From no-signalling to quantum states

Given the violation of Bell inequalities AspectEtAl1981 ; ShalmEtAl2015 ; ZeilingerEtAl2015 (see also, WuShaknov1950 ) and the paramount importance of entanglement for quantum information theory Shor1994 ; RaussendorfBriegel2001 ; AcinGisinMasanes2006 ; WisemanJonesDoherty2007 ; PawlowskiBrunner2011 ; deVicente2014 ; BravyiGossetKoenig2020 ; google2020 , characterising quantum correlations constitutes an important and ongoing research objective PawlowskiEtAl2009 ; CabelloSeveriniWinter2010 ; vanDam2013 ; FritzEtAl2013 ; JanottaHinrichsen2014 ; Popescu2014 . One line of this research has focused on the fact that while quantum mechanics is hardly reconcilable with the classical notion of local causality Bell1976 , quantum correlations obey the more general principle of no-signalling: the joint probability distributions for different local measurements $a,b$ with respective outcome sets $\{A\}$, $\{B\}$ marginalise to the same local distributions,

Importantly, Eq. (1) depends on the set of local measurements, consequently one must specify the possible choices of measurements on either subsystem in order to evaluate the constraints inherent to no-signalling. For instance, the PR-box correlations (for the CHSH scenario) restrict to just two measurements on either side PopescuRohrlich1994 . A physically more interesting scenario is that in which arbitrary local measurements are allowed. Remarkably, one can show that this already bounds the correlations to be quantum in the bipartite case BarnumEtAl2010 ; AcinEtAl2010 . Nevertheless, the collections of non-signalling distributions over the set of locally quantum observables do not correspond with quantum states KlayRandallFoulis1987 ; Wallach2002 . There are more non-signalling distributions than quantum states, i.e., while the correlations are as strong as quantum ones, the underlying distributions need not derive from a quantum state.

In this paper we identify two physical principles that characterise those non-signalling bipartite correlations which correspond with quantum states. To set the stage, we review some basic facts about correlations over non-signalling correlations in Sec. II. In Sec. III we reformulate the problem from the perspective of contextuality and identify no-disturbance as the key underlying principle. This subsumes no-signalling, but makes explicit the intimate relationship with non-contextuality. Based on this, we suggest an extension of the no-disturbance principle to dilated systems in Sec. IV. Thm. 2 shows that this strengthened principle almost singles out quantum states. In Sec. V we identify the missing piece of data by introducing a notion of time orientation. Our main result, Thm. 3, proves that under a related consistency condition with respect to unitary evolution in subsystems correlations indeed derive from quantum states. Sec. VI concludes.

Throughout, we denote by $\mathcal{L}(\mathcal{H})$ the algebra of linear operators on some finite-dimensional Hilbert space $\mathcal{H}$, by $\mathcal{P}(\mathcal{H})$ the lattice of projections on $\mathcal{H}$, and by $\mathcal{L}(\mathcal{H})_{\mathrm{sa}}$ the real-linear space of self-adjoint (Hermitian) operators, representing the observables of a system.

In contrast to restricted sets of observables such as those in PopescuRohrlich1994 , BarnumEtAl2010 study no-signalling for all locally quantum observables. Locally quantum here means that every local system is described in terms of an observable algebra $\mathcal{L}(\mathcal{H})_{\mathrm{sa}}$ of self-adjoint (Hermitian) operators corresponding to the respective quantum system described by the Hilbert space $\mathcal{H}$. However, rather than assuming the tensor product structure between Hilbert spaces, the composite system is described solely in terms of product observables, i.e., pairs $(a_{1},a_{2})$ where $a_{i}\in\mathcal{L}(\mathcal{H}_{i})_{\mathrm{sa}}$ for $i=1,2$. Using the spectral decomposition of observables $a_{i}=\sum_{j}A_{j}q_{i}^{j}\in\mathcal{L}(\mathcal{H}_{i})_{\mathrm{sa}}$ for $A_{j}\in\mathbb{R}$ and $q^{j}_{i}\in\mathcal{P}(\mathcal{H}_{i})$, bipartite correlations arise from measures $\mu:\mathcal{P}(\mathcal{H}_{1})\times\mathcal{P}(\mathcal{H}_{2})\rightarrow[0,1]$, $\mu(\mathbbm{1})=\mu(\mathbbm{1}_{1},\mathbbm{1}_{2})=1$ (here, $\mathbbm{1}_{i}\in\mathcal{L}(\mathcal{H}_{i})$ denotes the identity matrix). For the latter the no-signalling constraints in Eq. (1) read:

