 ∈ σ(A), λ 1 = λ 2 and 0 < ε < 1 2 |λ 1 − λ 2 |. Then the spectral subspaces H 1 = H A ((λ 1 − ε, λ 1 + ε)) and H 2 = H A ((λ 2 − ε, λ 2 + ε)) are non-trivial and orthogonal. Set H 3 to be the orthogonal complement of H 1 ⊕ H 2 , then the matrix of A written in the orthogonal decomposition Note that H 3 might be a trivial subspace. Since by Corollary 2.4 A and B commute with exactly the same projections, the matrix of At this point, let us emphasise that of course H j , A j and B j (j = 1, 2, 3) all depend on λ 1 , λ 2 and ε, but in order to keep our notation as simple as possible, we will stick with these symbols. However, if at any point it becomes important to point out this dependence, we shall use for instance B (λ1,λ2,ε) j instead of B j . Similar conventions apply later on.
Observe that by Corollary 2.8 we have Now, we pick two arbitrary points µ 1 ∈ σ(B 1 ) and µ 2 ∈ σ(B 2 ). Then obviously, the following two subspaces are non-zero subspaces of H 1 and H 2 , respectively: Similarly as above, we have the following matrix forms whereȞ j = H j H j (j = 1, 2): and Note thatȞ 1 orȞ 2 might be trivial subspaces. Again by Corollary 2.8, we have Let us point out that by construction gives the following identity for the strength functions, where I j denotes the identity on H j (j = 1, 2): Define and notice that we have the following two estimations for all rank-one projections P : and Note that the above estimations hold for any arbitrarily small ε and for all suitable choices of µ 1 and µ 2 (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 : Notice that as this set is an intersection of monotonically decreasing (as ε 0), compact, non-empty sets, it must contain at least one element. Also, observe that if µ 1 ∈ C 1 and µ 2 ∈ C 2 , then (11) and (12) hold for all ε > 0.
Observe that as we can do the above for any two disjoint elements of the spectrum σ(A), we can conclude that one of the following possibilities occur: or From here, we show that (13) implies A = B, and (14) implies B = A ⊥ . As the latter can be reduced to the case (13), by considering B ⊥ instead of B, we may assume without loss of generality that (13) (13) we notice that for every λ ∈ σ(A) there exists an 0 < ε λ < δ such that 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)} cover the whole spectrum σ(A), which is a compact set. Therefore we can find finitely many of them, let's say λ 1 , . . . , λ n so that Finally, we define the function h(λ) = λ j , where |λ − λ i | ≥ ε λi for all 1 ≤ i < j and |λ − λ j | < ε λj .
(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 with respect to the decomposition H = ⊕ i∈I H i : Without loss of generality we may assume from now on that there exists an i 0 ∈ I so that A i0 is not a scalar effect. (In case all of them are scalar, we simply combine two subspaces H i1 and H i2 so that σ(A i1 ) = σ(A i2 )). This implies either A i0 = B i0 , or B i0 = A ⊥ i0 . By considering B ⊥ instead of B if necessary, we may assume from now on that A i0 = B i0 holds.
Finally, let i 1 ∈ I \ {i 0 } be arbitrary, and let us consider the orthogonal decomposition H = ⊕ i∈I\{i0,i1} H i ⊕ K where K = H i0 ⊕ H i1 . Similarly as above, we obtain either As this holds for arbitrary i 1 , the proof is complete. Now, we are in the position to give an alternative proof of Molnár's theorem which also extends to the two-dimensional case.
Proof of Theorem 1.2 and Molnár'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, 1] → [0, 1] such that φ(λI) = g(λ)I for every λ ∈ [0, 1]. By Theorem 1.1 we conclude We only have to show that the same holds for scalar operators, because then the theorem is reduced to Ludwig's theorem. For any effect A and any set of effects S let us define the following sets if and only if t = 1 − s and s < 1 2 . Thus for all s < 1 2 we obtain which by (16) implies g(1 − s) = 1 − g(s) and g(s) < 1 2 , therefore we indeed have (⊥) for every effect.

Proof of Theorem 1.in two dimensions
In this section we prove our other main theorem for qubit effects. In order to do that we need to prove a few preparatory lemmas. We start with a characterisation of rank-one projections in terms of coexistence.
