Impossibility of mixed-state purification in any alternative to the Born Rule
Using the existing classification of all alternatives to the Measurement Postulates of Quantum Mechanics we study the properties of multi-partite systems in these alternative theories. We prove that in all these theories the Purification Principle is violated, meaning that some mixed states are not the reduction of a pure state in a larger system. This implies that simple operational processes, like mixing two states, cannot be described in an agent-free universe. This appears like an important clue for the problem of deriving the Born Rule within “unitary quantum mechanics” or the many-worlds interpretation. We also prove that in all such modifications the task of state tomography with local measurements is impossible, and present a simple toy theory displaying all these exotic non-quantum phenomena. This toy model shows that, contrarily to previous claims, it is possible to modify the Born rule without violating the No-Signalling Principle. Finally, we argue that the quantum measurement postulates are the most non-classical amongst all alternatives.
The postulates of quantum theory describe the evolution of physical systems by distinguishing between the cases where observation happens or not. However, these postulates do not specify what constitutes observation, and it seems that an act of observation by one agent can be described as unperturbed dynamics by another Frauchiger_single_2016 (). This opens the possibility of deriving the physics of observation within the picture of an agent-free universe that evolves unitarily. This problem has been studied within the dynamical description of quantum measurements Allahverdyan_understanding_2013 (), the decoherence program Zurek_probabilities_2005 () and the Many-Worlds Interpretation of quantum theory Deutsch_quantum_1999 (); Wallace_how_2010 ().
In this work, instead of presenting another derivation of the measurement postulates, we take a more neutral approach and analyze all consistent alternatives to the measurement postulates. In particular, we prove that in each such alternative there are mixed states which are not the reduction of a pure state on a larger system. This fact puts strong limitations on the type of “classical reality” that could emerge in these alternative theories, and in this way singles out the (standard) quantum measurement postulates including the Born Rule.
In our previous work Galley_2017_classification () we constructed a complete classification of all alternative measurement postulates, by establishing a correspondence between these and certain representations of the unitary group. However, this classification did not involve the consistency constraints that arise from the compositional structure of the theory; which governs how systems combine to form multi-partite systems. In this work we take into account compositional structure, and prove that all alternative measurement postulates violate two compositional principles: Purification Chiribella_probabilistic_2010 (); Chiribella_informational_2011 () and Local Tomography Hardy_quantum_2001 (); Barrett_information_2005 (). We also present a simple alternative measurement postulate (a toy theory) which illustrates these exotic phenomena. Additionally, this toy theory provides an interesting response to the claims that the Born rule is the only probability assignment consistent with no signalling Aaronson_quantum_2004 (); bao_grover_2016 (); Han_Quantum_2016 ().
In Section II we introduce a theory-independent formalism, which allows to study all alternatives to the measurement postulates. We also review the results of our previous work Galley_2017_classification (). In Section III we define the Purification and Local-Tomography Principles, and show that these are violated by all alternative the measurement postulates. In Section IV we describe a particular and very simple alternative measurement postulate, which illustrates our general results. In Section V we discuss our results in the light of existing work, and argue that the standard measurement postulates are the the most non-classical ones. All proofs are in the appendices.
Ii Dynamically-quantum theories
In this work we consider all theories that have the same pure states, dynamics and system-composition rule as quantum theory, but have a different structure of measurements and a different rule for assigning probabilities.
ii.1 States, transformations and composition postulates
The family of theories under consideration satisfy the following postulates, taken from the standard formulation of quantum theory.
Postulate (Quantum States).
Every finite-dimensional Hilbert space corresponds to a type of system with pure states being the rays of .
Postulate (Quantum Transformations).
The reversible transformations on the pure states are for all .
Postulate (Quantum Composition).
The joint pure states of systems and are the rays of .
ii.2 Measurement postulates
Before presenting the generalized measurement postulate we need to introduce the notion of outcome probability function, or OPF. For each measurement outcome of system there is a function that assigns to each ray in the probability for the occurrence of outcome . Any such function is called an OPF. Each system has a (trivial) measurement with only one outcome, which must have probability one for all states. The OPF associated to this outcome is called the unit OPF , satisfying for all . A -outcome measurement is a list of OPFs satisfying the normalization condition . As an example, the OPFs of quantum theory are the functions
for all Hermitian matrices satisfying . This implies that . (Here and in the rest of the paper we assume that kets are normalized.)
Postulate (Alternative Measurements).
