The measurement postulates of quantum mechanics are operationally redundant

Understanding the core content of quantum mechanics requires us to disentangle the hidden logical relationships between the postulates of this theory. Here we show that the mathematical structure of quantum measurements, the formula for assigning outcome probabilities (Born’s rule) and the post-measurement state-update rule, can be deduced from the other quantum postulates, often referred to as “unitary quantum mechanics”, and the assumption that ensembles on finite-dimensional Hilbert spaces are characterized by finitely many parameters. This is achieved by taking an operational approach to physical theories, and using the fact that the manner in which a physical system is partitioned into subsystems is a subjective choice of the observer, and hence should not affect the predictions of the theory. In contrast to other approaches, our result does not assume that measurements are related to operators or bases, it does not rely on the universality of quantum mechanics, and it is independent of the interpretation of probability.


systems
In this section we classify all alternative measurement postulates for the case of finite-dimensional single systems, that is, when the constraints associated to the composition of systems (the star product) are ignored. These results build up on the previous work [1] by two of us.
In this section we only consider finite-dimensional Hilbert spaces C d with d a positive integer. In this case we have U(d) ∼ = SU(d) × U(1); and since U(1) has a trivial action on rays, we only consider SU(d). Later on, when addressing the infinite-dimensional case d = ∞, we will work with U(d), since the condition det U = 1 is not well-defined when d = ∞.
Structure of measurements and mixed states Definition 1. F d is a set of functions f : PC d → [0, 1] which is closed under composition with unitaries U ∈ SU(d) closed under convex combinations and that contains the unit and the zero functions, respectively u(ψ) = 1 and 0(ψ) = 0, for all ψ ∈ PC d .
The unit function u represents an outcome that happens with probability one. For example, such unitprobability outcome can be the event corresponding to all outcomes of the measurement {f i }, which by normalization satisfy Analogously, the zero function 0 represents a formal outcome that has zero probability irrespectively of the state.
For what comes below, it is convenient to consider the set F d as embedded in the complex vector space CF d generated by itself. The fact that the group action (1) commutes with the mixing operation (2) can be extended to arbitrary linear combinations in CF d , providing a complex, linear representation of SU(d).
While only the elements of F d are outcome probability functions (OPFs), any element of CF d can be interpreted as the expectation value of an observable with complex outcome labels, in analogy to the algebra of observables in QM. While in QM the space CF d has dimension d 2 , here we leave the dimension unconstrained. However, in what follows, we show that the "possibility of state estimation" assumption implies that the linear space CF d is finite-dimensional. But before this, we recall that the probability of outcome f ∈ F d on an ensemble (ψ r , p r ) is given by The above follows from the rules of probability calculus.
Lemma 2. Suppose that the values of the outcomes f 1 , . . . , f k ∈ F d on any given ensemble (ψ r , p r ) determine the value of any other outcome g ∈ F d on that ensemble (ψ r , p r ). Then the functions {f 1 , . . . , f k , u} span the linear space CF d .
In other words, knowing the numbers f 1 [(ψ r , p r )], . . . , f k [(ψ r , p r )] allows us to determine the number g[(ψ r , p r )] without knowing the ensemble (ψ r , p r ). That is, the latter is some function of the former.
Proof. Once the OPFs f 1 , . . . , f k ∈ F d are given we can define the convex set Next we note that, the fact that the values of f 1 , . . . , f k determine the value of g on any ensemble (ψ r , p r ) means that there is a function ξ g : S d → [0, 1] such that r p r g(ψ r ) = ξ g r p r f 1 (ψ r ), . . . , r p r f k (ψ r ) .
