Impossibility of mixed-state purification in any alternative to the Born Rule Thomas D. Galley1, ∗ and Lluis Masanes1
arXiv:1801.06414v2 [quant-ph] 21 Feb 2018
1
Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom (Dated: February 22, 2018)
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 manyworlds 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.
I.
INTRODUCTION
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 [1]. 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 [2], the decoherence program [3] and the Many-Worlds Interpretation of quantum theory [4, 5]. 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 [6] 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 [7, 8] and Local Tomography [9, 10]. We also present a simple alternative measurement postulate (a toy the-
∗
[email protected]
ory) 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 [11–13]. 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 [6]. 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 nonclassical 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.
A.
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 Cd corresponds to a type of system with pure states being the rays ψ of Cd . Postulate (Quantum Transformations). The reversible transformations on the pure states Cd are ψ 7→ U ψ for all U ∈ SU(d).
2 Postulate (Quantum Composition). The joint pure states of systems CdA and CdB are the rays of CdA ⊗CdB ' Cd A d B . B.
C5. Consider measurements on system CdA with the help of an ancilla CdB . For any ancillary state φB ∈ CdB and any OPF in the composite FAB ∈ FdA dB there exists an OPF on the system FA0 ∈ FdA such that
Measurement postulates
FA0 (ψA ) = FAB (ψA ⊗ φB ) Before presenting the generalized measurement postulate we need to introduce the notion of outcome probability function, or OPF. For each measurement outcome x of system Cd there is a function F (x) that assigns to each ray ψ in Cd the probability F (x) (ψ) ∈ [0, 1] for the occurrence of outcome x. Any such function F 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 u, satisfying u(ψ) = 1 for all ψ. A k-outcome (1) measurement is a list of k OPFs (F . . . , F (k) ) satisP , (i) fying the normalization condition i F = u. As an example, the OPFs of quantum theory are the functions F (ψ) = tr(Fˆ |ψihψ|) ,
(1)
for all Hermitian matrices Fˆ satisfying 0 ≤ Fˆ ≤ I. This ˆ = I. (Here and in the rest of the paper we implies that u assume that kets |ψi are normalized.) Postulate (Alternative Measurements). Every type of system Cd has a set of OPFs Fd with a bilinear associative product ? : FdA × FdB → FdA dB satisfying the following consistency constraints: C1. For every F ∈ Fd and U ∈ SU(d) there is F 0 ∈ Fd such that F 0 (ψ) = F (U ψ) for all ψ ∈ Cd . That is, the composition of a unitary and a measurement can be globally considered a measurement. d
C2. For any pair of different rays ψ 6= φ in C there is F ∈ Fd such that F (ψ) 6= F (φ). That is, different pure states must be operationally distinguishable. C3. The ?-product satisfies uA ? uB = uAB and (FA ? FB )(ψA ⊗ φB ) = FA (ψA )FB (φB ) ,
(2)
for all FA ∈ FdA , FB ∈ FdB , ψA ∈ CdA , φB ∈ CdB . That is, tensor-product states ψA ⊗ φB contain no correlations. dA
dB
C4. For each φAB ∈ C ⊗ C and FB ∈ FdB there is an ensemble (ψAi , pi ) in CdA such that (FA ? FB )(φAB ) X = pi FA (ψAi ) , (uA ? FB )(φAB ) i
(4)
for all ψA . The derivation of these consistency constraints from operational principles is provided in Appendix B. Continuing with the example of quantum theory (1), the ?product in this case is (FA ? FB )(ψAB ) = tr(FˆA ⊗ FˆB |ψAB ihψAB |) .
(5)
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 Fˆ are allowed. In this work, when we refer to “all alternative measurement postulates” we do not include these trivial modifications.
C.
