Framed Hilbert space: hanging the quasi-probability pictures of quantum theory

Building on earlier work, we further develop a formalism based on the mathematical theory of frames that defines a set of possible phase-space or quasi-probability representations of finite-dimensional quantum systems. We prove that an alternate approach to defining a set of quasi-probability representations, based on a more natural generalization of a classical representation, is equivalent to our earlier approach based on frames, and therefore is also subject to our no-go theorem for a non-negative representation. Furthermore, we clarify the relationship between the contextuality of quantum theory and the necessity of negativity in quasi-probability representations and discuss their relevance as criteria for non-classicality. We also provide a comprehensive overview of known quasi-probability representations and their expression within the frame formalism.

However, the Wigner function representation is non-unique, and, moreover, Wigner's approach is not applicable to describing quantum systems with a finite set of distinguishable states. In recent years various analogs of the Wigner function for finite dimensional quantum systems have been proposed. An important subclass of such quasi-probability representations are those defined on a discrete phase space. We will use the term quasiprobability representation to refer to the broader class of representations where the (ontic) state space is neither required to be discrete nor required to carry any classical phase space structure 2 . Such representations have provided insight into fundamental properties of finite-dimensional quantum systems. For example, the representation proposed by Wootters identifies sets of mutually unbiased bases [4,5]. Inspired by the discovery that quantum resources lead to algorithms that dramatically outperform their classical counterparts, there has also been growing interest in the application of discrete phase space formalism to analyze the quantum-classical contrast for finite-dimensional systems. Examples include analyses of quantum teleportation [6], the effect of decoherence on quantum walks [7], quantum Fourier transform and Grover's algorithm [8], conditions for exponential quantum computational speedup [9,10], and quantum expanders [11].
As noted above, a central concept in studies of the quantum-classical contrast in the quasi-probability formalisms of quantum theory is the appearance of negativity. A nonnegative quasi-probability function is a true probability distribution, prompting some authors to suggest that the presence of negativity in this function is a defining signature of non-classicality. Unfortunately the application of any one of these quasi-probability representations in the context of determining criteria for the non-classicality of a given quantum task is limited in significance by the non-uniqueness of that particular representation. Ideally one would like to determine whether the task can be expressed as a classical process in any quasi-probability representation. Indeed the sheer variety of proposed quasi-probability representations prompts the question of whether there is some shared underlying mathematical structure that might provide a means for identifying the full family of such representations. Moreover, from an operational view, states alone are an incomplete description of an experimental arrangement. Hence, it is important to elucidate the ways in which a quasi- 2 The term quasi-probability is used to allow for the appearance of negative values in the states and/or measurements defined under the representation. probability representation of states alone can be lifted into an autonomous quasi-probability representation of both the states and measurements defining any set of experimental configurations.
In an earlier paper [12], we introduced the concept of frame representation which mathematically unifies the known quasi-probability representations of quantum states. Once the frame is identified for a particular quasi-probability representation, we showed how any dual frame leads to an autonomous representation of the operational quantum formalism (i.e., including both states and measurements). Then we proved that such representation must possess negative values in either the states or measurements (or both). However, because this approach explicitly made use of dual frames to construct the autonomous formulation of quantum mechanics, it remained possible that negativity could be avoided through some other representation that is not explicitly defined via a selection of a dual frame. In this paper we show that this is not possible. Specifically, we consider an alternative and more natural approach to the definition of the class of autonomous quasi-probability representations for finite-dimensional quantum mechanics and show that any such representation is equivalent to a frame representation and therefore can not avoid negativity.
The outline of the paper is as follows. In Section II, we give a comprehensive overview of the known quasi-probability functions representing quantum states. In Section III, we review the results of [12] showing how the mathematical theory of frames provides a formalism which underlies all known quasi-probability representations of finite dimensional quantum states. Furthermore, we explicitly construct the frames which give rise to several important quasi-probability representations. In Section IV we review the two ways, presented in [12], in which any frame representation of states can be extended to include a representation of measurements, and hence lifted to a fully autonomous formulation of finite dimensional quantum mechanics. In Section V we propose a more natural approach to the definition of quasi-probability representations and show that this approach is equivalent to the approach obtained under frame representations and therefore does not admit a non-negative representation. In Section VI, we look at the equivalence of non-negativity in quasi-probability representations and non-contextuality in ontological models. In Section VII we discuss the extension of our results to infinite-dimensional quantum systems and consider potential applications of our work in quantum information science.

II. REVIEW OF QUASI-PROBABILITY FUNCTIONS
Reviewed in this section are the existing quasi-probability representations of quantum states found in the literature. The original quasi-probability representation put forth by Wigner and later recognized as an equivalent formulation of the full quantum theory by Moyal [13] and others [3] is reviewed first. This phase space picture is only valid for infinite dimensional Hilbert spaces but it will be reviewed here because it has motivated all known generalizations for finite dimensions. Sections II A-II F are devoted to reviewing a representative sample of the known quasi-probability representations of finite dimensional quantum states.
In Section II G a summary of these and a few other quasi-probability representations is presented in a more concise manner.