Since the above equality holds for all ensembles, it also holds for the pure states ψ r g(ψ r ) = ξ g f 1 (ψ r ), . . . , f k (ψ r ) , for all r. Hence we have ξ g r p r f 1 (ψ r ), . . . , r p r f k (ψ r ) = r p r ξ g f 1 (ψ r ), . . . , f k (ψ r ) . (9) It follows that (9) also holds true if every appearance of ψ r is replaced by some ensemble (ψ s ), where s labels the possible states and their probabilities.
can take all values in S d by choosing the states and probabilities in a suitable way, this shows that ξ g is convex on the full set S d . This implies that ξ g can be affinely extended to all of R k , i.e. there is an affine function ξ g : R k → R which coincides with the previous function ξ g = ξ g inside the convex set S d . The affine nature of the function means for any c r ∈ R with r c r = 1 and x r ∈ R k , but the coefficients c r are not necessarily positive. Any affine function ξ g : R k → R can be written as where e g ∈ R k and c g ∈ R. Therefore we can write That is, any OPF g can be written as an R-linear combination of {f 1 , . . . , f k , u}, as in (12). Since every element of CF d is a complex-linear combination of such OPFs, every such element must thus be a complex-linear combination of {f 1 , . . . , f k , u}.
Corollary 3. The "possibility of state estimation" assumption implies that, for all finite d, the linear space CF d is finite-dimensional.
In what follows, we introduce a representation of pure states ψ that is linearly related to outcome probabilities. Because of this, this new representation encodes the equivalence relation between ensembles, and hence, the structure of mixed states arising from alternative measurement postulates.
Definition 4. For each pure state ψ ∈ PC d we define the linear form Ω ψ : CF d → C as with the natural SU(d) action This allows to write the probability of outcome f ∈ F d on ensemble (ψ r , p r ), in terms of the mixed state ω = r p r Ω ψr .
Hence, two different ensembles corresponding to the same mixed state (16) are indistinguishable. The next lemma gives us important information about the group representation CF d .
Lemma 5. The SU(d) action (1) on CF d decomposes as where N d j are the irreducible representations defined in Lemma 7. The finite set J contains zero and some positive integers (with no repetitions).
Before proving the above we mention that the quantum case is J = {0, 1}, and in section Non-quantum measurement postulate violating associativity of the main text the (non-quantum) case J = {0, 1, 2} is analyzed. Also, we have to mention that in this work we follow the notation of [2], where the group representations are labelled by the subspace they act on.
Proof. In this proof we establish the following four facts in the same order: (i) CF d decomposes into a finite sum of finite-dimensional irreducible representations (irreps), (ii) these irreps are of the type N d j , (iii) there are no repetitions, (iv) j = 0 is always included.
Fact (i). Lemma 2 shows that the SU(d) representation CF d is finite-dimensional. And these can always be decomposed into finite-dimensional irreps [2]. Also, we know that each finite-dimensional irrep of SU(d) corresponds to a d-row Young diagram λ. Hence we write where repeated values of λ can happen. Fact (ii). The fact that different elements of CF d are different functions PC d → C implies that the form Ω ψ (Definition 4) has support in each sub-space of (18). Indeed, if Ω ψ had no support in the sub-space V d λ , then any of the elements f + V d λ ⊆ CF d would correspond to the same function. Denote by SU(d, ψ) the subgroup of unitaries that leave the state ψ invariant U ψ = ψ, and note that -invariant subspace has dimension one. Which fixes the projection of the linear form Ω ψ onto each subspace of (17) up to a proportionality factor. Changing these proportionality factors modifies the structure of F d by the corresponding inverse linear transformation; but the space CF d remains identical.
To prove Fact (iii), suppose that there are two repeated irreps in (17). We can write the isomorphism with the understanding that the SU(d)-action in C 2 is trivial. Next, we invoke the above-shown unicity of Ω ψ to see that the projection of Ω ψ onto the subspace N d j ⊗ C 2 is of the form where Γ : C 2 → C is a linear form that depends on the above-mentioned proportionality factors. Given Γ it possible to find two different vectors v, v ∈ C 2 such that Γ(v) = Γ(v ). Then, taking any f ∈ N d j we can construct two different elements of CF d corresponding to the same function for all ψ, which is a non-sense.
To establish Fact (iv), we recall that the unit function u ∈ F d is always included (Definition 1). Since u is invariant under the action (1) the trivial irrep N d 0 must be included in the decomposition (17). Hence 0 ∈ J .
The SU(d) representations M d n and N d n In this subsection we introduce two families of SU(d) representations that allow to construct all alternative measurement postulates for single systems by using (17). For this, we recall that the projector onto the symmetric subspace of (C d ) ⊗n can be written as the average of all permutations π over n objects where π acts by permuting the n factor spaces of (C d ) ⊗n . Next we define an SU(d) representation that sometimes is named Sym n C d ⊗ Sym n C d * .