Mixed states and the Finiteness Principle
A source of systems that prepares state ψi ∈ Cd with probability pi is said to prepare the ensemble (ψi , pi ). Two ensembles (ψi , pi ) and (φj , qj ) are equivalent if they are indistinguishable X X pi F (ψi ) = qj F (φj ) (6) i
j
for all measurements F ∈ Fd . Note that distinguishability is relative to the postulated set of OPFs Fd . A mixed state ω is an equivalence class of indistinguishable ensembles, and hence, the structure of mixed states is also relative to Fd . To evaluate an OPF F on a mixed state ω we can take any ensemble (ψi , pi ) of the equivalence class ω and compute X F (ω) = pi F (ψi ) . (7) i
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.
(3)
for all FA ∈ FdA . That is, the reduced state on A conditioned on outcome FB on B (and renormalized) is a valid mixed state of A. In the next sub-section we fully articulate the notions of ensemble and mixed state.
Principle (Finiteness). Each mixed state of a finitedimensional system (Cd , Fd ) can be characterized by a finite number Kd of parameters. Recall that in quantum theory we have Kd = d2 −1. And in general, the distinguishability of all rays in Cd implies Kd ≥ 2d − 2. These Kd parameters can be chosen to be
3 a fix set of “fiducial” OPFs F1 , . . . , FKd ∈ Fd , which can be used to represent any mixed state ω as F1 (ω) F2 (ω) (8) ω ¯= . .. . FKd (ω)
case d ≥ 3: there is a non-constant F ∈ Fd . case d = 2: there is F ∈ F2 such that Fˆ has support on a sub-representation of the SU(2) action (12) with odd angular momentum.
III.
The fact that OPFs are probabilities implies that any OPF F ∈ Fd is a linear function of the fiducial OPFs X F = ci Fi . (9) i
In other words, the fiducial OPFs F1 , . . . , FKd ∈ Fd constitute a basis of the real vector space spanned by Fd . Using the consistency constraint C1, we define the SU(d) action X ¯ i,i0 (U ) Fi0 Γ (10) Fi =
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 [8, 14–16]. Remarkably, this principles are violated by all alternative measurement postulates.
A.
The Purification Principle
i0
on the vector space spanned by Fd . This associates to system (Cd , Fd ) a Kd -dimensional representation of the group SU(d). This, together with the consistency constraints, implies that only certain values of Kd are allowed. For example K2 = 3, 7, 8, 10, 11, 12, 14 . . . D.
Measurement postulates for single systems
In this subsection we review some of the results obtained in [6]. These provide the complete classification of all sets Fd 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 Fd . This is studied in Section III. Theorem (Characterization). If Fd satisfies the Finiteness Principle and C1 then there is a positive integer n and a map F 7→ Fˆ from Fd to the set of dn ×dn Hermitian matrices such that F (ψ) = tr Fˆ |ψihψ|⊗n (11) for all normalized vectors ψ ∈ Cd . Note that there are many different sets Fd with the same n. In particular, since the SU(d) action |ψihψ|⊗n 7→ U ⊗n |ψihψ|⊗n U ⊗n†
(12)
is reducible, the Hermitian matrices Fˆ can have support on the different irreducible sub-representations of (12), generating sets Fd with very different physical properties. All these possibilities are analyzed in [6]. Theorem (Faithfulness). If Fd satisfies the Finiteness Principle, C1 and C2, then
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 [17]. Principle (Purification). For each ensemble (ψi , pi ) in CdA there exists a pure state φAB in CdA ⊗ CdB for some dB satisfying X (FA ? uB )(φAB ) = pi FA (ψi ) , (13) i
for all FA ∈ FdA . Note that the original version of the Purification Principle introduced in [7] additionally demands that the purification state φAB is unique up to a unitary transformation on CdB . Also note that the following theorem does not require the Finiteness Principle. Theorem (No Purification). All alternative measurements postulates Fd 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 [3– 5, 18].