for all mutually orthogonal sets of projections $(q_{i}^{j})_{j}$, i.e., $\sum_{j}q^{j}_{i}=\mathbbm{1}_{i}$ and $q^{j}_{i}q^{k}_{i}=\delta_{jk}$.¹¹1For convenience, we restrict ourselves to (measures over) projective measurements. It is straightforward to extend the discussion to positive operator-valued measures (POVM), equivalently effect spaces. This allows to include the two-dimensional case in Thm. 1 (see BarnumEtAl2010 ; FrembsDoering2019b ). For later reference, we point out that the structure of observables implies that (in addition to no-signalling) $\mu$ is also independent of contexts, in a sense to be made precise in Sec. III.

Applying Gleason’s theorem Gleason1975 to the respective subsystems one shows that $\mu$ extends to a linear functional $\sigma_{\mu}:\mathcal{L}(\mathcal{H}_{1}\otimes\mathcal{H}_{2})\rightarrow\mathbb{C}$, equivalently, that there exists a linear operator $\rho_{\mu}$ such that $\sigma_{\mu}(a)=\mathrm{tr}[\rho_{\mu}a]$ for every $a\in\mathcal{L}(\mathcal{H}_{1}\otimes\mathcal{H}_{2})$. Since $\mu$ is positive, $\sigma_{\mu}$ is positive on all product observables $a_{1}\otimes a_{2}$ with $a_{1}\in\mathcal{L}(\mathcal{H}_{1})_{+}$ and $a_{2}\in\mathcal{L}(\mathcal{H}_{2})_{+}$. We call such normalised, i.e., $\mathrm{tr}[\rho_{\mu}]=1$, linear functionals positive on pure tensor (POPT) states.

Crucially, a POPT state $\sigma_{\mu}$ is generally not positive. Hence, $\mu$ defines a quantum state if and only if $\sigma_{\mu}$ is positive. Despite this discrepancy, BarnumEtAl2010 ; AcinEtAl2010 show that the correlations arising from POPT states do not exceed those arising from quantum states in the bipartite case.

The existence of unphysical POPT states gives rise to the problem of finding a sharper classification that rules out such states. This paper offers a solution to this problem. To this end, we first argue that rather than no-signalling, the natural principle underlying Eq. (2) is no-disturbance. We then extend the scope of the no-disturbance principle to dilations of local systems, and show that up to a consistency condition with respect to unitary evolution in subsystems this establishes a one-to-one correspondence between non-signalling joint probability distributions and bipartite quantum states.

We remark that distributions over product observables have been studied before, e.g. in KlayRandallFoulis1987 , out of the attempt to define a tensor product intrinsic to quantum logic.²²2For a recent contribution to this problem, see AbramskyBarbosa2021 (and references therein). We do not consider this problem here, but note that our result has the potential to give new impulse to this research programme. Another perspective on this problem is given by Wallach in Wallach2002 , who considers a generalisation of Gleason’s theorem to composite systems. Since it distracts from the physical significance of the present discussion, we study this problem in more generality elsewhere FrembsDoering2022b .

A crucial feature of locally quantum non-signalling joint probability distributions is that they satisfy a more general constraint than no-signalling, called no-disturbance. We introduce this notion in the following sections, before extending the no-disturbance principle to dilations in Sec. IV. Our approach naturally embeds within the analysis of the principal role of contextuality in quantum theory FreDoe19a . For a recent review of the wider subject, see Masse2021 .

A key idea of this work is to impose the no-disturbance condition in Eq. (6) not only between locally quantum observables, but to extend its scope to dilations of (at least) one of the two subsystems. We give some motivational background for this step first.