Every type of system has a set of OPFs with a bilinear associative product satisfying the following consistency constraints:
For every and there is such that for all . That is, the composition of a unitary and a measurement can be globally considered a measurement.
For any pair of different rays in there is such that . That is, different pure states must be operationally distinguishable.
The -product satisfies and
for all , , , . That is, tensor-product states contain no correlations.
For each and there is an ensemble in such that
for all . That is, the reduced state on conditioned on outcome on (and re-normalized) is a valid mixed state of . In the next sub-section we fully articulate the notions of ensemble and mixed state.
Consider measurements on system with the help of an ancilla . For any ancillary state and any OPF in the composite there exists an OPF on the system such that
for all .
A trivial modification of the Measurement Postulate consists of taking that of quantum mechanics (1) and restricting the set of OPFs in some way, such that not all POVM elements are allowed. In this work, when we refer to “all alternative measurement postulates” we do not include these trivial modifications.
ii.3 Mixed states and the Finiteness Principle
A source of systems that prepares state with probability is said to prepare the ensemble . Two ensembles and are equivalent if they are indistinguishable
for all measurements . Note that distinguishability is relative to the postulated set of OPFs . A mixed state is an equivalence class of indistinguishable ensembles, and hence, the structure of mixed states is also relative to . To evaluate an OPF on a mixed state we can take any ensemble of the equivalence class and compute
In general, ensembles can have infinitely-many terms, hence, the number of parameters that are needed to characterize a mixed state can be infinite too. When this is the case, state estimation is impossible, and for this reason we make the following assumption.
Each mixed state of a finite-dimensional system can be characterized by a finite number of parameters.
Recall that in quantum theory we have . And in general, the distinguishability of all rays in implies . These parameters can be chosen to be a fix set of “fiducial” OPFs , which can be used to represent any mixed state as
The fact that OPFs are probabilities implies that any OPF is a linear function of the fiducial OPFs
In other words, the fiducial OPFs constitute a basis of the real vector space spanned by . Using the consistency constraint C1, we define the action
on the vector space spanned by . This associates to system a -dimensional representation of the group . This, together with the consistency constraints, implies that only certain values of are allowed. For example
ii.4 Measurement postulates for single systems
In this subsection we review some of the results obtained in Galley_2017_classification (). These provide the complete classification of all sets satisfying the Finiteness Principle and the consistency constraints C1 and C2. These results ignore the existence of the -product, C3, C4 and C5. Hence, in alternative measurement postulates with a consistent compositional structure there will be additional restrictions on the valid sets . This is studied in Section III.
If satisfies the Finiteness Principle and C1 then there is a positive integer and a map from to the set of Hermitian matrices such that
for all normalized vectors .
Note that there are many different sets with the same . In particular, since the action
is reducible, the Hermitian matrices can have support on the different irreducible sub-representations of (12), generating sets with very different physical properties. All these possibilities are analyzed in Galley_2017_classification ().
If satisfies the Finiteness Principle, C1 and C2, then
there is a non-constant .
there is such that has support on a sub-representation of the action (12) with odd angular momentum.
Iii Features of all alternative measurement postulates
In this section we analyze the compositional structure of alternative measurement postulates. We do so by considering two well known physical principles which, together with other assumptions, have been used to reconstruct the full formalism of quantum theory Masanes_derivation_2011 (); Chiribella_informational_2011 (); Dakic_quantum_2009 (); Barnum_general_2014 (). Remarkably, this principles are violated by all alternative measurement postulates.
iii.1 The Purification Principle
This principle establishes that any mixed state is the reduction of a pure state in a larger system. This legitimises the “Church of the Larger Hilbert Space”, an approach to physics that always assumes a global pure states when an environment is added to the systems under consideration Nielsen_quantum_2011 ().
For each ensemble in there exists a pure state in for some satisfying
for all .
Note that the original version of the Purification Principle introduced in Chiribella_probabilistic_2010 () additionally demands that the purification state is unique up to a unitary transformation on . Also note that the following theorem does not require the Finiteness Principle.
Theorem (No Purification).
All alternative measurements postulates satisfying C1, C2, C3 and C4 violate the Purification Principle.