4 B.
The Local Tomography Principle
This principle has been widely used in reconstructions of quantum theory and the formulation of alternative toy theories [8, 14, 15, 29–31]. One of the reasons is that it endows the set of mixed states with a tensor-product structure [10]. This principle establishes that any bipartite state is characterized by the correlations between local measurements. That is, two different mixed states 0 ωAB 6= ωAB on CdA ⊗ CdB must provide different outcome probabilities 0 (FA ? FB )(ωAB ) 6= (FA ? FB )(ωAB )
(14)
for some local measurements FA ∈ FdA , FB ∈ FdB . Using the notation introduced in (9) we can formulate this principle as follows. Principle (Local Tomography). If {Fa } is a basis of FdA and {Fb } is a basis of FdB then {Fa ? Fb } is a basis of FdA dB , where a = 1, . . . , KdA and b = 1, . . . , KdB . 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 C3 ⊗ C3 = C9 . Theorem (No Local Tomography). All alternative measurements F3 and F9 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 SU(9) which are sub-representations of the action (12) have a sub-representation 13 ⊗ 13 when restricted to the subgroup SU(3) × SU(3). Here 13 denotes the trivial representation of SU(3). C.
Representation theoretic constraints imposed by the compositional structure
unique trivial representation (corresponding to the normalisation degree of freedom). The subspace acted on ¯ dA Γ ¯ dB is called the locally tomographic subspace. by Γ The above criterion is necessary, but not sufficient for composition. Theorem (Existence of locally tomographic subspace). Any alternative measurement structure F9 satisfying C1, C2 and the Finiteness Principle is such that the associ¯ 9 has a locally tomographic subrepated representation Γ resentation when restricted to SU(3) × SU(3). This theorem is proven in Appendix D. IV.
A TOY THEORY
In this section we present a simple alternative measurement postulate (Fd , ?) 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 dA and dB with sets of OPFs FdLA and FdLB . The composite (global) system has a set of OPFs FdGA dB . Definition (Local effects FdL ). To each d2 ×d2 Hermitian matrix Fˆ satisfying • 0 ≤ Fˆ ≤ S, P • Fˆ = i αi |φi ihφi |⊗2 for some |φi i ∈ Cd and αi > 0, P • S − Fˆ = i βi |ϕi ihϕi |⊗2 for some |ϕi i ∈ Cd and βi > 0, there corresponds the OPF
As shown in [6] all sets of OPFs Fd are associated to a representation of SU(d). The dynamical structure of quantum theory imposes some requirements on these representations; namely all representations corresponding to sets of OPFs Fd must have a trivial representation when restricted to S(U(d − 1) × U(1)). In Appendix B it is shown that the compositional structure implies that a system Cd with a set of OPFs Fd (where d = dA dB ) can be considered as a composite of two systems CdA and CdB ¯ d is such that: only if the associated representation Γ ¯d ¯ dA Γ ¯ dB ⊕ other terms . Γ |SU(dA )×SU(dB ) = Γ
(15)
¯d ¯d Here Γ |SU(dA )×SU(dB ) is the restriction of Γ to SU(dA ) × ¯d, Γ ¯ dA and Γ ¯ dB contain a SU(dB ). The representations Γ
F (ψ) = tr(Fˆ |ψihψ|⊗2 ) ,
(16)
where S is the projector onto the symmetric subspace of ˆ = S. Cd ⊗ Cd . The unit OPF corresponds to u That is, both matrices, Fˆ and S − Fˆ , have to be notnecessarily-normalized mixtures of symmetric product states. Example (Canonical measurement for d prime). For the case where d is prime there exists a canonical measurement which can be constructed as follows. Consider the (d+1) mutually unbiased bases (MUBs): {|φji i}di=1 where j runs from 1 to d + 1 [20]. Then we can associate an OPF to each Hermitian matrix 12 |φji ihφji |⊗2 . Since the
5 basis elements of these MUBs form a complex projective 2-design [21], by the definition of 2-design [22], we have the normalization constrain: 1 X j j ⊗2 (17) |φ ihφ | = S , 2 i,j i i and hence the set of OPFs forms a measurement. Definition (? product). For any pair of OPFs FA ∈ FdLA and FB ∈ FdLB the Hermitian matrix corresponding to their product FA ? FB ∈ FdGA dB is trFˆA trFˆB ˆ ˆ F\ AA ⊗ AB , (18) A ? F B = FA ⊗ FB + trSA trSB where SA and AA are the projectors onto the symmetric and anti-symmetric subspaces of CdA ⊗ CdA , and analogously for SB and AB . This product is clearly bilinear and, by using the identity SAB = SA ⊗ SB + AA ⊗ AB , we can check that uA ? uA = uAB . \ PWe observe⊗2that not all effects FA ? FB are of the form α |φ ihφ | . Hence the set of effects on the joint sysi AB i i i tem is not FdLA dB , but has to be extended to FdGA dB to include these joint product effects. Definition (Global effects
FdGA dB ).