Recall that by Thm. 1 a measure $\mu:\mathcal{P}(\mathcal{H}_{1})\times\mathcal{P}(\mathcal{H}_{2})\rightarrow[0,1]$ defines a POPT state $\mu(q_{1},q_{2})=\sigma_{\mu}(q_{1}\otimes q_{2})=\mathrm{tr}[\rho_{\mu}(q_{1}\otimes q_{2})]$ with corresponding linear operator $\rho_{\mu}\in\mathcal{L}(\mathcal{H}_{1}\otimes\mathcal{H}_{2})$. Similarly, for every $q_{1}\in\mathcal{P}(\mathcal{H}_{1})$ we define a positive linear operator $\rho_{\mu}(q_{1})\in\mathcal{L}(\mathcal{H}_{2})_{+}$ from $\mu(q_{1},q_{2})=:\mathrm{tr}_{\mathcal{H}_{2}}[\rho_{\mu}(q_{1})q_{2}]$ for all $q_{2}\in\mathcal{P}(\mathcal{H}_{2})$. It follows from Thm. 1 that $\rho_{\mu}$ extends to a positive linear map $\phi_{\mu}:\mathcal{L}(\mathcal{H}_{1})\rightarrow\mathcal{L}(\mathcal{H}_{2})$. On the other hand, for any single $q_{1}\in\mathcal{P}(\mathcal{H}_{1})$ we may assume $\rho_{\mu}(q_{1})\in\mathcal{L}(\mathcal{H}_{2})_{+}$ to arise via coarse-graining from some $|\psi_{\mu}(q_{1})\rangle\in\mathcal{K}:=\mathcal{H}_{2}\otimes\mathcal{H}_{E}$ on a larger system (to be thought of as the system together with an (environmental) ancillary system), by tracing out the extra degrees of freedom:

$|\psi_{\mu}(q_{1})\rangle$ is a purification of $\rho_{\mu}(q_{1})$, where the unitary $u\in\mathcal{L}(\mathcal{H}_{2}\otimes\mathcal{H}_{E})$ is chosen to separate the input from the ancillary system. We want to arrange the purifications for all $q_{1}\in\mathcal{P}(\mathcal{H}_{1})$ in an economical way. To start with, note that since $\rho_{\mu}(q_{1})\in\mathcal{L}(\mathcal{H}_{2})_{+}$ for every $q_{1}\in\mathcal{P}(\mathcal{H}_{1})$, $(\rho_{\mu}(q_{1}^{i}))_{i}$ defines a non-normalised ($\sum_{i}\rho_{\mu}(q_{1}^{i})=\rho_{\mu}(\mathbbm{1}_{1})=\mathrm{tr}_{\mathcal{H}_{1}}[\rho_{\mu}]]$) positive operator-valued measure (POVM) on $\mathcal{L}(\mathcal{H}_{2})$ for every set of mutually orthogonal projections $(q_{1}^{i})_{i}$ in $\mathcal{L}(\mathcal{H}_{1})$, i.e., for every context $V_{1}\in\mathcal{V}(\mathcal{H}_{1})$. We may thus write

for all $q_{1}^{i}\in\mathcal{P}(V_{1})$ and $V_{1}\in\mathcal{V}(\mathcal{H}_{1})$, mimicking the purification in Eq. (8). Eq. (9) is called a dilation of the POVM $(\rho_{\mu}(q_{1}^{i}))_{i}$ to the projection-valued measure (PVM) $q_{1}\mapsto q_{1}\otimes\mathbbm{1}_{\mathcal{H}_{E}}$. More generally, by Naimark’s theorem Naimark1943 ; Stinespring1955 every POVM $\rho^{V_{1}}_{\mu}$ admits a dilation to a PVM $\varphi_{\mu}^{V_{1}}:\mathcal{P}(V_{1})\rightarrow\mathcal{P}(\mathcal{K})$ such that $\rho_{\mu}=v^{*}\varphi_{\mu}^{V_{1}}v$, where $v:\mathcal{H}_{2}\rightarrow\mathcal{K}$ is a linear map.⁵⁵5Here, we concentrate the dependence on contexts in the mapping $V_{1}\mapsto\varphi^{V_{1}}_{\mu}$, by choosing $\mathcal{K}$ sufficiently large Naimark1943 ; Stinespring1955 ; Choi1975 and by absorbing any context dependence on $v_{V_{1}}$ into $\varphi_{\mu}^{V_{1}}$ for every $V_{1}\in\mathcal{V}(\mathcal{H}_{1})$.

Importantly, the dilations $V_{1}\mapsto\varphi^{V_{1}}_{\mu}$ generally depend on contexts $V_{1}\in\mathcal{V}(\mathcal{H}_{1})$. In contrast, recall that no-disturbance in Eq. (6) encodes non-contextuality constraints between product observables. By comparison, this suggests to extend the no-disturbance principle also to product observables with respect to the dilations $(\varphi^{V_{1}}_{\mu})_{V_{1}\in\mathcal{V}(\mathcal{H}_{1})}$.