A. Wigner phase space representation
The position operator Q and momentum operator P are the central objects in the abstract formalism of infinite dimensional quantum theory. The operators satisfy the canonical commutation relations [Q, P ] = i.
Since Q and P do not commute, the choice of the quantization map g(q, p) → g(Q, P ), for some function g, is not unique. This is the so-called "ordering problem". A class of solutions to this problem is the association e iξq+iηp → e iξQ+iηP f (ξ, η) for some arbitrary function f (See Table 1 of [2] for a review of the traditional choices for f ).
Consider the classical particle phase space R 2 and the continuous set of operators When f (ξ, η) = 1, the distribution µ Wigner ρ (q, p) := Tr(ρF (q, p)) is the celebrated Wigner function [1]. The Wigner function is both positive and negative in general. However, it otherwise behaves as a classical probability density on the classical phase space. For these reasons, the Wigner function and others like it came to be called quasi-probability functions.
The Wigner function is the unique representation satisfying the properties [14] Wig(1) For all ρ, µ Wigner ρ (q, p) is real.
Wig (2) For all ρ 1 and ρ 2 , Wig (3) For all ρ, integrating µ Wigner ρ along the line aq + bp = c in phase space yields the probability that a measurement of the observable aQ + bP has the result c.
Notice from Equation (2) (1) for f = 1. These properties are Wig (6) Let P c be the projector onto the eigenstate of aQ + bP with eigenvalue c. Then, The Wigner functions has many properties and applications [15] which are not of concern here. However, it is important to note that the wide variety of fruitful applications of the Wigner function is responsible for the interest in its generalization. The properties Wig(1)- Assume the the Hilbert space H has composite dimension d = d 1 d 2 · · · d k . The discrete phase space of the entire d dimensional system is the Cartesian product of two-dimensional prime phase spaces of the subsystems. The phase space is thus a ( Such as construction is formalized as follows: the discrete phase space . . , λ k ) of lines in the prime phase spaces. That is, a line is the set of d points satisfying the equation which is symbolically written aq + bp = c. The choice of phase point operators in [4] will be adopted. For d prime, the phase point operators are where ω is a d'th root of unity and X and Z are the generalized Pauli operators (see Appendix A). For composite d, the phase point operator in Φ d associated with the point α = (α 1 , α 2 , . . . , α k ) is given by where each A α i is the phase point operator of the point α i in Φ d i . The with m, n ∈ Z d and φ k are the eigenvectors of Z (see Appendix A). Then, the discrete Wigner function of a density operator ρ is where P is the parity operator (see Appendix A).
The authors call the operators qp = W qp P Fano operators and note that they satisfy † qp = qp , 2 qp = 1, The Fano operators play a role similar to Wootters' phase point operators; they form a complete basis of the space of Hermitian operators. The phase space implicitly defined through the definition of the discrete Wigner function (7)  Moreover, they are not even linearly independent which can easily be inferred since there are 4d 2 of them and a set of linearly independent operators contains a maximum of d 2 operators.

E. Wigner functions on the sphere
In [19], Heiss and Weigert are concerned with a set of postulates put forth by Stratonovich [20]. The aim of Stratonovich was to find a Wigner function type mapping, analogous to that of a infinite dimensional system on R 2 , of a d dimensional system on the sphere S 2 .
The first postulate is linearity and is always satisfied if the Wigner functions on the sphere satisfy µ sphere ρ (n) = Tr(ρ (n)), where n is a point on S 2 . The remaining postulates on this quasi-probability mapping are where g · ρ is the image of U g ρU † g and U : SU (g · n) = U g (n)U † g , g ∈ SU(2).
These postulates are the spherical analogies of properties Wig(4)-(6) (again, modulo the normalization condition). Heiss and Weigert provide a derivation of 2 2s , where s = d−1 2 is the spin, unique kernels satisfying these postulates. They are where C denotes the so-called Clebsch-Gordon coefficients; φ m (n) are the eigenvectors of the operator S · n, where S = (X, Y, Z); and l = ±1, for l = 1 . . . 2s and 0 = 1.
Heiss and Weigert relax the postulates Equations (9)-(12) on the kernel (n) to allow for a pair of kernels n and m . The pair individually satisfy Equation (9), while one of them satisfies Equation (10) and the other Equation (12). Together, the pair must satisfy the generalization of Equation (11) A pair of kernels, together satisfying Equation (14), is given by where γ l = ±1 for l = 1 . . . 2s and γ 0 = 1. The original postulates are satisfied when The major contribution of [19] is the derivation of a discrete kernel ν := nν , for The subset of points n ν is called a constellation. The linearity postulate is not explicitly stated since it is always satisfied under the assumption Equation (17) is called a duality condition. That is, it is only satisfied if ν and µ are dual bases for H(H). In particular, Although the explicit construction of a pair of discrete kernels satisfying Equations (15)- (18) might be computationally hard, their existence is a trivial exercise in linear algebra. Indeed, so long as ν is a basis for H(H), its dual, µ , is uniquely determined by where the Gram matrix G is given by The authors of [19] note that almost any constellation leads to a discrete kernel ν forming a basis for H(H). The term almost any here means that a randomly selected discrete kernel will form, with probability 1, a basis for H(H).