Definition 6. Let M d n be the linear space of complex matrices M acting on (C d ) ⊗n whose support is contained in the symmetric subspace And let the linear action of SU(d) on M d n be Lemma 7. The decomposition of M d n into SU(d) irreducible representations is where the subspace N d j,n is generated by applying the group action (25) to the element where |0 , |1 ∈ C d are any orthogonal pair. Also, the representation isomorphisms hold for all n, n ≥ j.
Isomorphism (28) allows us to use the shorthand notation N d j . Also, note that N d 0 is the trivial irrep, generated by the element N 0,n = P + ∈ M d n ; and N d 1 is the adjoint (quantum) irrep.
Proof. In order to obtain the decomposition (26) it is useful to define the trace map where tr n denotes the trace over the nth factor in (C d ) ⊗n . Note that, by symmetry, this partial trace is independent of the choice of factor: tr n M = tr 1 M . From now on, wherever is clear, we leave the dependence on d implicit.
Because the map (29) commutes with the SU(d) action, Schur's Lemma tells us that its kernel must be a subrepresentation of M n , which we denote by N n,n . It is proven in Lemma 23 that this representation is irreducible. Also, it is straightforward to check that the matrix N n,n defined in (27) is in the kernel of the map (29), that is tr n N n,n = 0 .
Combining the above with irreducibility we see that the subspace N n,n is generated by the action of the group on the single element N n,n . Because the map (29) is surjective, the orthogonal complement of N n,n ⊆ M n is a representation isomorphic to M n−1 , which in turn contains the irreducible representation N n−1,n−1 ⊆ M n−1 in the kernel of the trace map tr n−1 : M n−1 → M n−2 . Then, using Schur's Lemma again, there must be a subrepresentation N n−1,n ⊆ M n that is isomorphic to N n−1,n−1 ⊆ M n−1 , which proves isomorphism (28). Proceeding inductively, we obtain the full decomposition (26).
To conclude the proof of Lemma 7 we need to show that N j,n ∈ N j,n . By noting that is non-zero when j < n, we can proceed inductively to arrive at which is the case analyzed above (32). The isomorphisms (28) provided by Schur's Lemma conclude the proof.
The form Ω ψ in M d n and N d n In this section we introduce a simple choice for the linear form Ω ψ of Definition 4, for the cases CF d ∼ = M d n and CF d ∼ = N d n . As already mentioned, this form encodes the structure of the set of mixed states. Lemma 8. The linear form Ω ψ : M d n → C defined by is invariant under all stabilizer unitaries U ∈ SU(d, ψ) and has support in all irreps N d j ⊆ M d n .
Proof. It is straightforward to check that the form (35) satisfies (36). To see that (35) has support in each irrep N j,n ⊆ M n , we observe that, for each j, there is a pure state ψ such that tr N j,n |ψ ψ| ⊗n = 0 , where N j,n is defined in (27).
As mentioned above, these two constrains fix Ω ψ up to an irrelevant proportionality factor in each irrep. To obtain Ω ψ in the case CF d = N d n we proceed in the following maner. Since N d n is a subrepresentation of M d n we can take (35) and perform the orthogonal projection onto the subspace N d n,n ⊆ M d n , defined via (27) or via the kernel fo the map (29).

Supplementary note 2: Multipartite systems
In this section we describe and impose the consistency constraints associated to composite systems and the star product.

Closedness under system composition
We require that any family of OPF sets F 2 , F 3 , . . . and F ∞ must be closed under system composition. This means that the complete set of measurements F a of a system C a also includes the measurements that appear in the description of C a as part of a larger system C a ⊗ C b .
Definition 9 (Closedness under system composition). If F ab is the OPF set of C a ⊗ C b then the OPF set F a of C a is the following collection of functions for all f ∈ F ab and a fixed β ∈ PC b , and all a, b ∈ {2, 3, . . . , ∞}.
Note that the closure of F ab under 1 ⊗ SU(b) implies that the set F a defined via (38-39) does not depend on the choice of β. Also, it is straightforward to check that the OPF set F a so defined satisfies all the requirements of Definition 1.