This implies that in all alternative measurement postulates there are operational processes, such as mixing two states, which cannot be understood as a reversible transformation on a larger system. In such alternative theories, agents that perform physical operations cannot be integrated in an agent-free universe, as can be done in quantum theory, and the Church of the Larger Hilbert Space is illegitimate. An agent-free universe is also the starting point of the many-worlds interpretation of quantum theory. Hence, it seems like the No-Purification Theorem provides important clues for the derivation of the Born Rule within the many-worlds interpretation Everett_relative_1957 (); Zurek_probabilities_2005 (); Wallace_how_2010 (); Deutsch_quantum_1999 ().
iii.2 The Local Tomography Principle
This principle has been widely used in reconstructions of quantum theory and the formulation of alternative toy theories Dakic_quantum_2009 (); Masanes_derivation_2011 (); Chiribella_informational_2011 (); Barnum_local_2012 (); Masanes_existence_2013 (); Hohn_2017_toolbox (). One of the reasons is that it endows the set of mixed states with a tensor-product structure Barrett_information_2005 (). This principle establishes that any bi-partite state is characterized by the correlations between local measurements. That is, two different mixed states on must provide different outcome probabilities
for some local measurements , . Using the notation introduced in (9) we can formulate this principle as follows.
Principle (Local Tomography).
If is a basis of and is a basis of then is a basis of , where and .
A theory is said to violate local tomography if at least one composite system within the theory violates local tomography. Therefore, it is sufficient to analyze the particular bi-partite system .
Theorem (No Local Tomography).
All alternative measurements and satisfying C1, C2, C3 and the Finiteness Principle violate the Local-Tomography Principle.
The above result is proven in Appendix E by using a representation theoretic formulation of local tomography that we introduce in this work, and combining it with the following technical result. All non-quantum irreducible representations of which are sub-representations of the action (12) have a sub-representation when restricted to the subgroup . Here denotes the trivial representation of .
iii.3 Representation theoretic constraints imposed by the compositional structure
As shown in Galley_2017_classification () all sets of OPFs are associated to a representation of . The dynamical structure of quantum theory imposes some requirements on these representations; namely all representations corresponding to sets of OPFs must have a trivial representation when restricted to . In Appendix B it is shown that the compositional structure implies that a system with a set of OPFs (where ) can be considered as a composite of two systems and only if the associated representation is such that:
Here is the restriction of to . The representations and contain a unique trivial representation (corresponding to the normalisation degree of freedom). The subspace acted on by is called the locally tomographic subspace. The above criterion is necessary, but not sufficient for composition.
Theorem (Existence of locally tomographic subspace).
Any alternative measurement structure satisfying C1, C2 and the Finiteness Principle is such that the associated representation has a locally tomographic subrepresentation when restricted to .
This theorem is proven in Appendix D.
Iv A Toy Theory
In this section we present a simple alternative measurement postulate which serves as example for the results that we have proven in general (violation of the Purification and Local-Tomography Principles). In Appendix H it is proven that this alternative measurement postulate satisfies all consistency constraints (C1, C2, C3, C4, C5) except for the associativity of the -product. This implies that this toy theory is only fully consistent when dealing with single and bi-partite systems. However, most of the work in the field of general probabilistic theories (GPTs) focuses on bi-partite systems, because these already display very rich phenomenology. In the following we consider two local subsystems of dimension and with sets of OPFs and . The composite (global) system has a set of OPFs .
Definition (Local effects ).
To each Hermitian matrix satisfying
for some and ,
for some and ,
there corresponds the OPF
where is the projector onto the symmetric subspace of . The unit OPF corresponds to .
That is, both matrices, and , have to be not-necessarily-normalized mixtures of symmetric product states.
Example (Canonical measurement for prime).
For the case where is prime there exists a canonical measurement which can be constructed as follows. Consider the mutually unbiased bases (MUBs): where runs from to Bandyopadhyay_new_2001 (). Then we can associate an OPF to each Hermitian matrix . Since the basis elements of these MUBs form a complex projective 2-design Klappenecker_mutually_2005 (), by the definition of 2-design Zhu_clifford_2016 (), we have the normalization constrain:
and hence the set of OPFs forms a measurement.
Definition ( product).
For any pair of OPFs and the Hermitian matrix corresponding to their product is
where and are the projectors onto the symmetric and anti-symmetric subspaces of , and analogously for and .
This product is clearly bilinear and, by using the identity , we can check that .
We observe that not all effects are of the form . Hence the set of effects on the joint system is not , but has to be extended to to include these joint product effects.
Definition (Global effects ).
The set should include all product OPFs , all OPFs of understood as a single system, and their convex combinations.
The identity perfectly shows that the vector space is larger than the tensor product of the vector spaces and , by the extra term . This implies that this toy theory violates the Local-Tomography Principle.