F. Finite fields discrete phase space representation
Recall that when dim(H) = d is prime, Wootters defines the discrete phase space as a d × d lattice indexed by the group Z d . In [21], Wootters generalizes his original construction of a discrete phase space to allow the d × d lattice to be indexed by a finite field F d which exists when d = p n is an integer power of a prime number. This approach is discussed at length in the paper [5] authored by Gibbons, Hoffman and Wootters (GHW).
Similar to his earlier approach, Wootters defines the phase space, Φ d , as a d × d array of A translation in phase space, T α 0 , adds a constant vector, α 0 = (q 0 , p 0 ), to every phase space point:

in a striation is invariant under a translation by
any point contained in its ray, parameterized by the points (sq, sp). That is, The discrete Wigner function is where now the Hermitian phase point operators satisfy the following properties for a projector valued function Q, called a quantum net, to be defined later.
GHW (6) For any line λ, The projector valued function Q assigns quantum states to lines in phase space. This mapping is required to satisfy the special property of translational covariance, which is defined after a short, but necessary, mathematical digression. Notice first that properties GHW(4) and GHW(5) are identical to Woo(4) and Woo (4). Also note that if GHW (6) is to be analogous to Woo (6), the property of translation covariance must be such that the set {Q(λ)} when λ ranges over a striation forms a PVM.

The set of elements
where each x i is an element of the prime field Z p . The field trace 3 of any field element is given by There exists a unique field basis,Ẽ = {ẽ 0 , ...ẽ n−1 }, such that tr(ẽ i e j ) = δ ij . We callẼ the dual of E.
The construction presented in [5] is physically significant for a system of n objects (called particles) having a p dimensional Hilbert space. A translation operator, T α associated with a point in phase space α = (q, p) must act independently on each particle in order to preserve the tensor product structure of the composite system's Hilbert space. We expand each component of the point α into its field basis decomposition as in Equation (21 and with f any element of F d . Note that the basis we choose for p is a multiple of the dual of that chosen for q. Now, the translation operator associated with the point (q, p) is Since X and Z are unitary, T α is unitary.
We assign with each line in phase space a pure quantum state. The quantum net Q is defined such that for each line, λ, Q(λ) is the operator which projects onto the pure state associated with λ. As a consequence of the choice of basis for p in Equation (24), the state assigned to the line τ α λ is obtained through This is the condition of translational covariance and it implies that each striation is associated with an orthonormal basis of the Hilbert space. To see this, recall the property in Equation (20). From Equation (26), this implies that, for each s ∈ F d , T (sq,sp) must commute with Q(λ), where the line λ is any line in the striation defined by the ray consisting of the points (sq, sp). That is, the states associated to the lines of the striation must be common eigenstates of the unitary translation operator T (sq,sp) , for each s ∈ F d . Thus, the states are orthogonal and form a basis for the Hilbert space. That is, their projectors form a PVM which makes GHW(6) identical to Woo(6) when d is prime.
In [5], the author's note that, although the association between states and vertical and horizontal lines is fixed, the quantum net is not unique. In fact, there are d d−1 quantum nets which satisfy Equation (20). When d is prime, one of these quantum nets corresponds exactly to the original discrete Wigner function defined by Wootters in Section II B.

G. Summary of existing quasi-probability representations of quantum states
The phase space functions reviewed in Sections II B-II F form only a subset of the literature on finite dimensional phase space functions; there are indeed several others (for a recent review see [22]). More generally, there exist quasi-probability representations given by real-valued representations that do not necessarily reflect any preconceived classical phase space structure. For example in [23] Hardy shows that five axioms are sufficient to imply a special quasi-probabilistic representation which is equivalent to an operational form of quantum theory. In [24] Havel also proposes an kind of analog of the Wigner function called the "real density matrix".
An overview of all of the quasi-probability representations for finite dimensional quantum systems reviewed above (as well as a couple more) is presented in Table I. The table identifies the ontic space structure and the mathematical field which indexes it (if applicable).
The second to last column indicates whether or not the representation contains redundant information. The last column reveals the scope of quantum theory the paper aims to cover (notice that typically only states are considered).

III. FRAME REPRESENTATIONS OF QUANTUM STATES
In the previous section we reviewed some of the quasi-probability representations of quantum states found in the literature. In Section III A, we unify these known quasi-probability representations of quantum states into one concise mathematical definition.
Section III B contains an introduction to the mathematical theory of frames. In Section III C it is shown that frame theory is both necessary and sufficient to define any quasiprobability representation of quantum states. The frames for each of the quasi-probability representations reviewed in Section II are given in Section III D.