Lemma 3.1. For any A ∈ E(C 2 ) the following are equivalent: Proof. The case when A ∈ SC(C 2 ) is trivial, therefore we may assume otherwise throughout the proof.
(i)=⇒(ii): Suppose that A / ∈ P 1 (C 2 ), then by Corollary 2.6 there exists an ε > 0 such that {C ∈ E(C 2 ) : C ≤ εI} ⊆ A ∼ . Let B ∈ P 1 (C 2 ) ∩ A c , then we have B ∼ = B c = A c ⊆ A ∼ . But it is very easy to find a C ∈ E(C 2 ) such that C ≤ εI and C / ∈ B c , therefore we conclude B ∼ A ∼ . (ii)=⇒(i): If A ∈ P 1 (C 2 ), B ∈ E(C 2 ) and B ∼ A ∼ , then also B c A c , which is impossible.
Note that the above statement does not hold in higher dimensions, see the final section of this paper for more details. We continue with a characterisation of rank-one and ortho-rank-one qubit effects in terms of coexistence.
. Then the following are equivalent: (ii) There exists at least one B ∈ E(C 2 ) such that B ∼ A ∼ , and for every such pair of effects Moreover, if (i) holds, i.e. A or A ⊥ = tP with P ∈ P 1 (C 2 ) and 0 < t < 1, then we have B ∼ ⊆ A ∼ if and only if B or B ⊥ = sP with some t ≤ s ≤ 1.
(i)=⇒(ii): If we have B ∼ ⊆ (tP ) ∼ with some rank-one projection P , t ∈ (0, 1] and qubit effect B, then by Lemma 2.12 (b) we obtain P ∈ B c and B / ∈ SC(C 2 ). Furthermore, since B ∼ ∩F 1 (C 2 ) ⊆ (tP ) ∼ ∩F 1 (C 2 ), by Corollary 2.7 we obtain where we use the notation from the proof of Lemma 2.10. Thus, the discontinuity of T tP (α) at either α = 0, or α = π 2 , 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 ≤ 1.
(ii)=⇒(i): By Lemma 3.1, (ii) cannot hold for elements of P 1 (C 2 ), so we only have to check that if A, A ⊥ / ∈ F 1 (C 2 ) ∪ SC(C 2 ), then (ii) fails. Suppose that the spectral decomposition of A is λ 1 P + λ 2 P ⊥ where 1 > λ 1 > λ 2 > 0. Then by Lemma 2.12 (c) we find that (λ 1 P ) ∼ ⊆ A ∼ and (1 − λ 2 )P ⊥ ∼ ⊆ A ∼ (see Figure 1), but by the previous part neither ( Figure 1: The figure shows all effects commuting with A ∈ E(C 2 ) \ SC(C 2 ), whose spectral decomposition is For a visualisation of (tP ) ∼ ∩ F 1 (C 2 ) see Section 5. Before we proceed with the proof of Theorem 1.3 for qubit effects, we need a few more lemmas about rank-one projections acting on C 2 . Lemma 3.3. For all P, Q ∈ P 1 (C 2 ) we have 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 P − Q 2 = − det(P − Q). Applying a unitary similarity if necessary, we may assume without loss of generality that (1, 0) ∈ Im P . Obviously, there exist 0 ≤ ϑ ≤ π 2 and 0 ≤ µ ≤ 2π such that (cos ϑ, e iµ sin ϑ) ∈ Im Q. Thus the matrix forms of P and Q in the standard basis are and Q = P (cos ϑ,e iµ sin ϑ) = cos ϑ e iµ sin ϑ · cos ϑ e iµ sin ϑ * = cos 2 ϑ e −iµ cos ϑ sin ϑ e iµ cos ϑ sin ϑ where we used the notation of the Busch-Gudder theorem. Now, an easy calculation gives us det(P −Q) = − sin 2 ϑ and trP Q = cos 2 ϑ. Hence the second equation in (17) is proved, and the third one follows from trP ⊥ Q = 1 − trP Q.
Next, we examine this set.
