The Operational Choi-Jamiołkowski Isomorphism

The issue of ontology is a vexed one in physics. It is a well-established fact that ontology is always underdetermined by the empirical results, and therefore no amount of experimentation can give us certainty about the nature of the unobservable entities associated with our theory[14]. Furthermore, when one theory is replaced by another, it is often the case that the theoretical objects postulated by the original theory do not feature in the new theory, and thus unless we believe our current theory is the correct final theory of reality we should probably assume that the theoretical objects posited by our current theory will not feature in future theories[15]. Considerations such as these have led many physicists to adopt variants of anti-realism, positivism or empiricism, holding that physics should be concerned only with empirical predictions and should be silent on questions of ontology[16, 17].

However, it is important to recognise that the ontology we associated with our theories affects the way in which science progresses. For example, recall that classical physics can be written either in the standard Newtonian form, where we start with initial positions and calculate trajectories from those positions, or in a Lagrangian form where we define an ’action’ over an entire history and find the trajectory which optimizes the action [18, 19]. Historically most physicists have considered the Newtonian formulation of classical physics to be more fundamental than the Lagrangian one, and in this context it was natural for the founders of quantum mechanics to formulate the new theory in terms of a state evolving forwards in time, even though we now know that an alternative Lagrangian formulation in terms of path integrals exists[18, 20]. Quantum mechanics might look very different (and indeed might well be less difficult to ’interpret’) had we taken the Lagrangian formulation seriously from the start.

Moreover, simply becoming an operationalist and refusing to talk about ontology doesn’t confer immunity to these sorts of biases, because ontological assumptions are baked into the way in which we think about and do physics[18, 21, 4], and so those who decline to think seriously about ontology are effectively choosing to uphold the status quo ontological assumptions - which, in the case of quantum mechanics, entails upholding the assumption of an ontology based on states.

Thus reconsidering the ontology associated with our theories can be a useful way of flushing out unjustified assumptions and ultimately coming up with new physics. However, in adopting this strategy we risk getting caught up in the details of some specific ontology, which due to the problem of underdetermination will inevitably not be correct in every particular. Ideally, what we would like to do is extract the significant structural content of various different ontologies and discard the specifics, which don’t really contribute to the project of coming up with new physics and are essentially just window-dressing. This is exactly what the framework of generalized probabilistic theories allows us to do[5, 6, 7, 8, 9, 10, 11, 12, 13]: it offers a middle way between realism and empiricism, where we impose hierarchies within our theories by regarding certain features of the theory as consequences of other features, but do not actually specify the details of the ontology underlying the hierarchy.

Hierarchical structures of this kind occur in many places throughout the sciences.²²2I observe in passing that some philosophers have questioned the coherence of imposing a hierarchy of fundamentality on nature. I will not engage with this debate here, but will instead observe that whatever its metaphysical status, the ‘hierarchy of fundamentality’ picture is instrumentally useful in the actual practice of science as a means of rethinking the structural assumptions that go into our theories.. They are most straightforward in theories which come with an ontology attached, because in such cases a hierarchy of fundamentality can be read directly off the ontology. For example, the ontology of classical physics postulates objects which move in inertial reference frames and are subject to certain mechanical forces, but centrifugal force is not one of those mechanical forces - rather it is is a ‘fictitious’ force which arises when we try to describe an object in a reference frame which is accelerating relative to an inertial reference frame[22]. This ontology encodes a hierarchy where the centrifugal forces are to be understood as consequences of the more fundamental laws governing motion relative to inertial frames. Indeed, the hierarchy licences counterfactual assertions about the relationship between these features: if the laws governing motion relative to inertial frames had been relevantly different, the centrifugal forces would also have been different. Thus we observe that these hierarchies of fundamentality are roughly analogous to causal structures, with ‘causation’ to be understood in the counterfactual sense: in a given theory, feature A causes feature B if feature B would not have obtained if A had not obtained.³³3There are many subtleties that have to be dealt with in any counterfactual account of causation; I will not deal with any of them here, but refer to ref [23] for a general discussion of counterfactual approaches to causation. It is likely that further subtleties would arise in the application of this approach to intratheoretic causation; I will return to these in forthcoming work. Therefore I will henceforth refer to these hierarchies as ‘intratheoretic causal structures’ though I emphasize that the word ‘causal’ is here being used in a specialised sense, since the relata of the putative causal relation are not immanent.