A. Unification of existing quasi-probability representations of quantum states
Each of the quasi-probability functions discussed in Section II are linear representations of the density operator. These representations are also invertible as the density operator can be obtained from any quasi-probability function. Each quasi-probability function is a also member of the function space L 2 (Λ), where Λ represents a classical state space (e.g. the phase space, where applicable). These three properties will constitute the following minimal definition of a quasi-probability representation of quantum states alone [12]: This definition was chosen since all the known quasi-probability representations of states satisfy it. One might object that the restrictions imposed on this map are too strong.
Indeed there is no mention of linearity, invertibility, or L 2 spaces in any notion of classical probability. However, a classical probabilistic description describes an entire experimental arrangement. In Section V C we show that these ad hoc requirements follow from more natural assumptions on a representation on a complete experimental description. That is, Definition 1 is satisfied on the part of this more natural definition which represents quantum states.
Given Definition 1, any phase space function is then a particular type of quasi-probability representation.
Definition 2. If there exists a symmetry group on Λ, G, carrying a unitary representation U : G → U(H) and a quasi-probability representation satisfying the covariance property All phase space functions in the literature correspond to quasi-probability representations that satisfy this additional covariance condition. Table I shows that the range of validity in the Hilbert space dimension of these functions are often disjoint. Moreover, the construction of the phase spaces use varying mathematical structures: integers, finite fields and points on a sphere. Coupled with the fact that at least two known representations required redundancy, it may seem at first that Definition 1 is as far as one can can go in unifying the quasi-probability functions. However, a much stronger result is possible. In [12] it was shown that the mathematical theory of frames is both sufficient and necessary to describe any representation of quantum states satisfying Definition 1.

B. The mathematical theory of frames
A frame can be thought of as a generalization of an orthonormal basis [28]. However, the particular Hilbert space under consideration here is not H.

Definition 3. A frame for H(H) is a set of operators
for all A ∈ H(H) and some constants a, b > 0.
This definition generalizes a defining condition for an orthogonal basis for all A ∈ H(H).
for all A ∈ H(H), is a dual frame (to F).
The frame operator associated with the frame F is defined as If the frame operator satisfies S = a1, the frame is called tight. The frame operator is invertible and thus every operator has a representation The frame S −1 F is called the canonical dual frame. When |Λ| = d 2 , the canonical dual frame is the unique dual, otherwise there are infinitely many choices for a dual.
A tight frame is ideal from the perspective that its canonical dual is proportional to the frame itself. Hence, the reconstruction is given by the convenient formula which is to be compared with as an orthonormal basis. The mapping A → F (λ), A is usually called the analysis operation in the frame literature as it encodes the signal in terms of the frame. Here the notion of a signal is not appropriate and a more suggestive name has been chosen and formalized in the following definition.
where {F (λ)} is a frame, is a frame representation of H(H). Proof. The proof of this lemma also appears in [12]. As noted above, it is clear that a frame representation is a quasi-probability representation. So assume the mapping W : H(H) → L 2 (Λ) is a quasi-probability representation. Linearity and the Riesz representation theorem imply that W (A)(λ) = F (λ), A for some unique set F := {F (λ)} (not necessarily a frame).
Since H(H) is finite dimensional, the inverse W −1 is bounded. Thus W is bounded below by the bounded inverse theorem. That is, there exists a constant a > 0 such that Hence F is a frame.
Thus there is a unique frame which defines each of the quasi-probability functions reviewed in Section II. In the cases where the representation of the density operator is not redundant, the frame is just a basis. In the redundant cases, Leonhardt's even dimensional representation (Section II D) for example, the formalism of frame theory is necessary as a basis will not suffice.

D. Examples of quasi-probability representations of quantum states
The frames for each of the quasi-probability representations of quantum states reviewed in Section II are now given.

Wootters discrete Wigner function
Let d be a prime number.
The quasi-probability function µ Wootters ρ is a frame representation given by the frame Consider the group of translations on Λ with unitary representation T (q,p) = X q Z p . Then, Thus, by definition, the Wootters representation is a phase space representation.
Recall from Section II B that Wootters also considered non-prime dimensions. In that case, the phase point operators (Equation (5)) were a tensor product of phase point operators (Equation (4)) for prime dimensions. The same is true here for the frame in composite and each F Wootters (q (i) , p (i) ) is a frame as in Equation (III D 1).

Odd dimensional discrete Wigner functions
Let d be an odd integer.
The quasi-probability function µ odd ρ is a frame representation given by the frame F odd . The frame operator of F odd is S = d −11 . The unique dual frame of F odd is given by Consider the group of translations on Λ with unitary representation T (q,p) = X −q Z p . Then, By definition, this odd dimensional representation is a phase space representation.

Even dimensional discrete Wigner functions
Let d be an even integer. Then Λ = Z 2d × Z 2d . Consider the frame F even = {F even (q, p)}, The quasi-probability function µ even ρ is a frame representation given by the frame F even . The frame operator of F even is S = (2d) −11 . However, this implies the frame is only tight; it is not a basis and the dual is not unique. The canonical dual frame of F even is given by Then, Thus, by definition, this odd dimensional representation is a phase space representation.

Wigner functions on the sphere
Let d be any integer. Here the phase space is Λ = S 2 . Consider the frame F sphere := {F sphere (n)} given by where (n) is the same kernel given in Equation (13). The quasi-probability function µ sphere ρ is a frame representation given by the frame F sphere . From Equation (11), it follows that the frame operator of F sphere is S = 4πd −11 . Thus the frame is tight. Equation (12) is the group covariance property defining a phase space representation for the group SU(2). Now consider the discrete representation on sphere defined by Heiss and Weigert. Now the phase space Λ is a subset of points on the sphere which form a valid constellation.
Consider the frame F constellation := {F constellation (ν)}, where where ν is a kernel satisfying the postulates (15)- (18). The quasi-probability function µ constellation ρ is a frame representation given by the frame F constellation . As was the case for the other discrete representation in the previous examples, the frame operator of F constellation is S = d −11 . The unique dual frame of F constellation is given by S −1 F constellation , where here S −1 = d1. Thus, the dual frame to F constellation is what Heiss and Weigert call a dual kernel. Again, from Equation (18), this representation satisfies definition of a phase space representation.

