The Measurement Postulates of Quantum Mechanics are Redundant

The Measurement Postulates of Quantum Mechanics are Redundant

Lluís Masanes Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom    Thomas D. Galley tgalley1@perimeterinstitute.ca Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada    Markus P. Müller Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada
July 20, 2019
Abstract

Understanding the core content of quantum mechanics requires us to disentangle the hidden logical relationships between the postulates of this theory. The theorem presented in this work shows that the mathematical structure of quantum measurements, the formula for assigning outcome probabilities (Born’s rule) and the post-measurement state-update rule, can be deduced from the other quantum postulates, often referred to as “unitary quantum mechanics”. This is achieved by taking an operational approach to physical theories, and using the fact that the manner in which a physical system is partitioned into subsystems is a subjective choice of the observer, and hence, should not affect the predictions of the theory. Contrary to other approaches, our proof does not require assuming the universality of quantum mechanics, nor making sense of probability in a deterministic theory. In summary, this result unveils a deep connection between the dynamical and probabilistic parts of quantum mechanics, and it brings us one step closer to understand what this theory is telling us about the inner workings of Nature.

I Introduction

What sometimes is postulated as a fundamental law of physics is later on understood as a consequence of more fundamental principles. An example of this historical pattern is the rebranding of the symmetrization postulate as the spin-statistics theorem PhysRev.82.914 (). Another example, according to some authors, is the Born rule, the formula that assigns probabilities to quantum measurements. The Born rule has been derived within the framework of quantum logic Gleason_measures_1975 (); Cooke_elementary_1985 (); Pitowsky_infinite_1998 (); Wilce_quantum_2017 (), taking an operational approach Saunders_derivation_2002 (); Busch_quantum_2003 (); Caves_Gleason_2004 (), and using other methods Logiurato_born_2012 (); Han_Quantum_2016 (). But all these derivations assume, among other things, the mathematical structure of quantum measurements, that is, the correspondence between measurements and orthonormal bases, or more generally, positive-operator valued measures Helstrom76 ().

Taking one step further, the structure of measurements together with the Born rule can be jointly derived within the many-worlds interpretation of quantum mechanics (QM) Deutsch_quantum_1999 (); Wallace_how_2010 () and the framework of entanglement-assisted invariance Zurek_probabilities_2005 (); Zurek20180107 (); Barnum_no_2003 (); Schlosshauer_zureks_2005 (). But these derivations involve controversial uses of probability in deterministic multiverse scenarios, which have been criticized by a number of authors Barnum_quantum_2000 (); Kent_one_2009 (); Baker_measurement_2006 (); Hemmo_quantum_2007 (); Lewis_Peter_2010 (); Price_decisions_2010 (); Albert_probability_2010 (); Caves_note_2004 (); Schlosshauer_zureks_2005 (); Barnum_no_2003 (); mohrhoff_probabilities_2004 (). Also, these frameworks require the universality of QM, meaning that the measurement apparatus and/or the observer has to be included in the quantum description of the measuring process. While this is a meaningful assumption, it is interesting to see that it is not necessary, as proven in the present article.

In this work we take an operational approach, with the notions of measurement and outcome probability being primitive elements of the theory, but without imposing any particular structure on them. We use the fact that the subjective choices in the description of a physical setup in terms of operational primitives must not affect the predictions of the theory. For example, deciding to describe a tripartite system as either the bipartite system or as must not modify the outcome probabilities. Using these constraints we characterize all possible alternatives to the mathematical structure of quantum measurements and the Born rule, and we prove that there is no such alternative to the standard measurement postulates. This theorem has simple and precise premises, it does not require unconventional uses of probability theory, and it is independent of the interpretation of probability. A further interesting consequence of this theorem is that the post-measurement state-update rule must necessarily be that of QM.

The structure of this article is the following. Section II reviews the postulates of QM, introduces a new formalism that allows to specify any alternative to the measurement postulates, and uses this formalism to state the main result of this work: the measurement theorem. Section III illustrates this theorem with two interesting examples, and contrasts our result with Gleason’s theorem Gleason_measures_1975 (). Section IV provides a bird’s eye view of the proof of the theorem, which is fully detailed in the appendicies. Finally, Section V concludes with some important remarks.