We are already quite familiar with the case where the intratheoretic causal structure can be read off the ontology. But what if there is no general consensus regarding the ontology of a theory? Quantum mechanics is one such case – to this day it is not even clear whether quantum mechanics requires the existence of the wavefunction[24], let alone other proposed features such as spontaneous collapses[25], de Broglie-Bohm particles[26] or ‘many-worlds.’[27] And as a result the intratheoretic causal structure of quantum mechanics is much less transparent - witness the continuing controversy over the reality of the quantum state, which can be understood as a discussion about whether we should adopt an intratheoretic causal structure in which the characteristic features of quantum mechanics are derived from the properties of the quantum state, or whether we should take it that the quantum state itself is derived from some deeper properties.

Obviously it is open to us to resolve these questions by adopting an interpretation of quantum mechanics and drawing the conclusions implied by its ontology. For example, the Everett interpretation tells us that the quantum state is indeed the fundamental object of quantum mechanics[27], whereas Quantum Bayesianism suggests that the quantum state is simply a description of an agent’s degrees of belief about the outcomes of measurements[28]. But the study of generalized probabilistic theories offers an alternative, allowing us to draw conclusions about intratheoretic causal structure which are independent of any specific ontological assumptions. Research of this kind works by placing quantum mechanics in the context of a wider space of GPTs so that we can meaningfully consider counterfactual questions about how changing one aspect of a theory might lead to other changes[5, 6, 7, 8, 9, 10, 11, 12, 13]. This often involves proving statements of the form: ‘any generalized probabilistic theory which has feature $X$ must also have feature $Y$.’ For example, ref [12] proves that any GPT which obeys no-signalling must also obey a form of monogamy of correlations, thus demonstrating the existence of a (putatively causal) relationship between these features which holds regardless of what the underlying ontology might be. The framework of generalized probabilistic theories therefore provides exactly the formalism needed for us to study the counterfactual reasoning associated with intratheoretic causal structure in a way that does not depend on any particular ontology.

Of course, showing that certain features of quantum mechanics can be derived from other features in this way doesn’t actually prove anything about the intratheoretic causal structure of quantum mechanics, because there exists no unique axiomatization of quantum mechanics and each different axiomatization will suggest different intratheoretic causal relations. But that is just to restate the old problem of underdetermination of theory by data, and a similar response can be made: the point is not necessarily to know for certain, but to understand the implications of different possible causal structures as a way of exploring alternative ways forward for physics. One need not believe that any particular causal structure is the correct one, or indeed that a unique correct causal structure even exists, to see the value of the exercise.

I have suggested that one motivation for research in the space of generalized probabilistic theories is the idea that we don’t have much hope of coming to know the true ontology of our theories, so instead of focusing on the details of specific ontologies we should instead think in terms of intratheoretic causal structures, which plausibly we could get right even if the true ontology remains epistemically inaccessible. This suggestion clearly has common ground with the philosophical view known as structural realism[15], which proposes that since the specific ontologies of our theories will likely be discarded when we move to a new theory, we should not be epistemically committed to specific ontologies but instead commit to the structural relations implied by the ontologies of the theories, since these relations are often maintained when we move to a new theory. However, the relations to which structural realism refers are usually instantiated by specific objects in the world, whereas the intratheoretic relations that I discuss here hold between parts of a theory, so the domain of application of this form of structuralism is importantly different.

Although the mainstream literature on quantum mechanics is still very dominated by state-based approaches, in recent years a variety of interesting work has been done on non-standard temporal pictures. For example, there has been growing interest in retrocausal approaches to quantum theory[33, 34, 35], including a proof by Pusey and Leifer demonstrating that if quantum mechanics obeys a certain sort of time-symmetry then it must exhibit retrocausality[36] and a model due to Wharton which suggests a natural resolution to the quantum reality problem using the “all-at-once”-style analysis of action principles[37]. Elsewhere, Oreskhov and Cerf[38] have set out the process matrix formalism, which allows us to generalize the framework of operational theories in a way that does not depend on a predefined time or causal structure, thus giving us the mathematical resources to deal with theories that might contain indefinite causal order, causal loops or other structures that don’t fit into our familiar notions of time and causality. Similarly, Shrapnel and Costa have used the process matrix formalism to set out a generalisation of the ontological models approach which does not assume that information must be carried through time by a mediating state, and have used this generalisation to demonstrate that even without mediating states quantum mechanics must still exhibit a generalized form of contextuality[39]. Thus the ideas set out here add to a growing body of work on the ways in which quantum mechanics could be embedded into global and temporally non-local structures.