The joint probability of outcomes and on the entangled state can be written as
When we only consider sub-system outcome probabilities are given by
where the reduced state must necessarily be
All these reductions of pure bipartite states are contained in the convex hull of , as required by the consistency constraint . However, not all mixtures of can be written as one such reduction (20). That is, the Purification Postulate is violated. This phenomenon is graphically shown in Figure 1.
This toy model is restricted, in that not all mathematically allowed effects on the local state spaces are allowed effects. It also violates the principle of perfect distinguishability in that all the effects are noisy.
Multiple proofs have been put forward which claim to show that violations of the Born rule lead to signalling Aaronson_quantum_2004 (); bao_grover_2016 (); Han_Quantum_2016 (). However the authors often only consider modifications of the Born rule of a specific type. In Aaronson_quantum_2004 (); bao_grover_2016 () the authors only consider modifications of the Born rule of the following form:
where . Modified Born rules of this type are such that the measurements described have the same outcomes as the projective measurements of standard quantum theory. That is to say the outcomes of measurements are still associated to the basis elements . Another feature of these modifications is that probabilities are functions of a single amplitude (disregarding the normalisation, which can be removed using a -norm). Modifications of this form are very restricted. By modifying all the measurement postulates of quantum theory, we can create toy models like the one introduced which are non-signalling. This shows that by modifying the Born rule in a more general manner one can avoid issues of signalling.
In the case of the toy model it is immediate to see that it is consistent with no-signalling. The condition of no-signalling is equivalent to the existence of a well defined state-space for the subsystem (i.e. independent of action on the other subsystem). We see then that no-signalling is just a consequence of there existing a well defined reduced state and arguably can be considered an intrinsic feature of the operational framework, rather than a supplementary principle.
v.2 The quantum measurement postulates are the most non-classical
One consequence of the classification in Galley_2017_classification () is that the quantum measurement postulates are the ones which give the lowest dimensional state spaces. In this sense the quantum measurement postulates are the most non-classical, since they give rise to the state spaces with the most indistinguishable mixtures Mielnik_generalized_1974 ().
Together with the fact that all alternatives to the Born rule violate the purification principle (and hence have some irreducible classicality) this shows that the quantum measurement postulates are the least classical amongst all alternatives.
v.3 Toy model
The toy model can be obtained by restricting the states and measurements of two pairs of quantum systems and . In this sense we obtain a theory which violates both local tomography and purification. This method of constructing theories is similar to real vector space quantum theory, which can also be obtained from a suitable restriction of quantum states and also violates local tomography. The main limitation of the toy model is that it does not straightforwardly extend to more than two systems. There is a natural generalisation of the toy model to consider effects to be linear in for , however showing the consistency of the reduced state spaces and joint effects is more complex.
v.4 Theories which decohere to quantum theory
A recent result Lee_nogo_2017 () shows that all operational theories which decohere to quantum theory (in an analogous way to which quantum theory decoheres to classical theory) must violate either purification or causality (or both). The fact that all alternatives to the measurement postulates violate purification is a desirable feature if one wishes to consider them as possible successor theories to quantum theory. The requirement that a theory decoheres to quantum theory imposes additional constraints on the theory. Whether the theories described in this work meet these constraints is an open question.
v.5 Relation to work in GPTs
Many general results have been derived in the framework of GPTs showing features or properties of various natural families of theories which either obey the principle of local tomography Barrett_information_2005 (); Masanes_entanglement_2014 (); Lee_computation_2015 () or the purification principle Chiribella_probabilistic_2010 (); Chiribella_entanglement_2015 (); Lee_Generalised_2016 (); Barnum_Ruling_2017 (). In this paper show that there is a large family of bi-partite systems (and possibly full theories) which do not obey either of these principles. Whilst it is often convenient to assume one of these two properties when deriving results about general theories (or potential successor theories), it seems unduly restrictive considering that any modification (however small) to the measurement postulates of quantum theory lead to theories where these properties no longer hold.
We have studied composition in general theories which have the same dynamical and compositional postulates as quantum theory but which have different measurement postulates. We presented a toy model of a bi-partite system with alternative measurement rules, showing that composition is possible in such theories. We showed that all such theories violate two compositional principles: local tomography and purification.
vi.2 Future work
The toy model introduced in this work applies only to bi-partite systems and is simulable with quantum theory. Hence an important next step is constructing a toy model with alternative measurement rules which is consistent with composition of more than two systems. This requires a product which is associative. This construction may prove impossible, or it may be the case that all valid constructions are simulable by quantum theory. We suggest that there are three possibilities when considering theories with fully associative products.