From a physical point of view, we interpret the (conditional) probability distributions $\mu_{V_{2}}(q_{1},\cdot)$ in contexts $V_{2}\in\mathcal{V}(\mathcal{H}_{2})$ as states of incomplete information. We have seen that this interpretation arises naturally from the explicit dilations in Eq. (9), which express the state as arising from coarse-graining of ancillary degrees of freedom. This view is further corroborated if one allows not only projective measurements on the respective subsystems $\mathcal{L}(\mathcal{H}_{1})$ and $\mathcal{L}(\mathcal{H}_{2})$, but arbitrary positive operator-valued measures, e.g. as discussed in BarnumEtAl2010 .

Put the other way around, if $\mu$ did not satisfy no-disturbance for dilations in Def. 1—while potentially being non-disturbing (non-signalling) when restricted to the system $\mathcal{H}_{1}\otimes\mathcal{H}_{2}$—it would fail to be non-disturbing (non-signalling) for every choice of dilations $(\varphi^{V_{1}}_{\mu})_{V_{1}\in\mathcal{V}(\mathcal{H}_{1})}$ to a larger system. Such measures would at the very least prompt a substantial revision of the concept of mixed states in terms of states of information in quantum theory. As such, the extension to dilated systems in Def. 1 is a natural one, and arguably conservative compared with other related approaches, e.g. PawlowskiEtAl2009 ; vanDam2013 ; FritzEtAl2013 ; Popescu2014 .

Note also that Def. 1 does not change the fact that the composite system is described by means of the Cartesian product of contexts in Eq. (5), not the tensor product.

In order to state the implications of Def. 1, we remind the reader of some basic facts about Jordan algebras. Recall that the set of self-adjoint (Hermitian) matrices $\mathcal{L}(\mathcal{H})_{\mathrm{sa}}$ is closed under the anti-commutator $\{a,b\}=ab+ba$ for all $a,b\in\mathcal{L}(\mathcal{H})_{\mathrm{sa}}$. This defines the (special) Jordan algebra $\mathcal{J}(\mathcal{H})=(\mathcal{L}(\mathcal{H})_{\mathrm{sa}},\{\cdot,\cdot\})$.⁶⁶6A Jordan algebra is called special if it is isomorphic to the subalgebra of the self-adjoint part of an associative algebra. Otherwise, it is called exceptional. Moreover, we obtain a Jordan $*$-algebra by extending the operation $\{\cdot,\cdot\}$ to the complexified algebra $\mathcal{J}(\mathcal{H})=\mathcal{J}(\mathcal{H})_{\mathrm{sa}}+i\mathcal{J}(\mathcal{H})_{\mathrm{sa}}$. This expresses the fact that $\mathcal{J}(\mathcal{H})_{\mathrm{sa}}$ is the self-adjoint part of $\mathcal{L}(\mathcal{H})$. Finally, a Jordan $*$-homomorphism $\Phi:\mathcal{J}(\mathcal{H}_{1})\rightarrow\mathcal{J}(\mathcal{H}_{2})$ is a linear map $\Phi:\mathcal{L}(\mathcal{H}_{1})\rightarrow\mathcal{L}(\mathcal{H}_{2})$ such that $\Phi(\{a,b\})=\{\Phi(a),\Phi(b)\}$ and $\Phi^{*}(a)=\Phi(a^{*})$ for all $a,b\in\mathcal{J}(\mathcal{H}_{1})$.

We have the following key result, which extracts Jordan structure from Def. 1.

Of course, every quantum state has a purification which yields a dilation of the form in Thm. 2. Our argument works in the reverse direction: by requiring that measures $\mu$ have a non-contextual extension, i.e., that they satisfy the no-disturbance principle for at least one choice of dilations $V_{1}\mapsto\varphi^{V_{1}}_{\mu}$, $\mu$ is of the form in Thm. 2. Next, we show that, in this case, $\mu$ already defines a quantum state up to a choice of time orientation in local subsystems.