It has been recognised that the Choi-Jamiolkowski isomorphism is likely to play an important role in such non-standard temporal pictures - in particular, the process matrix is defined using the Choi-Jamiolkowski representations of the relevant CP maps[38]. The process matrix formalism therefore assigns to the isomorphism an implicit ontological significance, and so the results presented here can be understood as complementary to that line of research, since I have made the ontological significance of the isomorphism explicit and used an operational formulation to explore its consequences for intratheoretic causal structure. A variety of other authors have also suggested that the isomorphism should be understood in ontological terms, so for example it is noted in [40] that the Choi-Jamiołkowski representation for quantum operations ‘gives a nice way of unifying statics and dynamics in one framework: the future is entangled (or at least correlated) with the past,’ and likewise, ref [41] puts forward a theoretical model in which ‘one particle at N times is … equivalent to N particles at one time.’

Finally, a different version of an ‘operational Choi-Jamiołkowski isomorphism’ was put forward in ref [42], where it is shown that in any operational theory which a) is causal, and b) has the property that every mixed state has a purification, it is necessarily the case that there exists an isomorphism between transformations and bipartite states which has the same structural properties as the Choi-Jamiołkowski isomorphism. This paper is a good example of what I have described as the study of intratheoretic causal structure, but the structure proposed is quite different to the one I have suggested here. First, I have taken the ‘operational Choi-Jamiołkowski isomorphism’ to be fundamental and derived other features of quantum mechanics from it, whereas in ref [42] the operational Choi-Jamiołkowski is derived from other features of the theory, so the intratheoretic causal structure is in fact precisely reversed. Second, I have specifically avoided using the concept of state in my formulation of the operational Choi-Jamiołkowski isomorphism and instead derived properties of the state from the isomorphism together with constraints on possible correlations, whereas in ref [42] one of the axioms from which the rest of quantum mehcanics is derived is ‘every mixed state has a purification,’ so in this approach it seems natural to take the state as a fundamental object and not an emergent feature.

In attempting to understand some of the special features of quantum mechanics, it is often helpful to make comparisons between quantum theory and other possible theories. In this context it is common to employ the language of generalized probabilistic theories, where theories are defined entirely in terms of preparation procedures and measurements, eschewing abstract mathematical constructions. This has been a very active area of study in recent years, with many interesting results emerging to show how quantum mechanics relates to the broader field of possible operational theories. For example, it was shown that quantum mechanics is not the maximally non-local theory without signalling[10]: there exists a gap between the numerical bound on the set of non-local correlations which can be produced in a theory limited only by no-signalling, and the corresponding ‘Tsirelson’ bound on the set of non-local correlations which can be produced in quantum theory, and much effort has gone into trying to explain this difference (see for example refs [43, 44, 45, 11]).

In the framework of operational theories, a given theory is specified as a quadruple $(\mathcal{P},\mathcal{M},\mathcal{T},p)$ where $\mathcal{P}$ is a set of preparations, $\mathcal{M}$ is a set of measurements, $\mathcal{T}$ is a set of transformations, and the function $p(M^{x}|P,T)$ specifies, for every possible combination of preparation $P$, transformation $T$, and measurement outcome $M^{x}$, the probability of obtaining outcome $M^{x}$ to the measurement $M$ if it is performed on a system prepared according to $P$ and then subjected to transformation $T$. [46]

I will sometimes write $p(M^{x}|P)$ as shorthand for $p(M^{x}|P,\mathbb{I})$, so $p(M^{x}|P)$ denotes the probability of obtaining outcome $M^{x}$ to the measurement $M$ if it is performed directly on a system prepared according to $P$. For the operational theories considered in this paper, I stipulate that performing a transformation followed by a given measurement is equivalent to simply performing some other measurement, i.e. for any $P,T,\{M^{x}\}$, there exists some $\{N^{x}\}$ such that $p(M^{x}|P,T)=p(N^{x}|P,\mathbb{I})$; for any operational theory where this is not the case, we can trivially make it the case by expanding the set $\mathcal{M}$.