FdGA dB
The set should L \ include all product OPFs FA ? FB , all OPFs FdA dB of CdA dB understood as a single system, and their convex combinations. The identity SAB = SA ⊗ SB + AA ⊗ AB perfectly shows that the vector space FdGA dB is larger than the tensor product of the vector spaces FdLA and FdLB , by the extra term AA ⊗ AB . This implies that this toy theory violates the Local-Tomography Principle. The joint probability of outcomes FA and FB on the entangled state ψAB ∈ CdA ⊗ CdB can be written as
FIG. 1. Projection of C2 toy model state space. In this projection Pthe coloured points (blue and purple) are states of the form i pi |ψi ihψi |⊗2 . The blue points are projections of reduced states of a larger system obtained using formula (20). The left hand corner corresponds to the state |0ih0|⊗2 and the right hand corner to the state |1ih1|⊗2 . All pure states are projected onto the curved boundary. This figure shows that there are local mixed states (in purple) which are not reduced states (in blue).
V.
DISCUSSION
A.
No-signalling
Multiple proofs have been put forward which claim to show that violations of the Born rule lead to signalling [11–13]. However the authors often only consider modifications of the Born rule of a specific type. In [11, 12] the authors only consider modifications of the Born rule of the following form: |hk|ψi|n , 0 n k0 |hk |ψi|
p(k|ψ) = P
(21)
P where |ψi = k αk |ki. Modified Born rules of this type i are such that the measurements described have the same h ˆB ˆA ⊗2 trF trF ˆ ˆ outcomes as the projective measurements of standard (FA ?FB )(ψAB ) = tr FA ⊗ FB + trSA AA ⊗ trSB AB |ψAB ihψAB | quantum theory. That is to say the outcomes of meaWhen we only consider sub-system A outcome probabilsurements are still associated to the basis elements |ii. ities are given by Another feature of these modifications is that probabilih i ˆA ⊗2 F ties p(i|ψ) are functions of a single amplitude αi (disre(FA ? uB )(ψAB ) = tr FˆA ⊗ SB + tr A ⊗ A |ψ ihψ | A B AB AB trSA garding the normalisation, which can be removed using a i h (19) = trA FˆA ω ¯A , n-norm). Modifications of this form are very restricted. By modifying all the measurement postulates of quantum where the reduced state must necessarily be theory, we can create toy models like the one introduced SA ⊗2 ⊗2 which are non-signalling. This shows that by modifying ω ¯ A = trB SB |ψAB ihψAB | + . trAB AA ⊗AB |ψAB ihψAB | trSA the Born rule in a more general manner one can avoid (20) issues of signalling. All these reductions ω ¯ A of pure bipartite states ψAB are ⊗2 contained in the convex hull of |φA ihφA | , as required by In the case of the toy model it is immediate to see the consistency constraint C4. However, not all mixtures that it is consistent with no-signalling. The condition of |φA ihφA |⊗2 can be written as one such reduction (20). of no-signalling is equivalent to the existence of a well That is, the Purification Postulate is violated. This phedefined state-space for the subsystem (i.e. independent nomenon is graphically shown in Figure 1. of action on the other subsystem). We see then that noThis toy model is restricted, in that not all mathematisignalling is just a consequence of there existing a well cally allowed effects on the local state spaces are allowed defined reduced state and arguably can be considered effects. It also violates the principle of perfect distinan intrinsic feature of the operational framework, rather guishability in that all the effects are noisy. than a supplementary principle.