Possibility 1. (Logical Consistency of Postulates of Quantum Theory). The only measurement postulates which are fully consistent with the associativity of composition are the quantum measurement postulates.
If this were the case it would show that the postulates of quantum theory are not independent. Only the quantum measurement rules would be consistent with the dynamical and compositional postulates (and operationalism). However it may be the case that we can develop theories with alternative Born rules which compose with an associative product, but that all these theories are simulable by quantum theory.
Possibility 2. (Simulability of all Alternative Postulates). The only measurement postulates which are fully consistent with the dynamical postulates of quantum theory are simulable with a finite number of quantum systems.
Possibility 3. (Alternative measurement postulates which are not simulable). There exist measurement postulates which are fully consistent with the dynamical postulates of quantum theory and are not simulable with a finite number of quantum systems.
This final possibility would be interesting from the perspective of GPTs as it would show that there are full theories which can be obtained by modifying the measurement postulates of quantum theory. It would show that quantum theory is not, in fact, an island in theory space.
We are grateful to Jonathan Barrett and Robin Lorenz for helpful discussions. TG is supported by the Engineering and Physical Sciences Research Council [grant number EP/L015242/1]. LM is funded by EPSRC.
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Appendix A Operational principles and consistency constraints
a.1 Single system
As stated in the main section the allowed sets of OPFs are subject to some operational constraints. We introduce features obeyed by operational theories and derive the consistency constraints C1. - C5. from them. In the following we adopt a description of operational principles in terms of circuits, as in Chiribella_informational_2011 (). The basic operational primitives are preparations, transformations and measurements. These three are all procedures. We define them in terms of inputs, outputs and systems.
Definition (Preparation procedure).
Any procedure which has no input and outputs one or more systems is a preparation procedure.
Definition (Transformation procedure).
Any procedure which inputs one or more systems and outputs one or more systems is a transformation.
Definition (Measurement procedure).
Any procedure which inputs on or more systems and has no output is a measurement procedure.
In the above “no input” and “no output” refers to output or input of systems, typically there will be a classical input or output such as a measurement read-out.
Operational Implication (Composition of a procedure with a transformation).
The composition of a transformation with any procedure is itself a procedure of that kind.
For example the composition of a transformation and a measurement is itself a measurement.
Operational Assumption (Mixing).
The process of taking two procedures of the same kind and implementing them probabilistically generates a procedure of that kind.
An experiment is a sequence of procedures which has no input or output.
This principle entails that every experiment can be considered as a preparation and a measurement. The experiment is fully characterised by the probabilities
for all measurement outcomes and all preparations in the experiment.
In this approach a system is an abstraction symbolised by the wire between the preparation procedure and the measurement procedure. A system can be represented as:
a.2 Pairs of systems
Given the above definitions it is natural to ask when an experiment can be described using multiple systems. Let us consider a system which we represent using two wires:
These can only be considered as representing two distinct systems and it is possible to independently measure both systems.
Definition (Existence of subsystems).
A system can be considered as a valid composite system if a measurement on subsystem and a measurement on subsystem uniquely specify a measurement on independent of the temporal ordering Masanes_existence_2013 ().
If the above property is not met, then the system cannot be considered as a composite (and should be represented using a single wire). This principle is sometimes known as the no-signalling principle. Diagrammatically this entails that any preparation of a composite system is such that:
The most general form of an experiment with two systems is:
which can naturally be viewed as an experiment on a single system .
Definition (Separable procedures).
Two independent preparations and which are independently measured with outcomes and are such that:
In this case the joint procedures and are said to be separable.
By the definition of a preparation, any operational procedure which outputs a system is a preparation. Hence consider the case where the measurement is separable. The procedure of making a joint preparation and making a measurement on is a preparation of a state .
Operational Implication (Steering as preparation).
Operationally Alice can make a preparation of system by making a preparation of and getting Bob to make a measurement on system .
By the definition of a measurement any operational procedure which inputs a system and outputs no system is a measurement. Consider the case where the preparation is separable. Then the procedure of preparing system and jointly measuring and is a measurement procedure on .
Operational Implication (Measuring with an ancilla).
A valid measurement for Alice consists in adjoining her system to an ancillary system and carrying out a joint measurement.
In the case where both preparation and measurement are separable then the experiment can be viewed as two separate experiments. In the bi-partite case these are no further methods of generating preparations and measurements. Hence when determining whether a pair of systems is consistent with the operational properties which arise from composition, these are the only features which we need to consider.