The operational formulation of the Choi-Jamiołkowski isomorphism will make use of the following concept:

In particular, when the operational theory in question is quantum mechanics, every possible ensemble preparation can be described by a POVM $\{M_{i}\}$ and density operator $\rho$, where $p(i)=Tr(\rho M_{i})$ and $P_{i}$ is a preparation which produces the quantum state $\rho_{i}=\frac{\sqrt{\rho}M_{i}\sqrt{\rho}}{Tr(M_{i}\rho)}$ [47].

For brevity, I will also use the following definitions:

In its original form, the Choi-Jamiołkowski isomorphism is a one-to-one map between unnormalized quantum states and completely positive (CP) operators. Specifically, for a given CP operator $\epsilon_{B|A}$ mapping a Hilbert space $\mathscr{H}_{A}$ to a Hilbert space $\mathscr{H}_{B}$, the corresponding Choi-Jamiołkowski state $\rho^{\epsilon}$ is obtained by applying $\epsilon_{B|A}$ to half of the maximally entangled state $\rho_{AA^{\prime}}=\frac{1}{d}\sum_{i}\sum_{i^{\prime}}|i\rangle_{A}|i\rangle_{A^{\prime}}\langle i^{\prime}|_{A}\langle i^{\prime}|_{A^{\prime}}$, where $A^{\prime}$ is a quantum system with the same Hilbert space as $A$, and $d$ is the dimension of this Hilbert space: [47, 48]⁵⁵5The formula presented here is for the Choi version of the isomorphism; the Jamiołkowski version is $\rho^{\epsilon}_{AB}=(\epsilon_{B|A}\otimes\mathbb{I})(\frac{1}{d}\sum_{i}\sum_{i^{\prime}}|i\rangle_{A}|i\rangle_{A^{\prime}}\langle i^{\prime}|_{A}\langle i^{\prime}|_{A^{\prime}})^{T_{A}})$, where $T_{A}$ refers to the partial transpose in the basis $\{|i\rangle\}$. The key difference between the two formulations is that the Jamiołkowski version, unlike the Choi version, is basis-independent. But since I will not henceforth be working within the quantum formalism, this distinction is not important.

However, in ref [47] the isomorphism is reformulated as a map between conditional density operators and completely positive trace-preserving (CPTP) operators. This formulation, which I paraphrase below, is more transparent in its physical interpretation and hence I will take it as the starting point for the operational approach:

Using these concepts, we can define an operational version of the Choi–Jamiołkowski isomorphism.

In graphical terms, this says that for any scenario as in figure 1 there exists a scenario as in figure 2 and vice versa; likewise, for any scenario as in figure 3 there exists a scenario as in figure 4 and vice versa.

It is a particularly distinctive feature of quantum information that quantum states cannot be cloned - that is, there is no quantum operation which makes perfect copies of two non-orthogonal quantum states. In fact, the impossibility of cloning quantum states is a special case of the impossibility of broadcasting quantum states:

In ref [49], it is shown that corollary 6.3 can be derived directly from the monogamy of entanglement using the Choi-Jamiołkowski isomorphism:

Here I will derive a similar result, but my derivation will be presented in the context of general operational theories without presupposing the Hilbert space structure of quantum mechanics. Therefore before embarking on the proof, I need to provide definitions of broadcasting and monogamy which are suited to the operational context.

In order to define ‘broadcasting’ we will use the concept of as an ‘operational state’; note, again, that this concept will be used only to define broadcasting, and will not appear in the definitions of the features from which no-broadcasting will be derived.

Defining the relevant monogamy property for an operational context is less straightforward. In quantum mechanics, the monogamy of entanglement refers to the fact that the amount of entanglement a quantum system has with a given system limits the amount of entanglement it can have with any other system - in the most extreme case, a quantum system which is maximally entangled with a given quantum system cannot be entangled at all with any other quantum system. But entanglement is a property of quantum states, usually quantified using state-dependent measures like the ‘concurrence’ (or ‘tangle’)[51], and these measures do not have a straightforward physical interpretation, making it difficult to translate them to the framework of operational theories.

Therefore it is more helpful for us to focus on the ‘monogamy of correlations,’ which is a property of certain theories in which there are limits on how strongly a system can be correlated with other systems. The monogamy of correlations is defined purely in terms of observable statistics, and is therefore much more easily cast in an operational framework.