Extending no-disturbance to dilations via Def. 1 is not quite sufficient to ensure that a POPT state $\sigma_{\mu}=\mathrm{tr}[\rho_{\mu}\ \cdot]$ becomes positive and thus a quantum state—but it almost is. The difference is best expressed in terms of the Choi-Jamiołkowski isomorphism Jamiolkowski1972 . In particular, by Choi’s theorem the linear operator $\rho_{\mu}$ is positive if and only if a related map $\phi_{\mu}$ is completely positive Choi1975 . Moreover, by Stinespring’s theorem a map is completely positive if and only if it corresponds with an algebra homomorphism (a representation) on a larger system Stinespring1955 , i.e., $\phi_{\mu}:\mathcal{L}(\mathcal{H}_{1})\rightarrow\mathcal{L}(\mathcal{H}_{2})$ is completely positive if and only if there exists a Hilbert space $\mathcal{K}$, a linear map $v:\mathcal{H}_{2}\rightarrow\mathcal{K}$, and an algebra homomorphism $\Phi_{\mu}:\mathcal{L}(\mathcal{H}_{1})\rightarrow\mathcal{L}(\mathcal{K})$ such that $\phi_{\mu}(a)=v^{*}\Phi_{\mu}(a)v$. By Thm. 2, if $\mu$ satisfies no-disturbance for dilations, it is of this form with the only difference that $\Phi_{\mu}$ is a merely Jordan $*$-homomorphism.

The Jordan algebra $\mathcal{J}(\mathcal{H})$ does not completely determine the algebra $\mathcal{L}(\mathcal{H})$, since it lacks the antisymmetric part or commutator $[a,b]=ab-ba$ in the associative (and noncommutative for $\mathrm{dim}(\mathcal{H})\geq 2$) product $ab=\frac{1}{2}(ab+ba)+\frac{1}{2}(ab-ba)$ for all $a,b\in\mathcal{L}(\mathcal{H})$. In particular, the Jordan $*$-homomorphism $\Phi_{\mu}$ is an algebra homomorphism if and only if it also preserves commutators, i.e., $\Phi([a,b])=[\Phi(a),\Phi(b)]$ for all $a,b\in\mathcal{J}(\mathcal{H})$. This property has a distinctive physical meaning in terms the unitary evolution in local subsystems.

To see this, we describe the dynamics of a system represented by the observable algebra $\mathcal{L}(\mathcal{H})_{\mathrm{sa}}$ in terms of one-parameter groups of automorphisms $\mathbb{R}\mapsto\mathrm{Aut}(\mathcal{L}(\mathcal{H})_{\mathrm{sa}})$. Note that $\mathcal{L}(\mathcal{H})$ possesses a canonical action on itself by conjugation $\Psi:\mathbb{R}\times\mathcal{L}(\mathcal{H})_{\mathrm{sa}}\rightarrow\mathrm{Aut}(\mathcal{L}(\mathcal{H})_{\mathrm{sa}})$, $\Psi(t,a)b=e^{ita}be^{-ita}$ for all $a,b\in\mathcal{L}(\mathcal{H})_{\mathrm{sa}}$.⁷⁷7The map $\Psi$ ‘selects’ unitary (as opposed to anti-unitary Wigner_GruppentheorieUndQM ; Bargmann1964 ) symmetries on $\mathcal{J}(\mathcal{H})_{\mathrm{sa}}$. Inherent in this is an identification of self-adjoint elements in $\mathcal{J}(\mathcal{H})_{\mathrm{sa}}$ and generators of symmetries, called a dynamical correspondence. In other words, $\Psi$ ‘selects’ a canonical dynamical correspondence on $\mathcal{J}(\mathcal{H})_{\mathrm{sa}}$: $\Psi=\exp\circ\psi$ is the exponential of the (canonical) dynamical correspondence $\psi$ on $\mathcal{L}(\mathcal{H})$. For more details, see AlfsenShultz1998 ; Doering2014 . The latter expresses the unitary evolution in the system, thus promoting the parameter $t$ to a time parameter with $a$ playing the role of a Hamiltonian. Note however that without fixing a preferred Hamiltonian the value of $t$ has a priori no objective physical meaning, it is intrinsically relational. Nevertheless, the sign of $t$ turns out to be independent of (the choice of Hamiltonian) $a\in\mathcal{L}(\mathcal{H})_{\mathrm{sa}}$ AlfsenShultz1998 .