One may ask whether any further operational implications will emerge from considering more than two systems.
Operational Assumption (Associativity of composition).
The systems and are the same.
This implies that there are no new types of procedures which can be carried out on a single system by appending more than one system. Consider an experiment with multiple systems . For any partitioning of the experiment which creates a preparation of system , all regroupings of systems are equivalent. This is a preparation by steering of system conditional on a measurement on systems which can be viewed as a single system. Diagrammatically it tells us that all ways of partitioning an experiment with multiple systems are equivalent.
This entails the only constraints imposed by the operational framework will come from the assumptions and implications outlined above. There are no further operational implications which emerge from the above definitions and assumptions.
a.3 OPFs and the product
We assume the finiteness principle holds, and that for a set of OPFs there exists a finite linearly generating set . That is to say:
Consider a composite system . By the definition of a composite system above, for any OPF on and on there exists an OPF on . By the assumption that mixing is possible, the outcome is a valid outcome.
Operational Assumption (Mixing separable procedures).
If two parallel processes are separable, it is equivalent to mix them before they are considered as a joint process or after.
From the above assumption it follows that:
If we further assume that it is possible to mix with subnormalised probabilities, i.e. then the product is bi-linear. The identity follows from the fact that a separable measurement is a valid measurement on , and hence:
a.4 Consistency constraints
Consistency constraint C1. follows directly from the fact that the composition of a transformation and a measurement is a measurement.
A state corresponds to an equivalence class of indistinguishable preparation procedures.
From this definition it follows that two states cannot be indistinguishable. This implies C2.. In a system where some pure states are indistinguishable the manifold of pure states would no longer be the set of rays on (as required by the first postulate).
Principle (Uncorrelated pure states).
Given two systems and independently prepared in pure states and the joint state of the system is given by .
This principle, together with the definition of independent systems implies that:
Let us consider the steering scenario. In this case the process of both Alice and Bob making a measurement with outcome on a joint state can be considered as a measurement with outcome on system prepared in a certain manner.
An arbitrary preparation of system is given by . Hence the steering preparation implies that for each preparation of and for each local measurement outcome on there exists a state in which system is prepared. In the OPF formalism this means that for every and every there exists an ensemble such that
holds for all . The normalisation occurs due to the fact that summing over the measurement outcomes should give unity on both sides of the expression.
Let us consider the scenario consisting in measuring with an ancilla. In this case Alice and Bob carry out a joint measurement with outcomes on a system in an uncorrelated state . This should correspond to a valid measurement with outcome on system prepared in state . For each and for each there exists an such that
for all .
Appendix B OPF formalism, representation theory and local tomography
In this appendix we show that each set of OPFs is associated to a representation of the group . We also show that the representations associated to locally tomographic systems with sets of OPFs have certain features when restricted to the local subgroup . It is these features which will be used in the next appendix to show that all non-quantum measurement postulates lead to a violation of local tomography. In the following we assume the finiteness principle holds.
b.1 Single systems
To each set of measurement postulates obeying the finiteness principle there exists a unique representation of associated to that set.
We take a set of measurement postulates , where form a basis for .
We consider an OPF :
We can also consider as a single element:
This shows that and the map is a representation of . ∎
The representation of measurement postulates contains a unique trivial subrepresentation.
Consider a set of OPFs which form a measurement. The OPF is such that . Consider the basis where . We observe that . This implies that and that the representation has a trivial component. If the representation had another trivial component, it would necessarily be linearly dependent on the first. It would then be a redundant entry in the list of fiducial outcomes which is contrary to the property that they are linearly independent. ∎
b.2 Composite systems
The measurement structure contains all measurements of the form where and . Let and be bases for the two OPF spaces. Then the OPFs form a basis for the global OPFs . Hence a basis for is Hardy_foliable_2009 (); Masanes_lecture_2017 ().
b.2.1 Local tomography
For a locally tomographic bi-partite system with representation the restriction of to is:
A theory is locally tomographic if span . It follows immediately from the bilinearity of and the fact that span that the product for locally tomographic theories is a tensor product. Hence an arbitrary element of can be written:
Let us consider the action of an element of on an OPF in .
Let us consider
where we have used: which follows directly from the operational definitions of transformations, measurements and composite systems. It states that we can consider a separable transformation followed by a separable measurement as a separable measurement. We observe that is a well defined OPF on since and are well defined OPFs on and respectively.
b.2.2 Holistic systems
For a locally holistic bi-partite system with representation the restriction of to is: