1 Introduction

Gleason’s theorem [11] is a fundamental result in the foundations of quantum theory simplifying the axiomatic structure upon which the theory is based. The theorem shows that quantum states must correspond to density operators if they are to consistently assign probabilities to the outcomes of projective measurements in Hilbert spaces of dimension three or larger.Footnote 1

More explicitly, let \(\mathcal {P\left( H\right) }\) be the lattice of self-adjoint projections onto closed subspaces of a separable Hilbert space \(\mathcal {H}\) of dimension at least three. Consider functions \(f:\mathcal {P\left( H\right) }\rightarrow \left[ 0,1\right] \), which are additive for projections \(P_{1}\) and \(P_{2}\) onto orthogonal subspaces of \(\mathcal {H}\), i.e.

$$\begin{aligned} f\left( P_{1}\right) +f\left( P_{2}\right) =f\left( P_{1}+P_{2}\right) . \end{aligned}$$
(1)

Gleason concluded that the solutions of Eq. (1) necessarily admit an expression

$$\begin{aligned} f\left( \cdot \right) ={{\,\mathrm{Tr}\,}}\left( \rho \cdot \right) , \end{aligned}$$
(2)

for some positive-semidefinite self-adjoint operator \(\rho \) on \(\mathcal {H}\).

The result does not hold, however, in Hilbert spaces of dimension two since the constraints (1) degenerate in this case: the projections lack the “intertwining” property [11] present in higher dimensions. In 2003, Busch [3] and then Caves et al. [6] extended Gleason’s theorem to dimension two by considering generalised quantum measurements described by positive operator-valued measures, or POMs. In analogy with Gleason’s original requirement, a state is now defined as an additive probability assignment not only on projections but on a larger set of operators, the space \(\mathcal {E}\left( \mathcal {H}\right) \) of effectsFootnote 2 defined on a separable Hilbert space. Then, any function \(f:\mathcal {E\left( H\right) }\rightarrow \left[ 0,1\right] \) satisfying additivity,

$$\begin{aligned} f\left( E_{1}\right) +f\left( E_{2}\right) =f\left( E_{1}+E_{2}\right) , \end{aligned}$$
(3)

for effects \(E_{1},E_{2}\in \mathcal {E\left( H\right) }\) such that

$$\begin{aligned} E_{1}+E_{2}\in \mathcal {E\left( H\right) }, \end{aligned}$$
(4)

is found to necessarily admit an expression of the form given in Eq. (2).Footnote 3 The effects \(E_{1}\) and \(E_{2}\) are said to coexist since the condition in Eq. (4) implies that they occur in the range of a single POM. More recently, it has been shown that this Gleason-type theoremFootnote 4 also follows from weaker assumptions: it is sufficient to require Eq. (3) hold only for effects \(E_{1}\) and \(E_{2}\) that coexist in projective-simulable measurements obtained by mixing projective measurements [16].

Additive functions were first given serious consideration in 1821 when Cauchy [4] attempted to find all solutions of the equation

$$\begin{aligned} f\left( x\right) +f\left( y\right) =f\left( x+y\right) , \end{aligned}$$
(5)

for real variables \(x,\,y\in \mathbb {R}\). In addition to the obvious linear solutions, non-linear solutions to Cauchy’s functional equation are known to exist [12]. However, the non-linear functions f satisfying Eq. (5) cannot be Lebesgue measurable [2], continuous at a single point [8] or bounded on any set of positive measure [13]. Similar results also hold for Cauchy’s functional equation with arguments more general than real numbers, reviewed in [1], for example.

Recalling that the Hermitian operators on \(\mathbb {C}^{d}\) form a real vector space, it becomes clear that the Gleason-type theorems described above can be viewed as results about the solutions of Cauchy’s functional equation for vector-valued arguments: additive functions on subsets of a real vector space, subject to some additional constraints, are necessarily linear. Taking advantage of this connection, we use results regarding Cauchy’s functional equation to present an alternative proof of known Gleason-type theorems.

In Sect. 2, we spell out the conditions which single out linear solutions to Cauchy’s functional equation defined on a finite interval of the real line. The main result of this paper—an alternative method to derive Busch’s Gleason-type theorem—is presented in Sect. 3. We conclude with a summary and a discussion of the results in Sect. 4.