By differentiation, $\left.\frac{d}{dt}\right|_{t=0}\Psi(t,a)=i[a,\cdot]$, where $[\cdot,\cdot]$ is the commutator in the ambient algebra $\mathcal{L}(\mathcal{H})$. It follows that $\Phi_{\mu}$ preserves commutators if and only if it preserves the canonical unitary evolution $\Psi$ of the subsystem algebras. We call $\Psi$ the canonical time orientation on $\mathcal{L}(\mathcal{H})$ AlfsenShultz1998 ; Doering2014 ; Frembs2022a ; Frembs2022b , and $\Psi^{*}(t,a)=*\circ\Psi(t,a)=\Psi(-t,a)$ (cf. Prop. 15 in AlfsenShultz1998a ) the reverse time orientation, which corresponds with the opposite order of composition in $\mathcal{L}(\mathcal{H})$, equivalently the opposite sign for the commutator in the respective algebras,

where we set $a\cdot_{\pm}b:=\frac{1}{2}\{a,b\}\pm\frac{1}{2}[a,b]$ for all $a,b\in\mathcal{J}(\mathcal{H})$. Of course, $\mathcal{L}(\mathcal{H})_{+}=\mathcal{L}(\mathcal{H})$.

Two remarks on Def. 2 are in order. First, note that Eq. (11) requires consistency with respect to the commutators on $\mathcal{L}(\mathcal{H}_{1})$ and $\mathcal{L}(\mathcal{K})$. However, since $\mathcal{L}(\mathcal{H}_{2})$ arises from $\mathcal{L}(\mathcal{K})$ by restriction, the time orientation $\Psi_{2}$ on $\mathcal{L}(\mathcal{H}_{2})$ extends to a unique time orientation $\Psi^{\prime}_{2}$ on $\mathcal{L}(\mathcal{K})$. Second, we have used the reverse time orientation $\Psi^{*}_{1}$ on the first system. This choice of relative time orientation $(\Psi_{1}^{*},\Psi_{2})$ is intrinsic to Choi’s theorem Choi1975 (see also Frembs2022b ), which allows us to deduce positivity of $\sigma_{\mu}$ from complete positivity of $\phi_{\mu}$ in Thm. 2.

In fact, the conjunction of Def. 1 and Def. 2 yields our main result.

We finish this section with a few remarks. We defined the canonical time orientation $\Psi$ on $\mathcal{L}(\mathcal{H})=(\mathcal{J}(\mathcal{H}),\Psi)$ with respect to unitary symmetries. In turn, anti-unitary symmetries correspond with unitary symmetries on the algebra $\mathcal{L}_{-}(\mathcal{H})$ (and vice versa). Recalling that every anti-unitary operator is the product of a unitary and the time reversal operator further corroborates the notion of time-orientedness in Def. 2.

Moreover, note that Eq. (11) is invariant under changing time orientations, equivalently under time reversal in both subsystems (cf. Frembs2022b ). On the other hand, it follows from Thm. 3 that applying time reversal to one subsystem only will generally map outside the set of bipartite quantum states. This is reminiscent of the positive partial transpose (PPT) criterion for bipartite entanglement due to Peres Peres1996 ; Horodeckisz1996 . In fact, following this analogy the criterion can be given a sharp physical interpretation in terms of time orientations Frembs2022a .

Finally, note that Thm. 3 represents a generalisation of Gleason’s theorem to composite systems. We extend this perspective to the general setting of von Neumann algebras in FrembsDoering2022b .

We studied the physical principle of no-disturbance on joint probability distributions over product observables, which reduces to no-signalling as shown in Eq. (7). As such we identified no-disturbance as the key principle in BarnumEtAl2010 , which shows that correlations over product observables satisfying no-disturbance cannot exceed bipartite quantum correlations. However, no-disturbance is not sufficient to restrict to quantum states. Our main result, Thm. 3, resolves this issue: by extending the scope of no-disturbance to dilations in Def. 1, and by enforcing a consistency condition with respect to unitary evolution in local subsystems in Def. 2, we show that the resulting joint probability distributions correspond with bipartite quantum states unambiguously.

Our research naturally relates to other approaches, which seek to single out quantum correlations from more general non-signalling distributions, by imposing additional principles, e.g. PawlowskiEtAl2009 ; vanDam2013 ; FritzEtAl2013 ; Popescu2014 . Apart from the extension of the no-disturbance principle to dilated systems, which fits nicely with the usual interpretation of mixed states as states of incomplete information as argued above, our only additional assumption already has a clear physical meaning in terms of the arrow of time Frembs2022a .