In the finite-dimensional case, closedness under system composition (Definition 9) implies the following strong fact. For any set of measurements F ab of a bipartite system C a ⊗ C b , the measurement spaces of the subsystems are CF a ∼ = M a n for C a and CF b ∼ = M b n for C b , with the same n. In addition, using the fact that any pair of systems can be jointly described as a bipartite system, we conclude that all finite-dimensional systems C d must have OPF space M d n (with the same value for n). Lemma 10. For any pair of positive integers a, b, let F ab be the OPF set of C a ⊗ C b with decomposition (Lemma 5) Define F a as the set of functions for all f ∈ F ab and a fixed β ∈ PC b . Then we have the SU(a)-representation isomorphism where n = max J .
Note that, if we define F b by exchanging the role of the subsystems C a ⊗ C b in (42), then we obtain the SU(b)-representation isomorphism CF b ∼ = M b n with the same value for n as in (43). Using the fact that any pair of systems can be jointly described as a bipartite system, we arrive at the following.
Corollary 11. Closedness under system composition (Definition 9) implies that all finite-dimensional Hilbert Proof of Lemma 10. In order to establish the isomorphism (43) we analyze how the functions (42) transform under the subgroup SU(a) ⊗ 1. First, we do this in the case where J has finite cardinality, so that the decomposition (17) of CF ab has a largest irrep N ab n . We further split this analysis into the case where the function f in (42) belongs to the subspace f ∈ N ab n ⊆ CF ab , and the general case. Using the characterization of N ab n as the kernel of the map (29) we can say the following. For each f ∈ N ab n ⊆ CF ab there is a matrix F ∈ M ab n such that tr n F = 0 and for all α, β. (Note that, in order to improve clarity, we re-arranged the order of the tensor factors.) The matrices are contained in the subspace where the isomorphisms are of SU(a) ⊗ SU(b) representations. Even more, using the full support conditions (37) in each tensor factor, we conclude that the matrices |α α| ⊗n ⊗|β β| ⊗n generate the whole space (46). Next we analyze the SU(a) ⊗ SU(b) action on the function (44), which is the action on the intersection between the subspaces {F ∈ M ab n : tr n F = 0} and (46). This intersection can be characterized by writing the trace as tr n = tr An tr Bn , where tr An is the trace on the nth factor of M a n , and tr Bn is the trace on the nth factor of M b n . The above identity implies that if tr An F = 0 or tr Bn F = 0 then tr n F = 0. Therefore, the above-mentioned intersection contains all irreps N a n ⊗ N b j and N a j ⊗ N b n for j = 0, 1, . . . , n. This implies that the SU(a) ⊗ 1 action on the space of functions (44) with f ∈ N ab n ⊆ CF ab decomposes into the irreps N a 0 , . . . , N a n , with possible repetitions. In the general case f ∈ CF ab , the addition of all subspaces N ab j ⊆ F ab with j < n does not add any new irrep to the list N a 0 , . . . , N a n . Although it may increase the repetitions. Finally, we establish the desired isomorphism (43) by recalling Lemma 5. This tells us that any OPF set, like the F a defined through (42), has no repeated irreps.

The star product
In this subsection we introduce the star product, which contains the information of which measurements of a composite system F ab are local.
Definition 12. The star product is a map : F a × F b → F ab defined on any pair of OPF sets F a , F b , with the following properties: • preserves the local structure • preserves probability • commutes with local mixing operations • commutes with the local group action • and it is associative for any f , f x ∈ F a ; g ∈ F b ; h ∈ F c ; α ∈ PC a ; β ∈ PC b ; U ∈ U(a); a, b ∈ {2, 3, . . . , ∞}, and any probability distribution p x . Properties (48-53) must also hold when exchanging factors.
The -product allows us to write the reduced state of a bipartite pure state ψ ∈ C a ⊗ C b on the subsystem C a as the linear form f → Ω ψ (f u) for all f ∈ CF a .
Note that property (48) is weaker than the analog condition in the main text: The reason for writing the stronger condition in the main text is that it does not require u to be defined. The following lemma proves that, in our context, condition (48) implies condition (54).