Ii Main Result

In this section we present the main result of this work. But before doing so, we prepare the stage appropriately. This involves reviewing some of the postulates of QM, reconstructing the structure of mixed states from them, and introducing a general characterization of measurements that is independent of their mathematical structure.

ii.1 The standard postulates of QM

Postulate (states). To every physical system there corresponds a complex and separable Hilbert space , and the pure states of the system are the rays .

Separable Hilbert spaces have either finite or countably-infinite dimension. All countably infinite-dimensional Hilbert spaces are isomorphic rudin1991functional (), so we denote them all by ; and with a slight abuse of notation we include them in the family by allowing . Analogously, we include the unitary transformations of in the family of groups . In this document we represent states (rays) by normalized vectors

Postulate (transformations). The reversible transformations (for example, possible time evolutions) of pure states of are the unitary transformations with .

Postulate (composite systems). The joint pure states of systems and are the rays of the tensor-product Hilbert space .

Postulate (measurement). Each measurement outcome of system is represented by a linear operator on satisfying , where is the identity. The probability of outcome on state is

(1)

A (full) measurement is represented by the operators corresponding to its outcomes , which must satisfy the normalization condition .

The more traditional formulation of the measurement postulate in terms of (not necessarily positive) Hermitian operators is equivalent to the above. But we have chosen the above form because it is closer to the formalism used in the presentation of our results.

Postulate (post-measurement state-update). Each outcome is represented by a completely-positive linear map related to the operator via

(2)

for all . The post-measurement state after outcome is

(3)

A (full) measurement is represented by the maps corresponding to its outcomes whose sum is trace-preserving.

If the measurement is repeatable and minimally disturbing Chiribella_Sharpness_2014 () then are projectors and the above maps are of the form , which is the standard textbook “projection postulate”. Below we prove that the “measurement” and “post-measurement state-update” postulates are a consequence of the previous three postulates.

ii.2 The structure of mixed states

Mixed states are not mentioned in the standard postulates of QM, but their structure follows straightaway from the measurement postulate (1). Recall that a mixed state is an equivalence class of indistinguishable ensembles, and an ensemble is a probability distribution over pure states. Note that the notion of distinguishability depends on what the measurements are. For the particular case of quantum measurements (1), the probability of outcome when a source prepares state with probability is

(4)

where we define the density matrix

(5)

This matrix contains all the statistical information of the ensemble. Therefore, two ensembles with the same density matrix are indistinguishable.

The important message from the above is that a different measurement postulate would give different equivalence classes of ensembles, and hence, a different set of mixed states. An example of mixed states for a non-quantum measurement postulate is described in Section III.1.

ii.3 Formalism for any alternative measurement postulate

Before proving that the only possible measurement postulate is that of QM, we have to articulate what “a measurement postulate” is in general. In order to do so, we introduce a theory-independent characterization of measurements for single and multipartite systems. This is based on the concept of outcome probability function (OPF), introduced in Galley_classification_2017 () and defined next.

Definition (OPF). Each measurement outcome that can be observed on system is represented by the function being its corresponding probability for each pure state ; and we denote by the complete set of OPFs of system .

If instead of a single outcome we want to specify a full measurement with, say, outcomes, we provide the OPFs corresponding to each outcome; which must satisfy the normalization condition

(6)

for all states .

It is important to note that this mathematical description of measurements is independent of the underlying interpretation of probability: all we are assuming is that there exist experiments which yield definite outcomes (possibly relative to a given agent who uses this formalism), and that it makes sense to assign probabilities to these outcomes. For example, we could interpret them as Bayesian probabilities of a physicist who bets on future outcomes of experiments; or as limiting frequencies of a large number of repetitions of the same experiment, approximating empirical data. Whenever we have an experiment of that kind, the corresponding probabilities (whatever they mean) will be determined by a collection of OPFs.