6 B.
The quantum measurement postulates are the most non-classical
One consequence of the classification in [6] 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 nonclassical, since they give rise to the state spaces with the most indistinguishable mixtures [23]. 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.
C.
Toy model
The toy model can be obtained by restricting the states and measurements of two pairs of quantum systems (CdA )⊗2 and (CdB )⊗2 . 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 |ψihψ|⊗n for n > 2, however showing the consistency of the reduced state spaces and joint effects is more complex.
D.
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.
VI.
CONCLUSION A.
Summary
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.
B.
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.
Theories which decohere to quantum theory
A recent result [24] 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.
E.
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.
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 [10, 19, 25] or the purification principle [7, 26–28]. 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
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
7 of quantum theory and are not simulable with a finite number of quantum systems.
space.
VII.
ACKNOWLEDGEMENTS
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
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 1.
Single system
As stated in the main section the allowed sets of OPFs Fd 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 [8]. 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.
O
FIG. 4. Diagrammatic representation of a single system 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. Definition (Experiment). 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 p(O|P) ,
(A1)
for all measurement outcomes O and all preparations P in the experiment. P
O
FIG. 5. All experiments can be represented as a preparation and a measurement.
P
FIG. 2. Diagrammatic representation of a single system preparation procedure
Definition (Transformation procedure). Any procedure which inputs one or more systems and outputs one or more systems is a transformation.
In this approach a system is an abstraction symbolised by the wire between the preparation procedure and the measurement procedure. A system A can be represented as: A FIG. 6. Diagrammatic representation of a system.
T
FIG. 3. Diagrammatic representation of a single system transformation procedure.
Definition (Measurement procedure). Any procedure which inputs on or more systems and has no output is a measurement procedure.
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:
9 B
PA
OA
PB
OB
A FIG. 7. Diagrammatic representation of a pair of systems.
These can only be considered as representing two distinct systems A and B 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 A and a measurement on subsystem B uniquely specify a measurement on AB independent of the temporal ordering [30]. 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: OA
Operational Implication (Steering as preparation). Operationally Alice can make a preparation of system A by making a preparation of AB and getting Bob to make a measurement on system B.
PAB OB
OA =
PAB
FIG. 10. Separable two system experiment.
FIG. 11. Preparation by steering.
PAB
OB
OB
FIG. 8. Definition of a composite system in diagrammatic form.
The most general form of an experiment with two systems is:
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 B and jointly measuring A and B is a measurement procedure on A.
A OAB PAB
OAB B
PB
FIG. 12. Measurement using ancilla. FIG. 9. Generic two system experiment.
which can naturally be viewed as an experiment on a single system AB. Definition (Separable procedures). Two independent preparations PA and PB which are independently measured with outcomes OA and OB are such that: p(OA , OB |PA , PB ) = p(OA |PA )p(OB |PB ) , In this case the joint procedures PA , PB and OA , OB 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 B is a preparation of a state A.
Operational Implication (Measuring with an ancilla). A valid measurement for Alice consists in adjoining her system to an ancillary system B 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.
10 Operational Assumption (Associativity of composition). The systems (AB)C and A(BC) are the same.
4.
Consistency constraints a.
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 A, B, C.... For any partitioning of the experiment which creates a preparation of system A, all regroupings of systems B, C, ... are equivalent. This is a preparation by steering of system A conditional on a measurement on systems B, C, ... 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.
3.
C1.
Consistency constraint C1. follows directly from the fact that the composition of a transformation and a measurement is a measurement.
b.
C2.
Definition (State). 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 Cd (as required by the first postulate).
OPFs and the ? product c.
We assume the finiteness principle holds, and that for a set of OPFs F there exists a finite linearly generating set {Fi }K i=1 . That is to say: F =
X
ci Fi .
(A2)
i
Consider a composite system AB. By the definition of a composite system above, for any OPF FA on A and FB on B there exists an OPF FA ?FB on AB. By the assumption that mixing is possible, the outcome {pi , FAi } is a valid outcome.