2 Cauchy’s Functional Equation on a Finite Interval

In 1821 Cauchy [4] showed that a continuous function over the real numbers satisfying Eq. (5) is necessarily linear. It is important to note, however, that relaxing the continuity restriction does allow for non-linear solutions [12], as pathological as they may be.Footnote 5 Other conditions known to ensure linearity of an additive function include Lebesgue measurability [2], positivity on small numbers [9] or continuity at a single point [8]. We begin by proving a slight extension of these results, in which the domain of the function is restricted to an interval, as opposed to the entire real line.

Theorem 1

Let \(a\ge 1\) and \(f:\left[ 0,a\right] \rightarrow \mathbb {R}\) be a function which satisfies

$$\begin{aligned} f\left( x\right) +f\left( y\right) =f\left( x+y\right) , \end{aligned}$$
(6)

for all \(x,y\in \left[ 0,a\right] \) such that \(\left( x+y\right) \in \left[ 0,a\right] \). The function f is necessarily linear, i.e.

$$\begin{aligned} f\left( x\right) =f\left( 1\right) x, \end{aligned}$$
(7)

if it satisfies any of the following four conditions:

  1. (i)

    \(f\left( x\right) \le b\) for some \(b\ge 0\) and all \(x\in \left[ 0,a\right] \);

  2. (ii)

    \(f\left( x\right) \ge c\) for some \(c\le 0\) and all \(x\in \left[ 0,a\right] \);

  3. (iii)

    f is continuous at zero;

  4. (iv)

    f is Lebesgue-measurable.

Theorem 1 says that non-linear solutions of Eq. (6) cannot be bounded from below or above, continuous at zero or Lebesgue measurable. We will now prove the linearity of f for Case (i). The proofs for the remaining cases are given in Appendix A.

Proof

For any real number \(x\in \left[ 0,a\right] \), Eq. (6) implies that

$$\begin{aligned} f\left( x\right) =f\left( \frac{n}{n}x\right) =nf\left( \frac{1}{n}x\right) , \end{aligned}$$
(8)

where n is a positive integer. If we choose an integer \(m\in \mathbb {N}\) with \(m/n\in \left[ 0,a\right] \), then we have

$$\begin{aligned} f\left( \frac{m}{n}x\right) =mf\left( \frac{1}{n}x\right) =\frac{m}{n}f\left( x\right) . \end{aligned}$$
(9)

Any real number \(r\in (0,a/n)\) satisfies \(nr\in \left( 0,a\right) \) implying that

$$\begin{aligned} nf\left( r\right) =f\left( nr\right) \le b \end{aligned}$$
(10)

holds. Given any real number \(x\in \left[ 0,a\right] \) choose a rational number \(q_{x}\in \left[ 0,a\right] \) close to it, \(0<(x-q_{x})<a/n\). The difference between the additive function f(x) and the linear function f(1)x can be written as

$$\begin{aligned} \begin{aligned} f\left( x\right) -f\left( 1\right) x&=f\left( x-q_{x}+q_{x}\right) -f\left( 1\right) x\\&=f\left( x-q_{x}\right) +f\left( q_{x}\right) -f\left( 1\right) x\\&=f\left( x-q_{x}\right) +\left( q_{x}-x\right) f\left( 1\right) . \end{aligned} \end{aligned}$$
(11)

By letting \(r=\left( x-q_{x}\right) \) in Eq. (10), it follows that the modulus of the difference is given by

$$\begin{aligned} \left| f\left( x\right) -f\left( 1\right) x\right| \le \left| f\left( x-q_{x}\right) \right| +\left| \left( q_{x}-x\right) f\left( 1\right) \right| <\frac{\left( a+1\right) b}{n}, \end{aligned}$$
(12)

which, upon taking the limit \(n\rightarrow \infty \), implies that f(x) must have the form given in Eq. (7). \(\square \)

3 When Cauchy Meets Gleason: Additive Functions on Effect Spaces

The first Gleason-type theorem discovered in 2003 assumes additivity of the frame function not only on projections which occur in the same PVM but on the larger set of effects which coexist in the same POM.