Finite fields discrete phase space representation
Let d be a power of a prime number.
where Q is a quantum net. The quasi-probability function µ field ρ is a frame representation given by the frame F field . The frame operator of F field is S = d −11 . The unique dual frame of F field is given by S −1 F field , where here S −1 = d1. The dual frame to F field is a set of phase point operators.
Equation (26) shows this particular representation is constructed to be translationally covariant and is thus a phase space representation.

IV. FRAME REPRESENTATIONS OF QUANTUM THEORY
Quantum theory is an operational theory where each preparation is associated with a density operator ρ ∈ D(H). This association is not required to be injective; different preparations may lead to the same density operator. However, the mapping is assumed to be surjective; there exists a preparation which leads to each density operator. Similarly, each measurement procedure and outcome k is associated with an effect E k ∈ E(H). Again, this mapping need not be injective but it is surjective. Quantum theory prescribes the probability of outcome k via the Born rule Pr(k) = Tr(ρE k ) . Since the set of outcomes is mutually exclusive and exhaustive, each measurement procedure together will all the outcomes {E k } forms a positive operator valued measure (POVM). Table I shows that most proposed quasi-probability representations are representations of quantum states alone. In Sections IV A and IV B we show that there are two approaches within the frame formalism to lift any representation of states to a fully autonomous representation of finite dimensional quantum theory. In Section IV C, a set of internal consistency conditions for each of the two approaches is given that allows one to formulate quantum theory independently of the standard operator theoretic formalism. A short detour is taken in Section IV D to show that an operational formulation of quantum theory including transformations can be accommodated within the scope of the frame formalism. Finally in Section IV E a novel quasi-probability representation based on SIC-POVMs is presented which relates the frame formalism to a recent research topic in quantum information.

A. Deformed probability representations of quantum theory
The first frame representation approach consists of mapping both states and measurements to L 2 (Λ) via a particular choice of frame F. Formally, we have the following definition: Property (c) is a deformed law of total probability. It can be thought of as a deformation of the usual law of total probability. Recall from Lemma 1 that all quasi-probability representations of states are frame representations. Given a quasi-probability representation (of states), one can identify the unique frame which gives rise to it. Then, using that frame to represent the measurement operators, one obtains a deformed probability representation of quantum theory.

B. Frame representations of quantum theory
Notice that the deformed probability calculus (Proposition 1(c)) can be written where Recall that ξ E is the frame representation of E for the frame F. Hence ξ E can be identified as the frame representation of E using a frame D that is dual to F. The second frame representation approach consists of mapping states to L 2 (Λ) via a particular choice of frame (b) Λ dλ µ ρ (λ) = 1; and, Property (c) is formally identical to the usual law of total probability although the functions in the integrand are not presumed to be non-negative. Indeed, in [12] it was shown that a pair of dual frames cannot both consist solely of positive operators.
Recall from Lemma 1 that all quasi-probability representations of states are frame representations. Given a quasi-probability representation of states, one can identify the unique frame which gives rise to it. Then, using a dual frame to represent the measurement operators, one obtains a frame representation of quantum theory.

C. Internal consistency conditions
Consider first the deformed probability representations. Of course, for a particular choice of frame, not every function in L 2 (Λ) will correspond to a valid quantum state or effect. Here a set of internal conditions is provided, independent of the standard axioms of quantum theory, which characterize the valid functions in L 2 (Λ). The conditions can be found by noting that the frame representation Equation (31) is an isometric and algebraic isomorphism from H(H) to L 2 (Λ) equipped with inner product where D(λ, γ) := D(λ), D(γ) , and algebraic multiplication Now the condition for a function in L 2 (Λ) to be a valid state or effect can be stated. A pure state is a function µ pure ∈ L 2 (Λ) satisfying µ pure F µ pure = µ pure . A general state is a function µ ∈ L 2 (Λ) satisfying µ, µ pure D ≥ 0 for all pure states and Λ dλ µ(λ) = 1. A measurement is represented by a set {ξ k ∈ L 2 (Λ)} of effects which satisfies ξ k , µ pure D ≥ 0 for all pure states and for which k ξ k = ξ 1 , where ξ 1 is the identity element in L 2 (Λ) with respect to the algebra defined by F . That is, ξ 1 is the unique element satisfying Now consider the frame representations. Again for this approach, states and measurements in L 2 (Λ) must meet certain criteria to be valid. The conditions are similar to those in the deformed probability representation. Indeed the pure states and general states are equivalently characterized. However, a measurement is now represented by a set {ξ k ∈ L 2 (Λ)} which satisfies ξ k , µ pure ≥ 0 (now the usual pointwise inner product) for all pure states and for which k ξ k = ξ 1 , where ξ 1 is the identity element in L 2 (Λ) with respect to the algebra defined by E (which is defined in the same way as F with the roles of the frame and its dual reversed).