Principle (Uncorrelated pure states). Given two systems A and B independently prepared in pure states ψA and φB the joint state of the system AB is given by ψA ⊗ φB . This principle, together with the definition of independent systems implies that: (FA ? FB )(ψA ⊗ φB ) = FA (ψA )FB (φB )
d.
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: X X ( pi FAi ) ? FB = pi (FAi ? FB ) i
FA ? (
(A3)
i
X
pi FBi ) =
i
X
pi (FA ? FBi )
(A4)
i
If we further assume that it isPpossible to mix with subnormalised probabilities, i.e. i pi ≤ 1 then the ? product is bi-linear. The identity uA ? uB = uAB follows from the fact that a separable measurement is a valid measurement on AB, and hence: uAB =
X (FAi ? FBj ) = uA ? uB . i,j
(A5)
C3.
(A6)
C4.
Let us consider the steering scenario. In this case the process of both Alice and Bob making a measurement with outcome FA ? FB on a joint state φAB can be considered as a measurement with outcome FA on system A prepared in a certain manner. An arbitrary preparation of system A is given by {pi , ψAi }. Hence the steering preparation implies that for each preparation of AB and for each local measurement outcome on B there exists a state {pi , ψAi } in which system A is prepared. In the OPF formalism this means that for every φAB ∈ CdA dB and every FB ∈ FdB there exists an ensemble {pi φiA } such that (FA ? FB )(φAB ) X = pi FA (ψAi ) , (uA ? FB )(φAB ) i
(A7)
holds for all FA ∈ FdA . The normalisation occurs due to the fact that summing over the measurement outcomes FA should give unity on both sides of the expression.
11 e.
C5.
Let us consider the scenario consisting in measuring with an ancilla. In this case Alice and Bob carry out a joint measurement with outcomes FAB on a system in an uncorrelated state ψA ⊗ φB . This should correspond to a valid measurement with outcome FA0 on system A prepared in state ψA . For each FAB ∈ FdA dB and for each φB ∈ CdB there exists an FA0 ∈ FdA such that FAB (ψA ⊗ φB ) = FA0 (ψA ) ,
(A8)
dA
for all ψA ∈ C . Appendix B: OPF formalism, representation theory and local tomography
In this appendix we show that each set of OPFs Fd is associated to a representation of the group SU(d). We also show that the representations associated to locally tomographic systems with sets of OPFs FdA dB have certain features when restricted to the local subgroup SU(dA ) × SU(dB ). 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. 1.
Single systems
Lemma 1. To each set of measurement postulates Fd obeying the finiteness principle there exists a unique representation Γd of SU(d) associated to that set. Proof. We take a set of measurement postulates F, where {Fi } form a basis for F. X F (ψ) = ci Fi (ψ) , ∀ψ . (B1)
We can also consider U U 0 as a single element: X ¯ ki (U U 0 )Fk (ψ) F (U U 0 ψ) = ci Γ
¯ U 0 ) = Γ(U ¯ )Γ(U ¯ 0 ) and the map Γ ¯ : This shows that Γ(U ¯ U 7→ Γ(U ) is a representation of SU(d). ¯ d of measurement posLemma 2. The representation Γ tulates Fd contains a unique trivial subrepresentation. (i) Proof. Consider a set of OPFs } which form a meaP {F surement. The OPF u = F (i) is such that u(ψ) = 1 ∀ψ. Consider the basis {Fi } where F1 = u. We observe that u(ψ) = u(U ψ), ∀U ∀ψ. This implies that ¯ )u = u ∀U and that the representation Γ has a trivΓ(U ial 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.
2.
(F ◦ U )(ψ) = F (U ψ) =
X
ci Fi (U ψ)
(B2)
¯ j (U )Fj (ψ) . Γ i
(B3)
i
Where Fi (U ψ) = (Fi ◦ U )(ψ) =
X j
Composite systems
The measurement structure FAB contains all measurements of the form FA ? FB where FA ∈ FA and FB ∈ FB . Let {FAi } and {FBj } be bases for the two OPF spaces. Then the OPFs FAi ? FBj form a basis for the global OPFs k } [32, 33]. FA ? FB . Hence a basis for FAB is {FAi ? FBj , FAB a.