Lemma 13. Suppose that any ensemble (ψ r , p r ) satisfying is of the form ψ r = ϕ for all r. Then (48) implies Proof. First, let {g i } be a complete measurement and define the following probabilities and the (not necessarily pure) states for all f . Second, substitute i g i = u in (48) obtaining for all f . Third, use the premise of the lemma to conclude that for all i and f . Finally, substituting back the definition of Ω i we obtain which implies (56).
By "preservation of probability" (49) it is meant that the fact that all outcome probabilities add up to one is independent of whether we describe a system on its own or as part of a larger system Also, the joint outcome f A 0 B , where 0 B is the formal outcome with zero probability for all states, must have zero probability, which gives (50). The action of the * -product is not defined on the elements of CF a that are not in F a . However, the following lemma shows that one can define the action of the * -product to the rest of elements of CF a in such a way that the map is bilinear.
Lemma 14. Any star-product map : F a × F b → F ab as specified in Definition 12 can be extended to a bilinear map : CF a × CF b → CF ab with the same properties (48-53).
Proof. For any given g ∈ F b define the map Using Definition 12 we obtain the following properties for the map for any probability distribution p x . In Appendix 1 of [3] it is proven that it is possible to define a R-linear map ξ : RF a → RF ab which is identical to ξ inside F a . Finally, we can define the C-linear map ξ : CF a → CF ab in the natural way for any pair f 1 , f 2 ∈ RF a . The above construction can be repeated with an exchange of parties. Proving the desired result.
Lemma 15. In the case CF d ∼ = M d n we have the identity for all F ∈ M a n , G ∈ M b n , α ∈ PC a and β ∈ PC b . By noting that the set of matrices |α α| ⊗n ⊗|β β| ⊗n span the subspace P A + P B + M ab n P A + P B + ⊆ M ab n we can write (70) as for all F, G. This proves identity (69).

Supplementary note 3: Non-associativity of : M a × M b → M ab
The permutation group and Schur-Weyl duality In this section we review some well-known results of representation theory. The n-th tensor-power of a vector space C d can be decomposed as where λ runs over all partitions of n with at most d parts, V d λ are irreps of SU(d), and S n λ are the irreps of the group of permutations of n objects. The partition λ = (n) corresponds to the trivial representation of the group of permutations, and hence, all vectors in the subspace V d (n) ⊗ S n (n) are permutation-invariant. Because of this, this subspace and the corresponding projector P + = P (n) are called symmetric.
When considering a bipartite space C d = C a ⊗ C b , the symmetric projector can be written as where Q AB λ is the orthogonal projector onto the subspace of [V a λ ⊗ S n λ ] A ⊗ [V b λ ⊗ S n λ ] B that transforms trivially when applying the same permutation to A and B. Specifically, we can write it as where 1 V is the identity on the subspace V and is the "maximally entangled state" of the product space V ⊗ V . The invariance of |τ λ AB under identical permutations on A and B is analogous to the invariance of any maximally entangled state under transformations of the form U ⊗ U * , together with the fact that all irreps of the permutation group are real (self-dual).
In the tri-partite case C d = C a ⊗ C b ⊗ C c , the symmetric projector can be written as where Q ABC λ,µ,ν is the orthogonal projector onto the subspace of [V a λ ⊗S n λ ] A ⊗[V b µ ⊗S n µ ] B ⊗[V c ν ⊗S n ν ] C that transforms trivially when applying the same permutation on A, B, C. Therefore, the projector Q ABC λ,µ,ν is zero unless the irrep decomposition of S n λ ⊗ S n µ ⊗ S n ν contains the trivial S n (n) . Particularly, when one of the three partitions is (n), we recover the bipartite case Q ABC for all λ. That is, if one partition is (n) then the other two partitions have to be equal. And this is why, in the bipartite case, the projector Q AB λ only depends on one partition.

The irreducible representations of SU(d)
Using the Littlewood-Richardson rule [2], we can decompose V d λ ⊗ V d µ into irreps, and prove the following patterns.
Proof. If we denote by λ * the partition of the irrep V 4 * (n−1,1) , and by λ j the partition of N 4 j , then we have Applying the Littlewood-Richardson rule [2] to the Young tableaux λ and λ * , we see that all resulting tableaux have at most 4(n − 1) boxes, while the tableau of λ j has 4j boxes. Therefore, no tableau λ j with j ≥ n can appear in the product of λ and λ * .