The completeness of the set of OPFs implies the following three properties:

is closed under taking mixtures. Suppose that the random variable with probability determines which 2-outcome measurement we implement, and later on we forget the value of . Then the probability of outcome 1 for this “averaged” measurement is

(7)

which must be a valid OPF. Therefore, mixtures of OPFs are OPFs.

is closed under composition with unitaries. We can always perform a transformation before a measurement , effectively implementing the measurement

(8)

which then must be a valid OPF. Note that here we are not saying that all unitaries can be physically implemented, but only that the formalism must in principle include them.

is closed under systems composition. Since is complete, it also includes the measurements that appear in the description of as part of the larger system , for any background system . Formally, for each background state and global OPF there is local OPF which represents the same measurement outcome

(9)

for all .

Next we consider local measurements in multipartite systems. In order to do so, it is useful to recall that the observer always has the option of describing a systems as part of a larger system , without this affecting the predictions of the theory. In order to do so, the observer needs to know how to represent the OPFs of the small system as OPFs of the larger system . This information is contained in the star product, defined in what follows.

Definition (-product). Any pair of local OPFs, and , is represented as a global OPF via the star product , which satisfies

(10)

for all and . This product must be defined for any pair of (complex and separable) Hilbert spaces and .

In other words, the -product represents bi-local measurements, which in QM are represented with the tensor-product in the space of Hermitian matrices.

Since the option of describing system as part of a larger system is a subjective choice that must not affect the predictions of the theory, the embedding of into provided by the -product must preserve the structure of . This includes the mixing (convex) structure

(11)

as well as the action

(12)

And likewise for the other party . The -product must also preserve probability, in the sense that if and are full measurements satisfying the normalization condition (6) then we must have

(13)

for all rays of .

Pushing the same philosophy further, the observer has the option of describing the tripartite system as the bipartite system or the bipartite system , without this affecting the probabilities predicted by the theory. This translates to the -product being associative

(14)

That is, the probability of outcome is independent of how we choose to partition the global system into subsystems.

ii.4 The measurement theorem

Before stating the main result of this work, we specify what should be the content of any alternative measurement postulate, and state an operationally-meaningful assumption that is necessary to prove our theorem.

Definition (alternative measurement postulate). This is a family of OPF sets and equipped with a -product satisfying conditions (7-14).

In addition to the above, a measurement postulate could provide restrictions on which OPFs can be part of the same measurement (beyond the normalization condition). However, such rules would not affect our results.

Assumption (possibility of state estimation). Each finite-dimensional system has a finite list of outcomes such that knowing their value on any ensemble allows to calculate any other OPF on the ensemble .

It is important to emphasize that need not be outcomes of the same measurement; and also, this list need not be unique. For example, in the case of QM, we can specify the state of a spin- particle with the probabilities of outcome “up” in any three linearly independent directions. Also in QM, we have ; but here we are not assuming any particular relation between and . Now it is time to state the main result of this work, which essentially tells us that the only possible measurement postulates are the quantum ones.

Theorem (measurement). The only family of OPF sets and equipped with a -product satisfying the “possibility of state estimation” assumption and conditions (7-14), has OPFs and -product of the form

(15)
(16)

for all and , where the -operator satisfies , and analogously for .

Section IV provides a summary of the ideas and techniques used in the proof of this theorem. Full detail can be found in the appendices.

ii.5 The post-measurement state-update rule

At first sight, the above theorem says nothing about the post-measurement state-update rule. But actually, it is well-known DaviesLewis () that the only possible state-update rule that is compatible with the probability rule implied by the theorem (113-114) is the one stated above in postulate “post-measurement state-update rule”. We include a self-contained proof of the above in the appendices.

Iii Discussion

iii.1 Non-quantum measurement postulate violating associativity

In this section we present an example of alternative measurement postulate, which shows that it is possible to bypass the measurement theorem if we give up the associativity condition (14). It also illustrates how a different choice of measurement postulate produces a different set of mixed states.

Definition (non-quantum measurement postulate). An -outcome measurement on is characterized by Hermitian operators acting on and satisfying and

(17)

where is the projector onto the symmetric subspace of . The probability of outcome on the (normalized) state is given by

(18)

and the -product of two OPFs and of the form (18) is defined as

for any normalized .