Lemma 3.4. For all P ∈ P 1 (C 2 ) the following statements are equivalent: (i) s = sin π 4 , (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 . Proof. One could use the Bloch representation (see Section 5), however, let us give here a purely linear algebraic proof. Note that for any R 1 , R 2 ∈ P 1 (C 2 ) we have R 1 − R 2 = 1 if and only if R 2 = R ⊥ 1 . Without loss of generality we may assume that P has the matrix form of (18). Then for any 0 ≤ ϑ ≤ π 2 and R 1 , R 2 ∈ M P,sin ϑ we have R 1 = cos 2 ϑ e −iµ1 cos ϑ sin ϑ e iµ1 cos ϑ sin ϑ sin 2 ϑ and R 2 = cos 2 ϑ e −iµ2 cos ϑ sin ϑ e iµ2 cos ϑ sin ϑ sin 2 ϑ with some µ 1 , µ 2 ∈ R. Hence, we get Notice that the right-hand side is always less than or equal to 1. Moreover, for any µ 1 ∈ R there exist a µ 2 ∈ R such that R 1 − R 2 = 1 if and only if ϑ = π 4 . This completes the proof. Lemma 3.5. Let P, Q ∈ P 1 (C 2 ) and s, t ∈ (0, 1). Then the following are equivalent: Proof. The case when Q ∈ {P, P ⊥ } is trivial, so from now on we assume otherwise. Recall that two rank-one effects with different images are coexistent if and only if their sum is an effect, see [20,Lemma 2]. Therefore, (i) is equivalent to I − tP − sQ ≥ 0. Since tr(I − tP − sQ) = 2 − t − s > 0, the latter is 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 2 . Then a calculation gives From the latter we get that det(I − tP − sQ) ≥ 0 holds if and only if which, by (17) is equivalent to (ii).
Note that we have We need one more lemma.
Lemma 3.6. Let P, Q ∈ P 1 (C 2 ). Then there exists a projection R ∈ P 1 (C 2 ) such that Proof. Again, one could use the Bloch representation, however, let us give here a purely linear algebraic proof. We may assume without loss of generality that P and Q are of the form (18) and (19). Then for any z ∈ C, |z| = 1 the rank-one projection satisfies P − R = sin π 4 . In order to complete the proof we only have to find a z with |z| = 1 such that trRQ = 1 2 , 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.
Proof 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 4 into pairs of rank-one projections with the same distance, 3 we finish the proof by examining how φ acts on rank-one projections and rank-one effects.
By Theorem 1.1 we also obtain Now, we observe that φ preserves commutativity in both directions. Indeed we have the following for every A, B ∈ E(C 2 ) \ SC(C 2 ): 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 Therefore, by interchanging the φ-images of tP and I − tP for some 0 < t < 1 and P ∈ P 1 (C 2 ), we may assume without loss of generality that Hence we obtain the following for all rank-one projections P : Thus, again by interchanging the φ-images of P and P ⊥ for some P ∈ P 1 (C 2 ), and using Lemma 3.2, we may assume without loss of generality that for every P ∈ P 1 (C 2 ) there exists a strictly increasing bijective map f P : (0, 1] → (0, 1] such that STEP 2: We define the following set for any qubit effect of the form tP , 0 < t < 1, P ∈ P 1 (C 2 ): (For a visualisation of tP see Section 5.) Using Lemma 3.5 we see that tP = (tP ) ∼ \ ∪{(sP ) ∼ : t < s < 1} ∩ F 1 (C 2 ) (0 < t < 1, P ∈ P 1 (C 2 )).
By a straightforward calculation we get that Note that s(t, r) = sin π 4 holds if and only if t = r. By Lemma 3.4, this is further equivalent to the following: Notice that by (23) this is equivalent to the following: which is further equivalent to f P (t) = f P ⊥ (r).
We also observe that (25) implies therefore we notice that which is a consequence of the fact that the unique solution of the equation t = 1−t 1−t/2 , 0 < t < 1, is t = 2 − √ 2. STEP 3: Next, applying [12,Theorem 2.3] gives that there exists a unitary or antiunitary operator U : C 2 → C 2 such that we have U * φ(P )U ∈ {P, P ⊥ } (P ∈ P 1 (C 2 )).