Lemma 17. When restricting the irrep N 4 n of SU(4) to any SU(2) subgroup, the decomposition of N 4 n into irreps of SU(2) does not include any N 2 j with j > n.
Proof. The SU(2) irreps in N 4 n correspond to straight lines of weights in the weight diagram of N 4 n . The longest such line contains 2n + 1 weights. Therefore, the largest SU (2) irrep is N 2 n .

Proof of the main theorem
The following theorem shows that only in the quantum cas (that is n = 1) there is an associative star product : M a n × M b n → M ab n .
Theorem 18. If n ≥ 2 then there is no bilinear map : M a n × M b n → M ab n satisfying the star-product Definition 12.
Proof. Analysis of bipartite systems. Let us consider a bipartite system with Hilbert space C a ⊗ C b and dimensions a = 2 and b = 4. Using the decomposition (73) of the projector P AB + onto the symmetric subspace of (C a ⊗ C b ) ⊗n we can decompose M ab n into subspaces as each labeled by a pair of n-partitions (λ, µ). Since system A is 2-dimensional all n-partitions λ, µ have at most two rows. The action of SU(ab) might not be well-defined in some of these subspaces, but the action of the subgroup SU(a) ⊗ SU(b) ⊆ SU(ab) is well-defined in each (λ, µ) subspace from (81). Concretely, we have the following isomorphism of SU(a) ⊗ SU(b) representations which in particular gives Now, let us take the subspace (82) corresponding to λ = µ = (n−1, 1), and decompose it into two orthogonal subspaces defined in the following way: (i) consider the action of the subgroup 1 ⊗ SU(b) on the left-hand side of (84), (ii) decompose this action into irreps of SU(b), (iii) let W b licit be the direct sum of all irreps N b j for any j, and (iv) let W b illicit be the direct sum of the rest of irreps. The super-index b in these subspaces W b xxlicit reminds us that these are SU(b) representations.
Lemma 16 tells us that W b licit does not contain any N b j with j ≥ n. That is where the sum over j may contain some absences and repetitions. Combining this with Schur's Lemma and the commutativity constraint (52), we see that the image of the -product [M a n M b n ] ⊆ M ab n does not have support in W b illicit . In particular, In the next subsection we show that, if B is itself considered a bipartite system then the above subspace contains the irrep N b n , which is incompatible with (85) and (86). Analysis of tripartite systems. Now let us describe system B as a bipartite system CE with Hilbert space C b = C c ⊗ C e and dimensions c = e = 2. Combining the decompositions of the bipartite (73) and the tripartite (76) symmetric projectors, we can write Now, if we substitute the decomposition (87) into (86) and remove all terms except for the µ = (n) and ν = (n − 1, 1) one, the inclusion still holds Importantly, the projector Q ACE is a subrepresentation of (88), it does not contain the irrep N c n . Next we show that this is incompatible with associativity. Using Lemma 15 and recalling that we obtain the isomorphism Q ACE (n),(n−1,1),(n−1,1) [M a n u CE ] Q ACE (n),(n−1,1),(n−1,1) ∼ = M a n of SU(a) ⊗ 1 ⊗ 1 representations, which include the irrep N a n . By permuting the subsystems ACE we conclude that the 1 ⊗ SU(c) ⊗ 1 representation (89) also contains the irrep N c n , in contradiction with our previous conclusion! At this point we can contrast the above argument with the disregarded case n = 1. In this cse there is only one partition λ = µ = (1), and which implies that W b illicit is trivial. Therefore the above contradiction does not apply to the n = 1 case. Corollary 19 (measurement theorem). Any family of OPF sets F d with finite d, equipped with a -product, and satisfying the assumptions "possibility of state estimation" and "closedness under system composition", has OPFs and -product of the form for all normalized ϕ ∈ C a and ψ ∈ C a ⊗ C b , where the C a -matrix F satisfies 0 ≤ F ≤ 1, and analogously for G.

Supplementary note 4: Countably infinite-dimensional Hilbert spaces
Since all countably infinite-dimensional Hilbert spaces are isomorphic, we denote them all by C ∞ . The topological space of all one-dimensional subspaces of C ∞ is denoted by PC ∞ . Also, for any given subspace S ⊆ C ∞ we denote the corresponding orthogonal projector by Π S . The following lemma tells us that the measurements on C ∞ are of the quantum form (91) if and only if they have such form when restricted to any finite-dimensional subspaces of C ∞ .