A transformation is a superoperator (an operator acting on operators) Φ : D(H) → D(H).
Operationally, an experiment consists of preparations followed by transformations and ending in a measurement. Note that, in a purely mathematical sense, the transformations are somewhat redundant as they could be bundled with either the preparations (to make new preparations) or measurements (to make new measurements). Let µ ρ (λ) be a frame representation of a density operator ρ for a frame F. Let D be a dual frame of F and consider the action of a superoperator The frame representation of Φρ is µ Φρ (γ) = F (γ), Φρ . As was the case for including measurements into a frame representations, two approaches can be identified for including the transformations into the frame representation formalism.
Notice that Equation (35) is just the usual (perhaps infinite dimensional) matrix multiplication rule. It is the same rule for transitioning probability distributions via stochastic matrices in classical theories. However, as opposed to stochastic matrices, Φ qp could have negative entries.

E. Example: SIC-POVM representation
In [29], the authors conjecture 5 that the set forms a symmetric informationally complete positive operator valued measure (SIC-POVM).
The defining condition of a SIC-POVM is The set is called symmetric since the vectors have equal overlap. The POVM is formed by taking the projectors onto the one-dimensional subspaces spanned by the vectors. It is informationally complete since these d 2 projectors span H(H).

As of writing, it is an open question whether SIC-POVMs exist in every dimension.
However, there is numerical evidence for there existence for every dimension up to d = 45 [29] and analytic construction for a small number of dimensions.
Suppose then that for any dimension d, a SIC-POVM exists. Notice that a SIC-POVM forms a frame. Explicitly, let From the definition of the SIC-POVM, Equation (39), Since the frame forms a basis, the dual frame is unique and thus the inverse frame operator must satisfy

By inspection
Representing a quantum state via the frame or canonical dual yields the neat reconstruction where µ ρ (α) := F α , ρ is the frame representation of ρ.
Equation (40) was given in [30]. This equation fits naturally into the deformed probability representation formalism discussed in Section IV A. Notice that the dual frame satisfies If an arbitrary measurement {E k } is also represented via the SIC-POVM frame as ξ E k (β) := F β , E k , then Equation (40) is identical to the deformed law of total probability Equation (41) Since the SIC-POVM frame is made of projectors, the frame representation of the density operator is a true probability distribution. However, the dual frame operators are not positive. Thus in a SIC-POVM frame representation, the measurement objects (meant to represent conditional probabilities) contain negative values while the states are represented by true probabilities. The appearance of negativity is a necessary feature of frame representations, as was shown in [12].

V. NON-CLASSICALITY OF QUANTUM THEORY
In this section we prove the impossibility of representing quantum theory classically via a mapping from the quantum operators to classical probability functions.

A. Classical representations of quantum theory
In classical probability, we postulate that a physical system has a set of properties mathematically represented by a measure space Λ. The ontological states are represented by the Dirac measures which are the extreme points of the set of all probability measures on Λ.
The probability measures are the epistemic states representing our ignorance of the ontic state of the system. To measure the properties of the system we partition the space Λ into disjoint subsets { k }. The probability of the system to have properties in k (we will call this "outcome k") is where χ k (λ) ∈ {0, 1} is an idempotent indicator function associated with k . The measurement is equivalently specified by the set {χ k (λ)}, which is interpreted as the conditional probability of outcome k given the systems is known to have the properties λ. A measurement of this type is deterministic; it reveals with certainty the properties of the system.
Consider now an indeterministic measurement specified by the conditional probabilities Consider the problem of proving whether or not it is possible to construct a representation of quantum theory that satisfies these classical postulates. That is, we are interested if there exists a space Λ such that for every density operator there is a unique probability measure and every effect there is an unique indicator function on this space which reproduce the Born rule via the law of total probability. We can formalize this idea into the following definition. (b) Λ dλ µ ρ (λ) = 1; and, (c) Tr(ρE) = Λ dλ µ ρ (λ)ξ E (λ).
We submit Definition 8 as the most natural set of requirements for what can reasonably be called a "classical representation" of quantum theory.

B. Quasi-probability representations of quantum theory
In light of the above considerations, the most natural approach to obtaining a set of quasi-probability representations is to take the definition of a classical representation and simply relax the requirement of non-negativity. This motivates defining the following set of generalized representations.

Definition 9.
A quasi-probability representation of quantum theory is a pair of mappings µ and ξ whose domains are D(H) and E(H), respectively, and whose co-domains are a convex set of quasi-probability functions (real valued functions). These mappings satisfy, for all ρ ∈ D(H) and all E ∈ E(H), (a) µ ρ (λ) ∈ R and ξ E (λ) ∈ R; (b) Λ dλ µ ρ (λ) = 1; and, It should be obvious that a non-negative quasi-probability representation is just a classical representation. Now we show that a consequence of these requirements is that a quasiprobability satisfies a very important property which will be used later on.
Lemma 2. The mappings in a quasi-probability representation of quantum theory are affine 6 .
Comparing Definition 9 to Proposition 2 it would seem that a quasi-probability representation is equivalent to an frame representation, an vice versa. It is important to note however, via Definition 7, that a frame representation of quantum theory makes explicit reference to frames. Whereas, a quasi-probability representation of quantum theory is defined without any reference to frames. However, we will see in the next section that the two are indeed equivalent.