Theorem 2

(Busch [3]) Let \(\mathcal {E}_{d}\) be the space of effects on \(\mathbb {C}^{d}\) and \({{\,\mathrm{I}\,}}_{d}\) be the identity operator on \(\mathbb {C}^{d}\). Any function \(f:\mathcal {E}_{d}\rightarrow \left[ 0,1\right] \) satisfying

$$\begin{aligned} f\left( {{\,\mathrm{I}\,}}_{d}\right) =1, \end{aligned}$$
(13)

and

$$\begin{aligned} f\left( E_{1}\right) +f\left( E_{2}\right) =f\left( E_{1}+E_{2}\right) , \end{aligned}$$
(14)

for all \(E_{1},E_{2}\in \mathcal {E}_{d}\) such that \(\left( E_{1}+E_{2}\right) \in \mathcal {E}_{d}\), admits an expression

$$\begin{aligned} f\left( E\right) ={{\,\mathrm{Tr}\,}}\left( E\rho \right) , \end{aligned}$$
(15)

for some density operator \(\rho \), and all effects \(E\in \mathcal {E}_{d}\).

Theorem 2 rephrases the (finite-dimensional case of the) theorem proved by Busch [3] and the theorem due to Caves et al. [6]. Busch uses the positivity of the frame function f to directly establish its homogeneity whereas Caves et al. derive homogeneity by showing that the frame function f must be continuous at the zero operator. These arguments seem to run in parallel with Cases (ii) and (iii) of Theorem 1 presented in the previous section. In Sect. 3.2, we will give an alternative proof of Theorem 2 which can be based on any of the four cases of Theorem 1.

3.1 Preliminaries

To begin, let us introduce a number of useful concepts and establish a suitable notation. Throughout this section we will make use of the fact that the Hermitian operators on \(\mathbb {C}^{d}\) constitute \(\mathbb {H}^{d^{2}}\), a real vector space of dimension \(d^{2}\). We may therefore employ the standard inner product \(\left\langle A,B\right\rangle ={{\,\mathrm{Tr}\,}}\left( AB\right) \), for Hermitian operators A and B, in our reasoning as well as the norm \(\left\| \cdot \right\| \) which it induces.

A discrete POM on \(\mathbb {C}^{d}\) is described by its range, i.e. by a sequence of effects \(\llbracket E_{1},E_{2},\dots \rrbracket \) that sum to the identity operator on \(\mathbb {C}^{d}\). A minimal informationally-complete (MIC) POM \(\mathcal {M}\) on \(\mathbb {C}^{d}\) consists of exactly \(d^{2}\) linearly independent effects, \(\mathcal{M}=\llbracket M_{1},\ldots ,M_{d^{2}}\rrbracket \). Hence, MIC-POMs constitute bases of the vector space of Hermitian operators, and it is known that they exist in all finite dimensions [5].

Positive linear combinations of effects will play an important role below, giving rise to the following definition.

Definition 1

The positive cone of a set of Hermitian operators \(S=\left\{ H_{j},j\in J\right\} \) on \(\mathbb {C}^{d}\), where J is some index set, consists of the operators

$$\begin{aligned} \mathcal {C}\left( S\right) =\left\{ H=\sum _{j\in J}a_{j}H_{j},\,a_{j}\ge 0\right\} . \end{aligned}$$
(16)

Next, we introduce so-called “augmented” bases of the space \(\mathbb {H}^{d^{2}}\) which are built around sets of d projections \(\left\{ |e_{1}\rangle \langle e_{1}|,\ldots ,|e_{d}\rangle \langle e_{d}|\right\} \) where the vectors \(\left\{ |e_{1}\rangle ,\ldots |e_{d}\rangle \right\} \) form an orthonormal basis of \(\mathbb {C}^{d}\).

Definition 2

An augmented basis of the Hermitian operators on \(\mathbb {C}^{d}\) is a set of \(d^{2}\) linearly independent rank-one effects \(\mathcal {B}=\left\{ B_{1,}\ldots ,B_{d^{2}}\right\} \) satisfying

  1. (i)

    \(B_{j}=c|e_{j}\rangle \langle e_{j}|\) for \(1\le j\le d\), with \(0<c<1\) and an orthonormal basis \(\left\{ |e_{1}\rangle ,\ldots |e_{d}\rangle \right\} \) of \(\mathbb {C}^{d}\);

  2. (ii)

    \(\sum _{j=1}^{d^{2}}B_{j}\in \mathcal {E}_{d}\,\).