• For every finite-dimensional subspace S ⊂ C ∞ , there exists a self-adjoint operator F S fully supported on S, i.e. Π S F S Π S = F S , such that 0 ≤ F S ≤ 1 and f (ψ) = ψ|F S |ψ for all normalized ψ ∈ S.
Proof. Suppose the first statement, f (ψ) = ψ|F |ψ . Then, for every finite-dimensional subspace S, define F S = Π S F Π S . Now, it is clear that for all normalized ψ ∈ S, we have which is the second statement of the lemma. Conversely, suppose that for every finite-dimensional subspace S ⊂ C ∞ there exists F S satisfying Π S F S Π S = F S and f (ψ) = ψ|F S |ψ for all normalized ψ ∈ S. First we prove the following intermediate claim: Let (S (n) ) n∈N be any sequence of subspaces with dim S (n) = n and S (n) ⊂ S (n+1) such that for S := n∈N S (n) we get the norm closureS = C ∞ . Then there exists a unique bounded operator F on C ∞ such that f (ψ) = ψ|F |ψ for all normalized states ψ ∈ S.
This proves existence in our intermediate claim, now we would like to prove uniqueness. To this end, suppose that both F and G are bounded operators such that f (ψ) = ψ|F |ψ = ψ|G|ψ for all normalized ψ ∈ S.
Then the bounded operator ∆ := F − G satisfies ψ|∆|ψ = 0 for all ψ ∈ S. Since every vector in C ∞ can be approximated in norm to arbitary accuracy by elements in S, and since ∆ is continuous, this shows that ψ|∆|ψ = 0 for all ψ ∈ C ∞ , and thus ∆ = 0 since ∆ is bounded and the Hilbert space is complex [4]. This proves our intermediate claim. Since f (ψ) ∈ [0, 1] for all normalized ψ ∈ S, and all normalized vectors in C ∞ can be approximated in norm by normalized vectors in S, we have inf ψ ψ|F |ψ ≥ 0 and sup ψ ψ|F |ψ ≤ 1, where infimum and supremum are over all normalized vectors in C ∞ . Thus, 0 ≤ F ≤ 1, and hence F is self-adjoint.
Let ζ ∈ C ∞ be an arbitrary normalized vector. If ζ ∈ S then, by construction, f (ζ) = ζ|F |ζ . Now we want to show that this equation is also true if ζ ∈ S. In this case, define the sequence of subspaces T 1 := span{ζ} and T n+1 := span (S n ∪ {ζ}) for all n ∈ N. Clearly dim T n = n andT = C ∞ for T = n∈N T n . Thus, according to our intermediate claim, there is a bounded operator G such that f (ψ) = ψ|G|ψ for all normalized ψ ∈ T ; in particular, f (ζ) = ζ|G|ζ . But since S ⊂ T , we also have f (ψ) = ψ|G|ψ for all ψ ∈ S. But, according to our intermediate statement, F is the unique bounded operator satisfying this equation, hence F = G.
As a side remark, note that the operator sequence F (n) does not in general converge to F in operator norm.
Theorem 21. Suppose that for each finite d all OPFs f ∈ F d are of the form (91). Then the "closedness under system composition" assumption (Definition 9) implies that all OPFs f ∈ F ∞ are also of the form where the C ∞ -operator F satisfies 0 ≤ F ≤ 1.