Since either both U * φ(·)U and φ(·) satisfy our assumptions simultaneously, or none of them does, therefore without loss of generality we may assume that we have φ(P ) ∈ {P, P ⊥ } (P ∈ P 1 (C 2 )).

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. We start 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 1 (H) such that R ∼ A but R ∼ P .
Proof. Assume that A ∈ E(H) such that A ∼ ∩ F 1 (H) ⊆ P ∼ = P c holds. We have to show that then either A = P , or A = P ⊥ . Clearly, A is not a scalar effect. By Corollary 2.7 we obtain that Notice that the set supp Λ(P, ·) + Λ(P ⊥ , ·) := Q ∈ P 1 (H) : Λ(P, Q) + Λ(P ⊥ , Q) > 0 has two connected components (with respect to the operator norm topology), namely However, by the Busch-Gudder theorem we obtain that Since supp Λ(P, ·) + Λ(P ⊥ , ·) is a closed set, we obtain Notice that the left-hand side of (34) is connected if and only if A is not a projection, in which case it must be a subset of one of the components of the right-hand side. However, this is impossible because the left-hand side contains a maximal set of pairwise orthogonal rank-one projections. Therefore A ∈ P(H), and in particular supp Λ(A, ·) + Λ(A ⊥ , ·) 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 and only if for every C ∈ A ∼ \ SC(H) there exists a D ∈ B ∼ \ SC(H) such that C ∼ ⊆ D ∼ . Clearly, for every non-scalar effect B we have B ≺ B and B ⊥ ≺ B. In particular ≺ is a reflexive relation, but it is not antisymmetric. 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. 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 ∼ .
Let us now consider the orthogonal decomposition H = H 1 ⊕ H 2 ⊕ H 3 where With respect to this orthogonal decomposition we have Since coexistence is invariant under taking the ortho-complements, we may assume without loss of generality that H 3 = {0}. Let us set 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 Consider the effect D := ε · C and notice that 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 has been 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 decided to include the proof for the sake of completeness and clarity. where H 1 = Im P ∩ Im Q, H 2 = Im P ∩ KerQ, H 3 = KerP ∩ Im Q, H 4 = KerP ∩ KerQ and H = H 1 ⊕ H 2 ⊕ H 3 ⊕ H 4 . Note that some of these subspaces might be trivial. We observe that Hence we conclude that #{P, Q} cc = 2 #{j : Hj ={0}} . In particular, #{P, Q} cc = 2 if and only if P, Q ∈ {0, I}, and #{P, Q} cc = 4 if and only if either P / ∈ {0, I} and Q ∈ {I, 0, P, P ⊥ }, or Q / ∈ {0, I} and P ∈ {I, 0, Q, Q ⊥ }. Now, we easily conclude the following characterisation of rank-one and co-rank-one projections: This implies that φ({P : P or P ⊥ ∈ P 1 (H)}) = {P : P or P ⊥ ∈ P 1 (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 their orto-complementations does not change the property (35), we may assume without loss of generality that φ(P 1 (H)) = P 1 (H). It is easy to see that two rank-one projections commute if and only if either they coincide, or they are orthogonal to each other. Thus, as dim H ≥ 3, Uhlhorn's theorem [32] gives that there exist a unitary or antiunitary operator U : H → H such that φ(P ) = U P U * (P ∈ P 1 (H)).
Finally, note that for every projection Q ∈ P(H) we have Q c ∩ P 1 (H) = {P ∈ P 1 (H) : Im P ⊂ Im Q ∪ KerQ}, 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-separable Hilbert spaces. We will use the notation E f s (H) for the set of all effects whose spectrum has finitely many elements. Proof. We only have to observe the following for all A ∈ E(H) with #σ(A) = n ∈ N, where E 1 , . . . E n are the spectral projections and H j = Im E j (j = 1, 2, . . . n): 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 following steps: 2 we prove that φ has the form (2) on E f s (H) \ SC(H),
Of course, the properties of φ imply φ(A) ∼ = φ(A ∼ ) for all A ∈ E(H), and also From the latter it follows that Hence by Lemma 4.
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
Therefore we obtain and thus also In particular, we have φ(A) ∈ A cc (A ∈ E(H)).