Given any orthonormal basis \(\left\{ |e_{1}\rangle ,\ldots ,|e_{d}\rangle \right\} \) of \(\mathbb {C}^{d}\), we can construct an augmented basis for the space of operators acting on it. First, complete the d projectors

$$\begin{aligned} \Pi _{j}=|e_{j}\rangle \langle e_{j}|,\qquad j=1\ldots d, \end{aligned}$$
(17)

into a basis \(\left\{ \Pi _{j},1\le j\le d^{2}\right\} \) of the Hermitian operators on \(\mathbb {C}^{d}\), by adding \(d(d-1)\) further rank-one projections; this is always possible [5]. The sum

$$\begin{aligned} G=\sum _{j=1}^{d^{2}}\Pi _{j}, \end{aligned}$$
(18)

is necessarily a positive operator. The relation \({{\,\mathrm{Tr}\,}}G=d^{2}\) implies that G must have at least one eigenvalue larger than 1. If \(\Gamma >1\) is the largest eigenvalue of G, then \(G/\Gamma \) is an effect since it is a positive operator with eigenvalues less than or equal to one. Defining

$$\begin{aligned} B_{j}=\Pi _{j}/\Gamma ,\qquad j=1\ldots d^{2}, \end{aligned}$$
(19)

the set \(\mathcal {B}=\left\{ B_{1},\ldots ,B_{d^{2}}\right\} \) turns into an augmented basis. One can show that \(\mathcal {B}\) can never correspond to a POM. Nevertheless, the effects \(B_{j}\) are coexistent, in the sense that they can occur in one single POM, for example \(\llbracket B_{1},\ldots ,B_{d^{2}},{{\,\mathrm{I}\,}}-G/\Gamma \rrbracket \).

Given an effect, one can always represent it as a positive linear combination of elements in a suitable augmented basis.

Lemma 1

For any effect \(E\in \mathcal {E}_{d}\) there exists an augmented basis \(\mathcal {B}\) such that E is in the positive cone of \(\mathcal {B}\).

Proof

By the spectral theorem we may write

$$\begin{aligned} E=\sum _{j=1}^{d}\lambda _{j}|e_{j}\rangle \langle e_{j}|,\qquad \lambda _{j}\in \left[ 0,1\right] , \end{aligned}$$
(20)

for an orthonormal basis \(\left\{ |e_{j}\rangle ,1\le j\le d\right\} \) of \(\mathbb {C}^{d}\). Take \(\mathcal {B}\) to be an augmented basis with

$$\begin{aligned} B_{j}=c|e_{j}\rangle \langle e_{j}|, \end{aligned}$$
(21)

for \(1\le j\le d\) and some \(c\in \left( 0,1\right) \). Then we may express E as the linear combination

$$\begin{aligned} E=\sum _{j=1}^{d^{2}}e_{j}B_{j}, \end{aligned}$$
(22)

with non-negative coefficients

$$\begin{aligned} e_{j}={\left\{ \begin{array}{ll} \frac{1}{c}\lambda _{j} &{} j=1\ldots d,\\ 0 &{} j=(d+1)\ldots d^{2}, \end{array}\right. } \end{aligned}$$
(23)

showing that the positive cone of the basis \(\mathcal{B}\) indeed contains the effect E. \(\square \)

Finally, we need to establish that the intersection of the positive cones associated with an augmented basis and a MIC POM, respectively, has dimension \(d^{2}\).

Lemma 2

Let \(\mathcal {B}=\left\{ B_{1},\ldots ,B_{d^{2}}\right\} \) be an augmented basis and \(\mathcal {M}=\llbracket M_{1},\ldots ,M_{d^{2}}\rrbracket \) a MIC-POM on \(\mathbb {C}^{d}\). The effects in the intersection \(\mathcal{C}(\mathcal{B})\cap \mathcal{C}(\mathcal{M})\) of the positive cones of \(\mathcal {B}\) and \(\mathcal {M}\) span the real vector space \(\mathbb {H}^{d^{2}}\) of Hermitian operators on \(\mathbb {C}^{d}\).