This alternative theory violates the principles of “local tomographic” Hardy_quantum_2001 () and “purification” Chiribella_probabilistic_2010 (). This and other exotic properties of this theory are analyzed in detail in our previous work Galley_classification_2017 (); Galley_impossibility_2018 (). Also, the validity of marginal and conditional states imposes additional constraints on the matrices which are also worked out in Galley_impossibility_2018 (). It is easy to check that the above definition satisfies conditions (7-13) and violates associativity (14). Therefore, this provides a perfectly valid toy theory of systems that encompass either one or two components, but not more.

As we have mentioned in Section II.2, the structure of the mixed states depends on the measurement postulate. Here, the mixed state corresponding to ensemble is

(19)

Another non-quantum property of this toy theory is that the uniform ensembles corresponding to two different orthonormal bases, and are distinguishable

(20)

iii.2 Gleason’s theorem and non-contextuality

As mentioned in the introduction, Gleason’s theorem and many other derivations of the Born rule Gleason_measures_1975 (); Cooke_elementary_1985 (); Pitowsky_infinite_1998 (); Wilce_quantum_2017 (); Saunders_derivation_2002 (); Busch_quantum_2003 (); Caves_Gleason_2004 (); Logiurato_born_2012 (); Han_Quantum_2016 () assume the structure of quantum measurements. That is, the correspondence between measurements and orthonormal bases , or more generally, positive-operator valued measures Helstrom76 (). But in addition to this, they assume that the probability of an outcome does not depend on the measurement (basis) it belongs to. Note that this type of “non-contextuality” is already part of the content of Born’s rule.

To show that this “non-contextuality” assumption is by no means necessary, we review an alternative to the Born rule, presented in Aaronson_quantum_2004 (), which does not satisfy it. In this toy theory, we also have that measurements are associated to orthonormal bases and each outcome corresponds to an element of the basis. Then, the probability of outcome on state is given by

(21)

Since this example does not meet the premises of Gleason’s theorem (the denominator depends not only on but also on the rest of basis), there is no contradiction in that it violates the conclusion.

We stress that our results, unlike previous contributions Gleason_measures_1975 (); Cooke_elementary_1985 (); Pitowsky_infinite_1998 (); Wilce_quantum_2017 (); Saunders_derivation_2002 (); Busch_quantum_2003 (); Caves_Gleason_2004 (); Logiurato_born_2012 (); Han_Quantum_2016 (), do not assume this type of non-contextuality. In particular, our OPF framework perfectly accommodates the above example (21) with .

Iv Methods

This brief section provides a bird’s eye view of the proof of the measurement theorem. The argument starts by embedding the OPF set into a complex vector space so that physical mixtures (7) can be represented by certain linear combinations. Second, the “possibility of state estimation” assumption implies that, whenever is finite, this embedding vector space is finite-dimensional. This translates the action (8) on the set to a linear representation; and once in the land of representations we have a good map of the territory.

Third, the fact that the argument of the functions in is a ray (not a vector) imposes a strong restriction to the above-mentioned representation. All these restricted representations were classified by some of the authors in Galley_classification_2017 (). This amounts to a classification of all alternatives to the measurement postulate for single systems, that is, when the consistency constraints related to composite systems (9-14) are ignored. The next steps take composition into account.

Fourth, “closedness under system composition” (9) implies that all OPFs are of the form

(22)

where is a fixed positive integer. Recall that the case is QM and the case has been studied in Section III.1. In the final step, the representation theory of the unitary group is exploited to prove that, whenever , it is impossible to define a star product of functions (22) satisfying associativity (14). This implies that only the quantum case () fulfills all the required constraints (7-14).

V Conclusions

It may seem that conditions (7-14) are a lot of assumptions to claim that we derive the measurement postulates from the non-measurement ones. But, when taking an operational perspective, all these conditions are uncontroversial background assumptions, like the rules of probability calculus or the axioms of the real numbers, which most physics textbooks take for granted. In other words, specifying what do we mean by “measurement” is in a different category than stating that measurements are characterized by operators acting on a Hilbert space.