In order to facilitate comparison to the quantum case, I will define monogamy for a general operational theory using the CHSH quantity:

Note that is also possible to define a related CHSH quantity $\mathscr{B}_{AB}(P^{0},P^{1},T,M_{B}^{0},M_{B}^{1})$ for the scenario where an experimenter applies one of two ensemble preparations $P^{0},P^{1}$ to a system, then applies a fixed transformation $T$, then performs one of two measurements $M_{B}^{0},M_{B}^{1}$. This quantity is defined just as in equation LABEL:Belloperator, where the result of the measurement $M_{A}$ is replaced with the result of the probabilistic choice in the ensemble preparation. This definition ensures that if for some transformation $T$, some set of preparations $P_{AB}$, $P^{0},P^{1}$ and some set of measurements $M^{0}_{1},M^{1}_{1}$, $M^{0}_{2},M^{1}_{2}$, for any $x,y\in\{0,1\}$ the distributions $p_{P_{AB};M^{x}_{1}M^{y}_{2}}$ and $p_{P^{x};T;M^{y}_{2}}$ are the same, then $\mathscr{B}_{AB}(P_{AB},M_{A}^{0},M_{A}^{1},M_{B}^{0},M_{B}^{1})=\mathscr{B}_{AB}(P^{0},P^{1},T,M_{B}^{0},M_{B}^{1})$.

With a definition for the CHSH quantity in hand, we may now define two types of monogamy which might be obeyed by an operational theory:

It can be shown that this bound is necessarily obeyed by any operational theory which does not allow signalling (see section )[12].

Strong monogamy of correlations need not be obeyed by all non-signalling operational theories, but it is obeyed by quantum mechanics[52].

We will also require the notion of Bell nonlocality, which is again defined by analogy with the quantum case:

Using these concepts, we have the following theorem (see appendix .1 for the proof).

Note that if we impose the ontological Choi-Jamiołkowski isomorphism as a fundamental constraint on an operational theory, it follows by modus tollens that the strong monogamy of correlations together with the existence of Bell nonlocality implies the absence of broadcasting, as advertised in table 3.

The results of section 6.4 may also be used to derive the existence of quantum interference from the monogamy of correlations together with the existence of Bell non-locality.

To see this, let us limit ourselves to considering GPTs whose observables have the algebraic structure of the self-adjoint operators in a $C^{*}$ algebra. Not all GPTs meet this criterion⁸⁸8See ref [53] for a discussion of the general conditions under which a theory can be represented as a $C^{*}$ algebra, but nonetheless it is a reasonable heuristic framework since a large class of important GPTs, including classical mechanics and quantum mechanics, can indeed be thus represented. Then observe that for any theory represented as a $C^{*}$-algebra, it can be shown that if the $C^{*}$ algebra is commutative then the theory has a univeral broadcasting map [54, 13]. Thus if a GPT does not have a universal broadcasting map, any $C^{*}$ algebra that represents it must be non-commutative. Finally, note that theories represented by commutative $C^{*}$ algebras are classical phase space theories, whereas theories represented by non-commutative $C^{*}$ algebras have observables which cannot be simultaneously measured and hence exhibit superposition and interference[54].

Thus, if we limit ourselves to GPTs which have the algebraic structure of the self-adjoint operators in a $C^{*}$ algebra and which obey the operational Choi-Jamiołkowski isomorphism, it follows from theorem 3 that any GPT which exhibits strong monogamy of correlations and Bell non-locality must also exhibit superposition and interference. This means that superposition and interference need not be regarded as properties of the quantum state - with the help of the operational Choi-Jamiołkowski isomorphism they can be derived directly from properties of non-local correlations, with no need to appeal to a concept of state at all.

One of the most puzzling feature of quantum theory is the fact that it is contextual: that is, it is not possible to interpret the outcomes of quantum-mechanical measurements as if they simply reveal the pre-existing value of some property of the system being measured.

The original formulation of contextuality[55] tied the concept of contextuality to the concept of determinism, but the notion was subsequently generalized by Spekkens[56]. Spekkens’ form of contextuality is most easily defined within the ontological models framework, where one supposes that every system has a single real ‘ontic state,’ which determines the probabilities for the outcomes of any measurement on that system⁹⁹9The use of this faramework does not necessarily imply the endorsement of a description of quantum reality in terms of underlying states; the ontological models approach could simply be regarded as a helpful language in which I may express mathematical facts about the structure of quantum theory, such as its contextual character.. An ontological model thus consists of a space $\Lambda$ of ontic states $\lambda$, a set of probability distributions $\mu^{P}(\lambda)$ giving the probability that the system ends up in the state $\lambda$ when we perform the preparation procedure $P$, a set of response functions $\xi_{M,X}(\lambda)$ giving the probability that we obtain outcome $M^{x}$ when we perform measurement $M$ on a system whose ontic state is $\lambda$, and a set of column-stochastic matrices $T^{O}$ representing the way in which the ontic state is transformed when some operation $O$ is applied to the system. A valid ontological model must satisfy the following positivity and normalization conditions:

1. 1.

   $\forall P,\lambda\quad\mu_{P}(\lambda)\in[0,1]$

2. 2.

   $\forall P\quad\sum_{\lambda}\mu_{P}(\lambda)=1$

3. 3.

   $\forall M,X\quad\vec{\xi}_{M,X}(\lambda)\in[0,1]$

4. 4.

   $\forall M\quad\sum_{X}\vec{\xi}_{M,X}(\lambda)=1$

5. 5.

   $\forall O\quad T_{0}$ is a column-stochastic matrix. ¹⁰¹⁰10A column-stochastic matrix is a matrix whose entries are all non-negative and whose columns sum to one. Left-multiplication by a column-stochastic matrix is the most general possibility for the representation of operations in an ontological model, since such an operation must map ontic state $\lambda_{i}$ to $\lambda_{j}$ with some probability, which is specified by entry $(i,j)$ in the transformation matrix. Given that the entries are all probabilities, they must be nonnegative, and given that each transformation must map $\lambda_{i}$ to some state, the sum of the entries in a column must be one.

An ontological model is preparation contextual if it does not represent every quantum state by a unique probability distribution $\mu(\lambda)$; transformation contextual iff it does not represent every possible quantum operation $O$ by a unique transformation matrix $T_{O}$; and measurement contextual if it does not represent every possible quantum measurement element $M^{x}$ by a unique set of response functions $\xi_{M,X}(\lambda)$. Spekkens proved[56] that any ontological model of quantum mechanics which reproduces all the correct measurement statistics must exhibit preparation contextuality, but examples such as the Kochen-Specker model [57] show that this is not true for measurement contextuality. ¹¹¹¹11Transformation contextuality has not been studied so thoroughly, and most existing explicit ontological models omit transformations, so to my knowledge the question of whether or not quantum mechanics necessarily exhibits transformation contextuality remains unresolved.

I now generalize the definition of preparation contextuality to the framework of operational theories. (Similar generalizations can be made for measurement and transformation contextuality, but I will not need these concepts here).

I will also employ a definition of the no-signalling principle suited for an operational framework. This generalization is a straightforward one, since it is just the operational version of the no-signalling principle as it is used in the context of relativistic cryptography[58]:

Thus we have the following theorem (see appendix .1 for the proof).

Note that if we impose the ontological Choi-Jamiołkowski isomorphism as a fundamental constraint on an operational theory, it follows by modus tollens that if the theory exhibits Bell nonlocality but obeys the no-signalling principle, it must be preparation contextual; that is, preparation contextuality is a necessary feature of any such operational theory describing a world which is non-local but also non-signalling.

’Heisenberg’s Uncertainty Principle’ is one of the most well-known features of quantum physics, not least because the fact that there exist incompatible measurements whose results cannot be simultaneously predicted with certainty seems strongly in conflict with our classical intuitions. This provides a clear imperative to seek an explanation for the existence of such apparently arbitrary limitations on the properties that quantum systems can possess.

To be more precise, there are in fact a number of different uncertainty relations which characterize tradeoffs between the precision with which the outcomes of various sets of measurements can be predicted. Most of these relations were originally formulated in terms of standard deviations and commutators, but this approach has been criticized on the grounds that the use of commutators makes the relations state-dependent, which should not be the case given that in quantum mechanics there always exists a dynamical evolution which transforms any initial pure state into any any other pure state[59]. Thus more recently there has been a move towards expressing uncertainty relations as a constraint on the Shannon entropies corresponding to the probability distributions over outcomes associated with various possible measurements[59, 60, 61]. However these entropic uncertainty relations have their own limitations - in particular, they are not capable of distinguishing between different possible combinations of outcomes[62] - and therefore in certain cases it is more useful to consider fine-grained uncertainty relations, which directly constrain particular sets of outcomes of different measurements.