Proof

Since the effects in a POM sum to the identity, we have

$$\begin{aligned} \frac{1}{d}{{\,\mathrm{I}\,}}_{d}=\sum _{j=1}^{d^{2}}\frac{1}{d}M_{j}. \end{aligned}$$
(24)

With each of the coefficients in the unique decomposition on the right-hand side being finite and positive (as opposed to non-negative), the effect \({{\,\mathrm{I}\,}}_{d}/d\) is seen to be an interior point of the positive cone \(\mathcal {C}\left( \mathcal {M}\right) \). At the same time, the effect \({{\,\mathrm{I}\,}}_{d}/d\) is located on the boundary of the cone \(\mathcal{C}(\mathcal{B})\) since its expansion in an augmented basis has only d non-zero terms. Let us define the operator

$$\begin{aligned} E_{\delta }=\frac{1}{d}{{\,\mathrm{I}\,}}_{d}+\delta \sum _{j=d+1}^{d^{2}}B_{j}=\frac{1}{cd}\sum _{j=1}^{d}B_{j}+\delta \sum _{j=d+1}^{d^{2}}B_{j}, \end{aligned}$$
(25)

which, for any positive \(\delta >0\), is an interior point of the cone \(\mathcal {C}\left( \mathcal {B}\right) \): each of the positive coefficients in its unique decomposition in terms of the augmented basis \(\mathcal{B}\) is non-zero; we have used Property 1 of Definition 2 to express the identity \({{\,\mathrm{I}\,}}_{d}\) in terms of the basis \(\mathcal{B}\). For sufficiently small values of \(\delta \), the operator \(E_{\delta }\) is also an interior point of the open ball \(\mathfrak {B}_{\varepsilon }\left( {{\,\mathrm{I}\,}}_{d}/d\right) \) with radius \(\varepsilon \) about the point \(I_{d}/d\) since

$$\begin{aligned} \left\| E_{\delta }-\frac{1}{d}{{\,\mathrm{I}\,}}_{d}\right\| =\delta \left\| \sum _{j=d+1}^{d^{2}}B_{j}\right\| <\varepsilon \end{aligned}$$
(26)

holds whenever

$$\begin{aligned} 0<\delta <\varepsilon \left\| \sum _{j=d+1}^{d^{2}}B_{j}\right\| ^{-1}\,. \end{aligned}$$
(27)

Being an interior point of both the positive cones \(\mathcal {C}\left( \mathcal {B}\right) \) and \(\mathcal {C}\left( \mathcal {M}\right) \), the operator \(E_{\delta }\) is at the center of an open ball \(\mathfrak {B}_{\gamma }\left( E_{\delta }\right) \), located entirely in the intersection \(\mathcal{C}(\mathcal{B})\cap \mathcal{C}(\mathcal{M})\) (cf. Fig. 1). Since the ball \(\mathfrak {B}_{\gamma }\left( E_{\delta }\right) \) has dimension \(d^{2},\) the effects contained in it must indeed span the real vector space \(\mathbb {H}^{d^{2}}\) of Hermitian operators. \(\square \)

Fig. 1
figure 1

Sketch of the construction of the open ball \(\mathfrak {B}_{\gamma }\left( E_{\delta }\right) \) of dimension \(d^{2}\); the positive cones \(\mathcal {C}\left( \mathcal {M}\right) \) (solid border) and \(\mathcal {C}\left( \mathcal{B}\right) \) (dotted border) intersect in the cone \(\mathcal{C}(\mathcal{B})\cap \mathcal{C}(\mathcal{M})\) (shaded cone); the intersection entirely contains the \(d^{2}\)-dimensional ball \(\mathfrak {B}_{\gamma }\left( E_{\delta }\right) \) around \(E_{\delta }\) (solid circle) sitting inside the ball \(\mathfrak {B}_{\varepsilon }\left( {{\,\mathrm{I}\,}}_{d}/d\right) \) of radius \(\varepsilon \) around \({{\,\mathrm{I}\,}}_{d}/d\) (dashed circle); the distance between \(E_{\delta }\) and \({{\,\mathrm{I}\,}}_{d}/d\) (solid line) is given in Eq. (26)

Full size image

Combining Theorem 1 with Lemmas 1 and 2 will allow us to present a new proof of Busch’s Gleason-type theorem.