Proof. Let us fix a finite-dimensional subspace S ⊂ C ∞ . Denote the dimension of S by d. Let us fix an orthonormal basis ψ 1 , . . . , ψ d of S, an orthonormal basis α 1 , . . . , α d of C d , and a normalized vector β ∈ C ∞ . The Hilbert spaces C ∞ and C d ⊗ C ∞ are isomorphic in a very non-unique way; so let X : for all i = 1, . . . , d (this does not determine X uniquely; we will pick any such X arbitrarily). Hence, for any vector ψ ∈ S there is α ∈ C d such that ψ = X(α ⊗ β). And for any OPF f of C ∞ ,the OPF G := f • X must be well-defined, since F ∞ is closed under composition with unitaries. In particular, Note that due to the mentioned isomorphism both, f and g, belong to F ∞ . At this point we invoke "closedness under system composition" (Definition 9). This tells us that for any for all α ∈ PC d . This together with Corollary 19 implies that there is a C d -matrix H such that 0 ≤ H ≤ 1 and h(α) = α|H|α . Next we decompose H in the chosen orthonormal basis of C d , obtaining H = d i,j=1 h ij |α i α j |. Also, we use the coefficients h ij to define the (C d ⊗ C ∞ )-operator F S = d i,j=1 h ij |ψ i ψ j |, which is supported on the subspace S and satisfies 0 ≤ F S ≤ 1.
Finally, for any given normalized ψ ∈ S, we decompose it in the chosen S-basis ψ = Combining this with (96) and (97) we obtain In summary, for any given finite-dimensional subspace S, we have constructed a C ∞ -operator F S satisfying the premises of Lemma 20. This gives us the conclusion of Theorem 21.

Supplementary note 5: The post-measurement state-update Rule
Until now we have been concerned with the outcome probabilities of quantum measurements. In this section, we characterize the transformation that the quantum state undergoes during the measurement process.
Lemma 22 (quantum post-measurement state-update rule). The only post-measurement state-update rule compatible with the quantum probability assignment (91-92) is such that each measurement outcome is represented by a completely-positive linear map Λ. The probability of this outcome is given by and the post-measurement state after outcome Λ is ρ = Λ(|ψ ψ|) trΛ(|ψ ψ|) . (100) In this statement each outcome is characterized by a map Λ, while in Corollary 19 each outcome is characterized by s POVM elements F . This two mathematical descriptions of an outcome are connected via trΛ(|ψ ψ|) = ψ|F |ψ , for all ψ.
The remainder of this section constitutes the proof of Lemma 22. While the mathematics of this proof is certainly not new, we give the details in terms of the context and formalism of this paper.
Proof. Corollary 19 states that any measurement has OPFs {f i } of the form f i (ψ) = tr[F i |ψ ψ|], where {F i } are positive operators satisfying i F i = 1, that is, a POVM. This implies that all the statistical information of any ensemble (ψ r , p r ) is given by the corresponding density matrix ρ = r p r |ψ r ψ r |. The associated linear form Ω ρ : CF d → C is given by Ω ρ (f i ) = tr(ρF i ), relating the usual density matrix formalism to the general formalism of this paper.
At this point we still have not said anything about the post-measurement state update rule. But whatever this rule is, let σ(F i , ρ) be the post-measurement state (that is, its density matrix) after outcome F i , when the initial state is ρ. And define the map Λ Fi which takes the original state ρ to the post-measurement state times its corresponding probability: Next, consider another given measurement with POVM {G j }, and define the POVM {H j,i } to be that corresponding to the successive implementation of the measurements {F i } and {G j }. (This must correspond to a valid measurement, because the whole point of talking about a post-measurement state is that one can make further measurements on it.) Then, using the rules of probability calculus and the above formulas we obtain tr[H j,i ρ] = P (j, i) = P (j|i)P (i) = tr[G j σ(F i , ρ)]tr[F i ρ] for all i, j and ρ. This equation implies that the map Λ Fi (ρ) is linear in ρ.
By definition, the map Λ Fi takes every valid density matrix to a non-negative multiple of another valid density matrix, hence, the map Λ Fi is positive and trace-non-increasing. To recover formulas (99) and (100) we use the fact that tr σ(F i , ρ) = 1, which gives trΛ Fi (ρ) = tr[F i ρ] = P (F i |ρ). In summary, the probability of an outcome is the trace of the unnormalized post-measurement state given by the map Λ associated to the outcome F under consideration. This allows to fully characterize an outcome with the corresponding map Λ, with no reference to a POVM element F .
Finally, we show that each outcome map Λ is not just positive, but completely positive. As argued in the main text, we use the fact that one can always regard a system C d as part of a larger system C d ⊗ C b . Then, the outcome map Λ must remain a valid outcome map when extended to the larger system Λ ⊗ I, where I is the identity map on the Hermitian operators acting on C b . This is the definition of complete positivity.