It is rather remarkable that none of the three measurement postulates (structure, probabilities and state-update) can be modified without having to redesign the whole theory. In particular, that the probability rule is deeply ingrained in the main structures of the theory. This fact shows that one need not appeal to any supplementary principles beyond operational primitives to derive the Born rule, nor do we need to make any assumptions about the structure of measurements, unlike previous work Saunders_derivation_2002 (); Aaronson_quantum_2004 (); Zurek_probabilities_2005 (); Logiurato_born_2012 (); Wallace_how_2010 (); Han_Quantum_2016 (). Finally, having cleared up unnecessary postulates in the formulation of quantum mechanics, we find ourselves closer to its core message.

Vi Acknowledgements

We are grateful to Jonathan Barrett and Robin Lorenz for discussions about the toy theory of Section III.1, which was independently studied by them. LM acknowledges financial support by the Engineering and Physical Sciences Research Council [grant number EP/R012393/1]. TG acknowledges support by the Engineering and Physical Sciences Research Council [grant number EP/L015242/1]. This research was supported in part by Perimeter Institute for Theoretical Physics; research at Perimeter Institute is supported by the Government of Canada through Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science. This publication was made possible through the support of a grant from the John Templeton Foundation; the opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation.

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Appendix A Alternative measurement postulates for single systems

In this section we classify all alternative measurement postulates for the case of finite-dimensional single systems, that is, when the constraints associated to the composition of systems (the star product) are ignored. These results build up on the previous work Galley_classification_2017 () by two of us.

In this section we only consider finite-dimensional Hilbert spaces with a positive integer. In this case we have ; and since has a trivial action on rays, we only consider . Later on, when addressing the infinite-dimensional case , we will work with , since the condition is not well-defined when .

a.1 Structure of measurements and mixed states

Definition 1.

is a set of functions which is closed under composition with unitaries

(23)

closed under convex combinations

(24)

and that contains the unit and the zero functions, respectively and , for all .

The unit function represents an outcome that happens with probability one. For example, such unit-probability outcome can be the event corresponding to all outcomes of the measurement , which by normalization satisfy

(25)

Analogously, the zero function represents a formal outcome that has zero probability irrespectively of the state.

For what comes below, it is convenient to consider the set as embedded in the complex vector space generated by itself. The fact that the group action (23) commutes with the mixing operation (24)

(26)

can be extended to arbitrary linear combinations in , providing a complex, linear representation of . While only the elements of are outcome probability functions (OPFs), any element of can be interpreted as the expectation value of an observable with complex outcome labels, in analogy to the algebra of observables in QM. While in QM the space has dimension , here we leave the dimension unconstrained. However, in what follows, we show that the “possibility of state estimation” assumption implies that the linear space is finite-dimensional. But before this, we recall that the probability of outcome on an ensemble is given by

(27)

The above follows from the rules of probability calculus.

Lemma 2.

Suppose that the values of the outcomes on any given ensemble determine the value of any other outcome on that ensemble . Then the functions span the linear space .

In other words, knowing the numbers , …, allows us to determine the number without knowing the ensemble . That is, the latter is some function of the former.

Proof.

Once the OPFs are given we can define the convex set

(28)

Next we note that, the fact that the values of determine the value of on any ensemble means that there is a function such that

(29)

Since the above equality holds for all ensembles, it also holds for the pure states

(30)

for all . Hence we have

(31)

It follows that (31) also holds true if every appearance of is replaced by some ensemble , where labels the possible states and their probabilities. Since can take all values in by choosing the states and probabilities in a suitable way, this shows that is convex on the full set . This implies that can be affinely extended to all of , i.e. there is an affine function which coincides with the previous function inside the convex set . Affinity means

(32)

for any with and , but the coefficients are not necessarily positive. Any affine function can be written as

(33)

where and . Therefore we can write

(34)

That is, any OPF can be written as an -linear combination of , as in (34). Since every element of is a complex-linear combination of such OPFs, every such element must thus be a complex-linear combination of . ∎

Corollary 3.

The “possibility of state estimation” assumption implies that, for all finite , the linear space is finite-dimensional.

In what follows, we introduce a representation of pure states that is linearly related to outcome probabilities. Because of this, this new representation encodes the equivalence relation between ensembles, and hence, the structure of mixed states arising from alternative measurement postulates.

Definition 4.