3.2 An Alternative Proof of Busch’s Gleason-Type Theorem

Recalling that the trace of the product of two Hermitian operators constitutes an inner product on the vector space of Hermitian operators, Theorem 2 essentially states that the frame function f acting on an effect can be written as the inner product of that effect with a fixed density operator. To underline the connection with the inner product we adopt the following notation. Let \(\mathcal {A}=\left\{ A_{1},\ldots A_{d^{2}}\right\} \) be a basis for the Hermitian operators on \(\mathbb {C}^{d}\). We describe the effect E by the “effect vector” \(\mathbf {e}=\left( e_{1},\ldots ,e_{d^{2}}\right) ^{T}\in \mathbb {R}^{d^{2}}\), given by its expansion coefficients in this basis,

$$\begin{aligned} E=\sum _{j=1}^{d^{2}}e_{j}A_{j}\equiv \mathbf {e}\cdot \mathbf {A}, \end{aligned}$$
(28)

where \(\mathbf {A}\) is an operator-valued vector with \(d^{2}\) components. Theorem 2 now states that the frame function is given by a scalar product,

$$\begin{aligned} f\left( E\right) =\mathbf {e}\cdot \mathbf {c}, \end{aligned}$$
(29)

between the effect vector \(\mathbf {e}\) and a fixed vector \(\mathbf {c}\in \mathbb {R}^{d^{2}}\). Let us determine the relation between the density matrix \(\rho \) in (15) in the theorem and the vector \(\mathbf {c}\) in (29). Consider any orthonormal basis \(\mathcal {W}=\left\{ W_{j}\right\} \) of the Hermitian operators on \(\mathbb {C}^{d}\) and let \(\mathbf {e}^{\prime }\in \mathbb {R}^{d^{2}}\) be the vector such that \(E=\mathbf {e^{\prime }}\cdot \mathbf {W}\). Then we may write

$$\begin{aligned} f\left( E\right)&=\mathbf {e}\cdot \mathbf {c}=\mathbf {e}^{\prime }\cdot \mathbf {c}^{\prime }={{\,\mathrm{Tr}\,}}\left( \sum _{j=1}^{d^{2}}e_{j}^{\prime }W_{j}\sum _{k=1}^{d^{2}}c_{k}^{\prime }W_{k}\right) \nonumber \\&={{\,\mathrm{Tr}\,}}\left( E\sum _{j=1}^{d^{2}}c_{j}^{\prime }W_{j}\right) \,; \end{aligned}$$
(30)

here \(\mathbf {c^{\prime }}\in \mathbb {R}^{d^{2}}\) is a fixed vector given by \(\mathbf {c}^{\prime }=C^{-T}\mathbf {c}\) and \(C^{-T}\) is the inverse transpose of the change-of-basis matrix C between the bases \(\mathcal {B}\) and \(\mathcal {W}\), i.e. the matrix satisfying \(C\mathbf {h}=\mathbf {h}^{\prime }\) for all Hermitian operators \(H=\mathbf {h}\cdot \mathbf {B}=\mathbf {h^{\prime }}\cdot \mathbf {W}\). By the definition of a frame function the operator

$$\begin{aligned} \rho \equiv \sum _{j=1}^{d^{2}}c_{j}^{\prime }W_{j}=\sum _{j=1}^{d^{2}}\left( C^{-T}\right) _{jk}c_{k}W_{j} \end{aligned}$$
(31)

must be positive semi-definite (since f is positive) and have unit trace [due to Eq. (13)] i.e. be a density operator.

We will now prove that a frame function always admits an expression as in Eq. (29).

Proof

By Lemma 1, there exists an augmented basis \(\mathcal {B}=\left\{ B_{1},\ldots ,B_{d^{2}}\right\} \) for any \(E\in \mathcal {E}_{d}\) such that

$$\begin{aligned} E=\mathbf {e}\cdot \mathbf {B}\equiv \sum _{j=1}^{d^{2}}e_{j}B_{j}, \end{aligned}$$
(32)

with coefficients \(e_{j}\ge 0\), as in Eq. (28).