The derivation of this section is inspired by the work of ref [62], where it is shown that it is possible to derive the Tsirelson bound from the following fine-grained entropic uncertainty relation, which applies to any pair of orthogonal two-outcome measurements $M$ and $M^{\prime}$ on a two-dimensional quantum system (such as the binary spin-half observables $X$ and $Z$):

Here, $p(m|M)_{\rho}$ denotes the probability that I obtain outcome $m\in\{0,1\}$ when I perform measurement $M$ on a system in quantum state $\rho$.

In the operational framework, equation 2 becomes a constraint applying to all preparations in the operational theory which prepare systems of dimension two, according to the operational definition of ‘dimension’ which I provide in section 8.2. I also reverse the argument of ref [62], in order to derive the uncertainty relation from the Tsirelson bound. I then note that it was shown in ref [11] that the Tsirelson bound can be derived from the principle of information causality:

Thus, if we take it that information causality is a fundamental constraint on quantum theory, as suggested in [11], the fact that quantum states obey this particular fine-grained uncertainty relation in quantum theory may be understood as a consequence of the fundamental constraint. Moreover, in quantum mechanics, the inequality in equation 2 is tight (for example, the inequality is saturated if we choose measurements $X$, $Z$ and prepare the system to be measured in an eigenstate of $\frac{1}{\sqrt{2}}(X+Z)$ or $\frac{1}{\sqrt{2}}(X-Z)$) and thus argue in this particular case information causality may be understood as setting the exact limits of quantum uncertainty.

Since the fine-grained uncertainty relation of equation 2 applies only for systems of dimension two, we need an operational way to single out systems of dimension two. To do so, we must first add some structure to the usual framework of operational theories. Thus far, we have assumed that for a given operational theory $(\mathcal{P},\mathcal{M},\mathcal{T},p)$ it is possible to perform any preparation followed by any transformation followed by any measurement, but in a theory with a meaningful concept of dimension this is not so, because we can only perform measurements which match the dimension of the system I have prepared. Thus in this section I consider an operational theory to be a collection of subtheories $(\mathcal{P},\mathcal{M},\mathcal{T},p)$, such that within any given subtheory it is possible to perform any preparation followed by any transformation followed by any measurement, but it is not possible to perform a preparation from one subtheory followed by a transformation or measurement from another subtheory. I then define the dimension of a sub-theory as follows.

Obviously, the choice to associate $d$ dimensions with $d^{2}-1$ parameters is inspired by quantum theory: a $d$-dimensional quantum state is described by a set of $d$ complex parameters, or $d^{2}$ continuous real parameters, minus $1$ to account for normalization. But this particular mapping is a naming convention only - I could equally well have chosen to say that a subtheory described by a set of $d$ parameters is $d$-dimensional, which might have seemed the natural choice if it were not for the example of quantum theory. And therefore this definition does not take for granted any particular structure for the state space of the subtheory, other than the continuity implied by the second property.

The fine-grained uncertainty relation also applies only for pairs of orthogonal measurements, and therefore we require a notion of ‘orthogonal measurement’ for the operational framework:

This definition implies that given a preparation $P$, and any set of $q$ orthogonal measurements $\{M_{i}\}$ each having two outcomes $M_{i}^{0},M_{i}^{1}$, we need $q$ independent parameters to specify the set of $q$ probabilities $p(M_{i}^{0}|M_{i},P)$. Therefore I can set $q$ out of the $d^{2}-1$ parameters needed to specify the full set of outcome probabilities to be equal to the $q$ probabilities $p(M_{i}^{0}|M_{i},P)$. In particular, for a two-dimensional system, given any set of three orthogonal measurements $\{M_{1},M_{2},M_{3}\}$ each having two outcomes, specifying the three probabilities $p(M_{1}^{0}|M_{1},P),p(M_{2}^{0}|M_{2},P),p(M_{3}^{0}|M_{3},P)$ is sufficient to fix the probabilities for any other measurement which may be performed after the preparation $P$.

Using these concepts, we have the following theorem (see appendix .3 for the proof).