For each pure state we define the linear form as

(35)

with the natural action

(36)

This allows to write the probability of outcome on ensemble ,

(37)

in terms of the mixed state

(38)

Hence, two different ensembles corresponding to the same mixed state (38) are indistinguishable. The next lemma gives us important information about the group representation .

Lemma 5.

The action (23) on decomposes as

(39)

where are the irreducible representations defined in Lemma 7. The finite set contains zero and some positive integers (with no repetitions).

Before proving the above we mention that the quantum case is , and in Section III.A of the main text the (non-quantum) case is analyzed. Also, we have to mention that in this work we follow the notation of Fulton91 (), where the group representations are labelled by the subspace they act on.

Proof.

In this proof we establish the following four facts in the same order: (i) decomposes into a finite sum of finite-dimensional irreducible representations (irreps), (ii) these irreps are of the type , (iii) there are no repetitions, (iv) is always included.

Fact (i). Lemma 2 shows that the representation is finite-dimensional. And these can always be decomposed into finite-dimensional irreps Fulton91 (). Also, we know that each finite-dimensional irrep of corresponds to a -row Young diagram . Hence we write

(40)

where repeated values of can happen.

Fact (ii). The fact that different elements of are different functions implies that the form (Definition 4) has support in each sub-space of (40). Indeed, if had no support in the sub-space , then any of the elements would correspond to the same function.

Denote by the subgroup of unitaries that leave the state invariant , and note that

(41)

for any . According to (36), the action of on a subspace of (40) leaves the projection of onto the subspace invariant. This implies that contains an invariant vector under the action of . But Lemma 1 from Galley_classification_2017 () tells us that the only irreps with an -invariant vector are for Hence, all irreps in (40) are of the form . At this point it is worth mentioning that the irreps are real.

In addition, Lemma 1 from Galley_classification_2017 () tells us that in each irrep the -invariant subspace has dimension one. Which fixes the projection of the linear form onto each subspace of (39) up to a proportionality factor. Changing these proportionality factors modifies the structure of by the corresponding inverse linear transformation; but the space remains identical.

To prove Fact (iii), suppose that there are two repeated irreps in (39). We can write the isomorphism

(42)

with the understanding that the -action in is trivial. Next, we invoke the above-shown unicity of to see that the projection of onto the subspace is of the form

(43)

where is a linear form that depends on the above-mentioned proportionality factors. Given it possible to find two different vectors such that . Then, taking any we can construct two different elements of corresponding to the same function

(44)

for all , which is a non-sense.

To establish Fact (iv), we recall that the unit function is always included (Definition 1). Since is invariant under the action (23) the trivial irrep must be included in the decomposition (39). Hence . ∎

a.2 The representations and

In this subsection we introduce two families of representations that allow to construct all alternative measurement postulates for single systems by using (39). For this, we recall that the projector onto the symmetric subspace of can be written as the average of all permutations over objects

(45)

where acts by permuting the factor spaces of . Next we define an representation that sometimes is named .

Definition 6.

Let be the linear space of complex matrices acting on whose support is contained in the symmetric subspace

(46)

And let the linear action of on be

(47)
Lemma 7.

The decomposition of into irreducible representations is

(48)

where the subspace is generated by applying the group action (47) to the element

(49)

where are any orthogonal pair. Also, the representation isomorphisms

(50)

hold for all .

Isomorphism (50) allows us to use the shorthand notation . Also, note that is the trivial irrep, generated by the element ; and is the adjoint (quantum) irrep.

Proof.

In order to obtain the decomposition (48) it is useful to define the trace map

(51)
(52)

where denotes the trace over the th factor in . Note that, by symmetry, this partial trace is independent of the choice of factor: . From now on, wherever is clear, we leave the dependence on implicit.

Because the map (51) commutes with the action,

(53)

Schur’s Lemma tells us that its kernel must be a subrepresentation of , which we denote by . It is proven in Lemma 23 that this representation is irreducible. Also, it is straightforward to check that the matrix defined in (49) is in the kernel of the map (51), that is

(54)

Combining the above with irreducibility we see that the subspace is generated by the action of the group on the single element .

Because the map (51) is surjective, the orthogonal complement of is a representation isomorphic to , which in turn contains the irreducible representation in the kernel of the trace map . Then, using Schur’s Lemma again, there must be a subrepresentation that is isomorphic to , which proves isomorphism (50). Proceeding inductively, we obtain the full decomposition (48).