For each value \(j\in \left\{ 1,\dots ,d^{2}\right\} \), we write the restriction of the frame function f to the set of effects of the form \(xB_{j}\), for \(x\in \mathbb {R}\), as

$$\begin{aligned} f\left( xB_{j}\right) =F_{j}\left( x\right) , \end{aligned}$$
(33)

where \(F_{j}:\left[ 0,a_{j}\right] \rightarrow \left[ 0,1\right] \) and \(a_{j}=\max \left\{ x|xB_{j}\in \mathcal {E}_{d}\right\} \). By Eq. (14) we have that \(F_{j}\) satisfies Cauchy’s functional equation, i.e. \(F_{j}\left( x+y\right) =F_{j}\left( x\right) +F_{j}\left( y\right) \). Due to the assumption in Theorem 2 that \(f:\mathcal {E}_{d}\rightarrow \left[ 0,1\right] \), each \(F_{j}\) must satisfy Condition (i) of Theorem 1 which implies

$$\begin{aligned} f\left( xB_{j}\right) =F_{j}\left( x\right) =F_{j}\left( 1\right) x=f\left( B_{j}\right) x. \end{aligned}$$
(34)

Thus we find

$$\begin{aligned} f\left( E\right) =\sum _{j=1}^{d^{2}}f\left( e_{j}B_{j}\right) =\sum _{j=1}^{d^{2}}e_{j}f\left( B_{j}\right) =\mathbf {e}\cdot \mathbf {f}_{\mathcal {B}}, \end{aligned}$$
(35)

where the j-th component of \(\mathbf {f}_{\mathcal {B}}\in \mathbb {R}^{d^{2}}\) is given by \(f\left( B_{j}\right) \), by repeatedly using additivity and Eq. (34). Note that Eq. (35) is not yet in the desired form of Eq. (29) since the vector \(\mathbf {f}_{\mathcal {B}}\) depends on the basis \(\mathcal {B}\) and thus the effect E.

Let \(\mathcal {M}=\llbracket M_{1},\ldots ,M_{d^{2}}\rrbracket \) be a MIC-POM on \(\mathbb {C}^{d}\). Since the elements of \(\mathcal {M}\) are a basis for the space \(\mathbb {H}^{d^{2}}\), the Hermitian operators on \(\mathbb {C}^{d}\), we have for any \(E\in \mathcal {E}_{d}\)

$$\begin{aligned} E=\mathbf {e^{\prime \prime }}\cdot \mathbf {M}, \end{aligned}$$
(36)

for coefficients \(e_{j}^{\prime \prime }\in \mathbb {R}\) which may be negative. There exists a fixed change-of-basis matrix D such that

$$\begin{aligned} D\mathbf {e}=\mathbf {e^{\prime \prime }}, \end{aligned}$$
(37)

for all \(E\in \mathcal {E}_{d}\). Now we have

$$\begin{aligned} \begin{aligned} f\left( E\right)&=\mathbf {e}\cdot \mathbf {f}_{\mathcal {B}}\\&=\left( D\mathbf {e}\right) \cdot \left( D^{-T}\mathbf {f}_{\mathcal {B}}\right) \\&=\mathbf {e^{\prime \prime }}\cdot \left( D^{-T}\mathbf {f}_{\mathcal {B}}\right) . \end{aligned} \end{aligned}$$
(38)

Any effect G in the intersection of the positive cones \(\mathcal {C}\left( \mathcal {B}\right) \) and \(\mathcal {C}\left( \mathcal {M}\right) \) can be expressed in two ways,

$$\begin{aligned} G=\mathbf {g}\cdot \mathbf {B}=\mathbf {g}^{\prime \prime }\cdot \mathbf {M}, \end{aligned}$$
(39)

where both effect vectors \(\mathbf {g}\) and \(\mathbf {g}^{\prime \prime }\) have only non-negative components. Eqs. (35) and (38) imply that

$$\begin{aligned} \begin{aligned} \mathbf {g^{\prime \prime }}\cdot \mathbf {f}_{\mathcal {M}}&=f\left( G\right) =\mathbf {g}^{\prime \prime }\cdot \left( D^{-T}\mathbf {f}_{\mathcal {B}}\right) . \end{aligned} \end{aligned}$$
(40)