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 the orthogonal projection onto M . Consider two arbitrary effects A, B ∈ (P M ) ∼ ∩ E f s (H) which therefore have the following matrix representations: Obviously, Note that by (39), the polynomial p A acts injectively on σ(A), therefore and of course, similarly for B. We observe that by Lemma 2.2 the following two equations hold: and It is important to observe that by (38) the set in (41) is the φ-image of (40). Thus we obtain the following equivalence if A M / ∈ SC(M ): such pair. But a similar statement holds for all two-dimensional subspaces, therefore it is easy to show that 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 the following for every two-dimensional subspace M : From here it is rather straightforward to obtain STEP 3: Observe that (43) holds for every A ∈ F(H), therefore an application of Theorem 1.1 and Corollary 2.5 completes the proof in the separable case. As for the general case, let us consider an arbitrary effect A ∈ E(H) \ E f s (H) and an orthogonal decomposition H = ⊕ i∈I H i such that each H i is a separable invariant subspace of A. By (38) and Lemma 2.1 (b), each H i is an invariant subspace also for φ(A), in particular, we have Without loss of generality we may assume from now on that there exists an i 0 ∈ I so that A i0 is not a scalar effect. Now, let i ∈ I, F ∈ F(H) and Im F ⊆ H i be arbitrary. Then by (43) we have In particular, , therefore by Theorem 1.1 we get that for all i we have either By considering A ⊥ instead of A if necessary, we may assume that we have A i0 = A i0 . Finally, for any i 1 ∈ I \ {i 0 } let us consider the orthogonal decomposition Similarly as above, we then get A i0 ⊕ A i1 = A i0 ⊕ A i1 , and the proof is complete.
Recall the remarkable angle doubling property of the Bloch representation, namely, we have P − Q = sin θ if and only if the angle between the vectors ρ(P ) − 1 2 e 0 and ρ(Q) − 1 2 e 0 is exactly 2θ. Next, we call a positive (semi-definite) element of B sa (C 2 ) a density matrix if its trace is 1, or in other words, if it is a convex combination of some rank-one projections. Therefore ρ maps the set of all 2 × 2 density matrices onto the closed ball of the three-dimensional affine subspace {(1/2, x 1 , x 2 , x 3 ) : x j ∈ R, j = 1, 2, 3} with centre at (1/2, 0, 0, 0) and radius 1/2. Hence, we see that the cone of all positive (semidefinite) 2 × 2 matrices is mapped onto the infinite cone spanned by (0, 0, 0, 0) and the aforementioned Figure 2: Illustration of ρ(E(C 2 )) ∩ S µ . The circle is ρ(P 1 (C 2 )) ∩ S µ .
If one illustrates the set ρ ((A) ∼ ) ∩ ρ F 1 (C 2 ) ∩ S µ with A, A ⊥ / ∈ SC(C 2 ) ∪ F 1 (C 2 ) in the way as above, then one gets a set on the boundary of the cone which is bounded by a continuous closed curve containing the ρ-images of the spectral projections.

Final remarks and open problems
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 2 ≤ dim H < ∞ and A ∈ E(H). Then the following are equivalent: (i) 0, 1 ∈ σ(A), (ii) there exists no effect B ∈ E(H) such that B ∼ A ∼ .
Proof. If dim H = 2, then (i) ⇐⇒ (ii) was proved in Lemma 3.1, so from now on we will assume 2 < dim H < ∞. Also, as the case when A ∈ SC(H) is trivial, we assume otherwise throughout the proof.
(i)=⇒(ii): Suppose that 0, 1 ∈ σ(A) and consider an arbitrary effect B with B ∼ ⊆ A ∼ . By Lemma 2.12, A and B commute. If 0 = λ 1 ≤ λ 2 ≤ · · · ≤ λ n−1 ≤ λ n = 1 are the eigenvalues of A, then the matrices of A and B written in an orthonormal basis of joint eigenvectors are the following: In particular, choosing i = 1, j = n implies either µ 1 = 0 and µ n = 1, or µ 1 = 1 and µ n = 0. Assume the 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 on the 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 ⊥ . We only proved the above lemma and Corollary 2.11 in the finite dimensional case. The following two questions would be interesting to examine: 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 towards the better understanding of coexistence.