C. Equivalence of frame representations and quasi-probability representations
In [12] it was shown that there does not exist a frame of positive operators that is dual to another frame of positive operators. This establishes that frame representations of quantum theory, as described in Section IV B, must poses negativity. However, this proof does not rule out the possibility of non-negative quasi-probability representations (or, equivalently, classical representations) as defined above. We now close this loophole and thereby extend the significance of the no-go theorem in [12] via the following lemma which shows that the set of quasi-probability representations and the set of frame representations are equivalent. Proof. A frame representation clearly satisfies Definition 9. Suppose then that µ and ξ form a quasi-probability representations of quantum theory. First consider µ. According the procedure described in [31], this mapping can be extended to a linear map on H(H).
Then the Riesz representation theorem implies that there exists a unique operator valued where S is the frame operator of F. Since the range of the frame operator is another By Definition 4, D is dual to F.

D. Non-classicality of quantum theory
Using Lemma 3 in combination with the negativity theorem of [12] it is easy to prove the following. Proof. Lemma 3 establishes that a classical representation is a quasi-probability representation which requires that the dual of a positive frame be positive. Such a pair of frames does not exist as proven in [12]. For the reader unfamiliar with frames this fact can also be proven directly, without making use of frames, as follows.
It is much easier to see the contradiction when Λ is assumed to be finite. It follows from [31] (see also [32]) that µ ρ (λ) = ρ, F (λ) , where F is some effect-valued function.
This is true for all ρ including the rank-1 projectors: the extreme points of D(H). Equation Now assume Λ is infinite. Property 8(c) still requires µ must map the rank-1 projectors to the Dirac measures. Recall for a finite dimensional simplex that each vector has a unique decomposition into extreme points. The same is true here for the probability measures on Λ; each measure has a unique decomposition into Dirac measures [36]. Since, in general, ρ ∈ D(H) has infinitely many non-equivalent decompositions into rank-1 projectors, it is impossible for µ to be injective.
Theorem 1 establishes the necessity of negativity in quasi-probability representations of quantum theory. In the next section another notion of non-classicality, contextuality, will be compared with negativity.

VI. CONNECTION WITH CONTEXTUALITY
Within the quantum formalism there are many notions of non-classicality. In the previous section, negativity was proven to be such a notion. In this section the idea of contextuality is considered as an alternative candidate. In Section VI A the traditional notion of contextuality is reviewed. A generalization due to Spekkens is presented in Section VI B. In Section VI C the somewhat misleading claim in [33] that "negativity and contextuality are equivalent" is clarified. Finally, a more general model of quantum theory, which allows both negativity and contextuality is considered in Section VI D. 7 Spekkens [33] has also made these observations but follows a different route from here on to obtain a contradiction. See Section VI for details of the connection to Reference [33]. 8 The injectivity of this map can be proven from Property8(c) as in the proof of Lemma 3. 9 Compare this to the "Ontological excess baggage theorem" of Hardy [34]. Also note that Busch, Hellwig and Stulpe have come to the same conclusion for finite representations: quantum states can be mapped to probability vectors and quantum observables to classical random variables at the expense of quantum effects being mapped to non-classical (i.e. negative valued) objects [35].

A. Traditional definition of contextuality
The traditional definition of contextuality evolved from a theorem which appears in a paper by Kochen and Specker [37]. The Kochen-Specker theorem concerns the standard quantum formalism: physical systems are assigned states in a complex Hilbert space H and measurements are made of observables represented by Hermitian operators. The theorem establishes a contradiction between a set of plausible assumptions which together imply that quantum systems possess a consistent set of pre-measurement values for observable quantities. Let H be the Hilbert space associated with a quantum system and A ∈ H(H) be the operator associated with an observable A. The function f ψ (A) represents the value of the observable A when the system is in state ψ. One assumption used to derive the contradiction is that for any function F , f ψ (F (A)) = F (f ψ (A)). This is plausible because, for example, we would expect that the value of A 2 could be obtained in this way from the value of A.
Assuming that physical systems do possess values which can be revealed via measurements, the Kochen-Specker theorem leads to the following counterintuitive example [38]. This fact is often expressed via the phrase "quantum theory is contextual".

B. Generalized definition of contextuality
The original notion of contextuality only applies to measurements in standard quantum theory. It does not apply to general operational theories, and, in particular, the mathematical model of open quantum systems (i.e., consisting of mixed states, POVM measurements and completely-positive maps). This problem was addressed by Spekkens in Reference [39] as follows. One may supplement the operational theory with an ontic state space Λ. Then the preparation procedures become probability densities µ P (λ) while the measurement procedures become conditional probabilities ξ M,k (λ). The probabilities of the outcomes of the measurements is required to satisfy the law of total probability Such a supplemented operation model is called an ontological model. The ontological model That is, the representation of the preparation procedure is independent of context. Similarly the ontological model is measurement non-contextual if The terminology "contextual" is again shorthand for the inability of an operational theory to admit a (preparation or measurement) non-contextual ontological model. However, the term "contextual" is also used to describe specific ontological model which do not satisfy Equations (45) and (46) [40]. In Reference [39], it was proven that quantum theory is both preparation and measurement contextual.
Since the standard quantum formalism is an instance of an operational model, Spekkens' notion of non-contextuality is a generalization of the traditional notion initiated by Kochen and Specker. Moreover, considering only measurements, one can see that Spekkens generalizes the notion of non-contextuality from outcomes of individual measurements being independent of the measurement context to probabilities from outcomes of measurements being independent of the measurement context.