Since, by Lemma 2, there are \(d^{2}\) linearly independent effects G in the intersection \(\mathcal {C}\left( \mathcal {M}\right) \cap \mathcal {C}\left( \mathcal {B}\right) \), we conclude that

$$\begin{aligned} D^{-T}\mathbf {f}_{\mathcal {B}}=\mathbf {f}_{\mathcal {M}}. \end{aligned}$$
(41)

Combining this equality with Equation (38) we find, for a fixed MIC-POM \(\mathcal {M}=\llbracket M_{1},\ldots ,M_{d^{2}}\rrbracket \) and any effect \(E\in \mathcal {\mathcal {E}}_{\mathbb {C}^{d}}\), that the frame function f takes the form

$$\begin{aligned} f\left( E\right) =\mathbf {e}^{\prime \prime }\cdot \mathbf {f}_{\mathcal {M}}. \end{aligned}$$
(42)

Here \(\mathbf {f}_{\mathcal {M}}\equiv \mathbf {c}\) is a fixed vector since it does not depend on E. \(\square \)

Note that Eq. (34) may also be found using the other three cases of Theorem 1. For Case (ii), we observe that each of the functions \(F_{j},j=1\ldots d^{2}\), is non-negative by definition. Alternatively, each function \(F_{j}\) can be shown to be continuous at zero [Case (iii)] using the following argument which is similar to the one given in [5]. Assume \(F_{j}\) is not continuous at zero. Then there exists a number \(\varepsilon >0\) such that for all \(\delta >0\) we have

$$\begin{aligned} F_{j}\left( x_{0}\right) >\varepsilon , \end{aligned}$$
(43)

for some \(0<x_{0}<\delta <1\). For any given \(\varepsilon \) choose \(\delta =1/n<\varepsilon \), there is a value of \(x_{0}<\delta \) such that \(F_{j}\left( x_{0}\right) >\varepsilon \). However, \(nx_{0}<1\), which leads to

$$\begin{aligned} F_{j}\left( nx_{0}\right) =nF_{j}\left( x_{0}\right)>n\varepsilon >1, \end{aligned}$$
(44)

contradicting the the existence of an upper bound of one on values of \(F_{j}\). Finally, each of the functions \(F_{j}\) is Lebesgue measurable (Case (iv)) which follows from the monotonicity of the function.

4 Summary and Discussion

We are aware of two papers linking Gleason’s theorem and Cauchy’s functional equation. Cooke et al. [7] used Cauchy’s functional equation to demonstrate the necessity of the boundedness of frame functions in proving Gleason’s theorem. Dvurečenskij [10] introduced frame functions defined on effect algebras but did not proceed to derive a Gleason-type theorem in the context of quantum theory.

In this paper, we have exploited the fact that additive functions are central to both Gleason-type theorems and Cauchy’s functional equation. Gleason-type theorems are based on the assumption that states assign probabilities to measurement outcomes via additive functions, or frame functions,on the effect space. Linearity of the frame functions has been shown to follow from positivity and other assumptions, which are well-known in the context of Cauchy’s functional equation. Altogether, the result obtained here amounts to an alternative proof of the extension of Gleason’s theorem to dimension two given by Busch [3] and Caves et al. [5].

Other Gleason-type theorems are known which are stronger, in the sense that they depend on assumptions weaker than those of Theorem 2. The smallest known set of assumptions requires Eq. (14) to only be valid for effects \(E_{1}\) and \(E_{2}\) which coexist in a projective-simulable POM [15], i.e. a POM which may be simulated using only classic mixtures of projective measurements, as opposed to any POM. Since the proof given in [16] relies on Theorem 2, the alternative proof presented in Sect. 3.2 also gives rise to a new proof of the strongest existing Gleason-type theorem.

We have not been able to exploit the structural similarity between the requirements on frame functions and on the solutions of Cauchy’s functional equation in order to yield a new proof of Gleason’s original theorem. Additivity of frame functions defined on projections instead of effects does not provide us with the type of continuous parameters which are necessary for the argument developed here. It remains an intriguing open question whether such a proof does exist.