To conclude the proof of Lemma 7 we need to show that . By noting that

(55)

is non-zero when , we can proceed inductively to arrive at

(56)

which is the case analyzed above (54). The isomorphisms (50) provided by Schur’s Lemma conclude the proof. ∎

a.3 The form in and

In this section we introduce a simple choice for the linear form of Definition 4, for the cases and . As already mentioned, this form encodes the structure of the set of mixed states.

Lemma 8.

The linear form defined by

(57)

is invariant under all stabilizer unitaries

(58)

and has support in all irreps .

Proof.

It is straightforward to check that the form (57) satisfies (58). To see that (57) has support in each irrep , we observe that, for each , there is a pure state such that

(59)

where is defined in (49). ∎

As mentioned above, these two constrains fix up to an irrelevant proportionality factor in each irrep. To obtain in the case we proceed in the following maner. Since is a subrepresentation of we can take (57) and perform the orthogonal projection onto the subspace , defined via (49) or via the kernel fo the map (51).

Appendix B Multipartite systems

In this section we describe and impose the consistency constraints associated to composite systems and the star product.

b.1 Closedness under system composition

We require that any family of OPF sets and must be closed under system composition. This means that the complete set of measurements of a system also includes the measurements that appear in the description of as part of a larger system .

Definition 9 (Closedness under system composition).

If is the OPF set of then the OPF set of is the following collection of functions

(60)
(61)

for all and a fixed , and all .

Note that the closure of under implies that the set defined via (60-61) does not depend on the choice of . Also, it is straightforward to check that the OPF set so defined satisfies all the requirements of Definition 1.

b.2

In the finite-dimensional case, closedness under system composition (Definition 9) implies the following strong fact. For any set of measurements of a bipartite system , the measurement spaces of the subsystems are for and for , with the same . In addition, using the fact that any pair of systems can be jointly described as a bipartite system, we conclude that all finite-dimensional systems must have OPF space (with the same value for ).

Lemma 10.

For any pair of positive integers , let be the OPF set of with decomposition (Lemma 5)

(62)

Define as the set of functions

(63)
(64)

for all and a fixed . Then we have the -representation isomorphism

(65)

where .

Note that, if we define by exchanging the role of the subsystems in (64), then we obtain the -representation isomorphism with the same value for as in (65). Using the fact that any pair of systems can be jointly described as a bipartite system, we arrive at the following.

Corollary 11.

Closedness under system composition (Definition 9) implies that all finite-dimensional Hilbert spaces have OPF space with the same .

Proof of Lemma 10.

In order to establish the isomorphism (65) we analyze how the functions (64) transform under the subgroup . First, we do this in the case where has finite cardinality, so that the decomposition (39) of has a largest irrep . We further split this analysis into the case where the function in (64) belongs to the subspace , and the general case.

Using the characterization of as the kernel of the map (51) we can say the following. For each there is a matrix such that and

(66)

for all . (Note that, in order to improve clarity, we re-arranged the order of the tensor factors.) The matrices

(67)

are contained in the subspace

(68)

where the isomorphisms are of representations. Even more, using the full support conditions (59) in each tensor factor, we conclude that the matrices generate the whole space (68).

Next we analyze the action on the function (66), which is the action on the intersection between the subspaces and (68). This intersection can be characterized by writing the trace as

(69)

where is the trace on the th factor of , and is the trace on the th factor of . The above identity implies that if or then . Therefore, the above-mentioned intersection contains all irreps and for . This implies that the action on the space of functions (66) with decomposes into the irreps , with possible repetitions.

In the general case , the addition of all subspaces with does not add any new irrep to the list . Although it may increase the repetitions.

Finally, we establish the desired isomorphism (65) by recalling Lemma 5. This tells us that any OPF set, like the defined through (64), has no repeated irreps. ∎

b.3 The star product

In this subsection we introduce the star product, which contains the information of which measurements of a composite system are local.

Definition 12.

The star product is a map defined on any pair of OPF sets , with the following properties:

  • preserves the local structure

    (70)
  • preserves probability

    <