C. On the equivalence of non-negativity and non-contextuality
Recall that, in quantum theory, a preparation is specified by a density operator ρ and a measurement outcome by an effect E. Thus a (preparation and measurement) non-contextual ontological model of quantum theory would require and the law of total probability Assuming the probabilities satisfy the usual normalization conditions these conditions are equivalent to those proposed in Definition 8 identifying a classical representation. Spekkens noticed this equivalence in [33] and has therefore independently obtained a proof of negativity. Similarly, our direct proof of the non-existence of a positive dual frame to a positive frame gives a new independent proof that quantum theory satisfies Spekkens' generalized notion of contextuality.
Note that the terminology "negativity and contextuality are equivalent" used in [33], is somewhat misleading. A quasi-probability representation that takes on negative values is not equivalent to some contextual ontological model. Within the formalism of quasi-probability representations of quantum theory, a classical representation (Definition 8) is a special case (a non-negative one). A classical representation of quantum theory is also a special case of a general operational model (a non-contextual ontological model). The key point is that non-negative quasi-probability representations and non-contextual ontological models are the same thing, and one can establish the non-existence of such a classical representation for quantum theory (Theorem 1) starting from either formalism. In other words, a classical representation does not exist but one can relax the constraints on Definition 8 to achieve a representation which may contain negativity or contextuality (or both).

D. General models for quantum theory
A broader framework is to consider a very general class of models not requiring any classical features. That is, suppose we define a general model as just a pair of relations generalizing the (well-defined) mappings µ and ξ of Definition 9. In that case, quantum states and measurements are represented by functions over an ontic space which allows for negativity, contextuality and a non-canonical (deformed) probability calculus. Within such a framework we can consider the following propositions: the model is non-negative (NN); the model uses the law of total probability to reproduce the Born rule (LTP); and the model is non-contextual (NC). This paper establishes that the logical conjunction NN ∧ LTP ∧ NC is false. We can deny NN to obtain a quasi-probability representation. Alternatively, we can deny LTP to obtain, for example, a deformed probability representation (Section IV A). We have already seen how these two cases are intimately related. Finally, one can instead deny NC while retaining LTP and NN, which explicitly shows that negativity and contextuality are not equivalent. An example of such a contextual model is discussed in Reference [40], section V A.

VII. DISCUSSION
Although we conjecture that our no-go theorem will hold when H is infinite dimensional, it is clear the technique in the proof of Lemma 3 will not suffice in that case. The proof makes explicit use of the finite dimensionality of H. However, the alternate proof of Theorem 1 does not make explicit use of this fact. The key fact used in the alternate proof is the non-simplex structure of the quantum state space D(H). This fact is true regardless of the dimension of H. However, it was also necessary that the quasi-probability representation have the form µ ρ = ρ, F . This form is guaranteed for infinite dimensions if we assume that the quasi-probability representation is bounded : there exist b > 0 such that for all ρ ∈ D(H), If we demand boundedness on physical grounds, then our proofs hold for physically reasonable quasi-probability representations in infinite dimensional Hilbert spaces.
Finally, we would like to comment on potential applications of the frame formalism and our analysis of the relationship between non-negativity and non-contextuality. The question of "What is non-classical about quantum mechanics?" has taken a more practical and welldefined meaning within the context of quantum information theory. On the one hand, the necessary and sufficient conditions on the quantum resources required to outperform classical information are not well understood [41,42]. On the other hand, the rapid developments in quantum control in various experimental settings, such as trapped ions, superconducting circuits, quantum optics, and both liquid and solid-state NMR, and their application as quantum information processors, has renewed interest in understanding the extent to which specific experimental achievements may be taken as evidence for truly coherent, quantum behaviour [43,44]. It is our hope that the formalism developed in this work will lead to more systematic and operationally meaningful criteria for characterizing both of these issues.

A set of effects {E k } which satisfy
is called a POVM (positive operator valued measure). The set of all POVMs is denoted POVM(H).
5. An effect P satisfying is called a projector. The set of all projects is denoted P(H). projectors which satisfy the orthogonality condition P k P j = P k δ kj , for any k and j.
Here are some examples of important Hermitian operators used in this paper. Consider the operator Z whose spectrum is spec(Z) = {ω k : k ∈ Z d }, where ω k = e The operators Z, X and Y are often called generalized Pauli operators since they are indeed the usual Pauli operators when d = 2. The parity operator is defined by P φ k = φ −k . Notes: *Wootters' original discrete Wigner function [4] is usually understood to be valid for prime dimension but, as discussed in Section II B, is easily extended to any dimension by combining prime dimensional phase spaces. **Leonhardt [17] also defines a discrete Wigner function valid for odd dimensional which is equivalent to the other odd dimensional cases [16]. ***The phase space of Heiss and Weigert is any subset of d points on the sphere which can indexed arbitrarily.