Local tomography
Lemma 3. For a locally tomographic bi-partite sys¯ dA dB the restriction of Γ ¯ dA dB to tem with representation Γ SU(dA ) × SU(dB ) is: ¯ dA dB ¯ dA Γ ¯ dB Γ |SU(dA )×SU(dB ) = Γ
i
We consider an OPF F ◦ U :
(B6)
ik
(B7)
Proof. A theory is locally tomographic if {FAi ? FBj } span FAB . It follows immediately from the bilinearity of ? and the fact that {FAi ? FBj } span FAB that the ? product for locally tomographic theories is a tensor product. Hence an arbitrary element of FAB can be written: X FAB = γij (FAi ⊗ FBj ) (B8) ij
Hence F ◦ U (ψ) =
X
¯ j (U )Fj (ψ) ci Γ i
(B4)
¯ dA dB (UAB )FAB FAB ◦ UAB = Γ
(B9)
ij
Let us consider the action of an element of SU(dA ) × SU(dB ) on an OPF in FAB .
Consider F (U U 0 ψ) =
X
=
X
¯ j (U )Fj (U 0 ψ) ci Γ i
ij
ijk
¯ j (U )Γ ¯ kj (U 0 )Fk (ψ) ci Γ i
(B5)
¯ dA dB (UA ⊗ UB )FAB = FAB ◦ (UA ⊗ UB )(ψAB ) Γ X = γij (FAi ⊗ FBj )((UA ⊗ UB )(ψAB )) . (B10) ij
12 Let us consider (FAi ⊗ FBj )((UA ⊗ UB )(ψAB )) = (FAi ⊗ FBj ) ◦ (UA ⊗ UB ) (ψAB ) =(FAi ◦ UA ) ⊗ (FBj ◦ UB )(ψAB ) ,
(B11)
where we have used: (FAi ⊗ FBj ) ◦ (UA ⊗ UB ) = (FAi ◦ UA ) ⊗ (FBj ◦ UB ) 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 (FAi ◦UA )⊗(FBj ◦UB ) is a well defined OPF on CdA dB since (FAi ◦ UA ) and (FBj ◦ UB ) are well defined OPFs on CdA and CdB respectively. X ¯ dA (UA ))ik FAk (ψA ) FAi ◦ UA (ψA ) = (Γ (B12)
k
FBj ◦ UB (ψB ) =
X
¯ dB (UB ))j FBl (ψB ) . (Γ l
(B13)
l
FAB ◦ (UA ⊗ UB )(ψAB ) X ¯ dA (UA ))i ⊗ (Γ ¯ dB (UB ))j F l = γij (Γ k l B ijkl
=
dA dB ¯ dA dB ¯ dA dB Γ |SU(dA )×SU(dB ) = ΓLT|SU(dA )×SU(dB ) ⊕ ΓH|SU(dA )×SU(dB ) (B18)
FAB ◦ (UA ⊗ UB ) X LT = γij ((FAi ◦ UA ) ? (FBj ◦ UB )) ij
+
X
=
X
k γkH FAB ◦ (UA ⊗ UB )
k LT ¯ i ¯ j (UB )FBm ) γij (ΓlA (UA )FAl ) ? (Γ mA
ijlm
+
X
n γkH ΓH (UA ⊗ UB )FAB
kn LT H ¯ A (UA ) ⊗ Γ ¯ B (UB )FAB =Γ + ΓH (UA ⊗ UB )FAB
(B19)
¯ dA dB contains a trivial representation We observe that Γ ˆA ? u ˆ B (= uAB ) is invariant under since the element u its action. This element is also part of the locally to¯ dA Γ ¯ dB mographic subspace, and left invariant under Γ ¯ dA and Γ ¯ dB must both have a trivial subshowing that Γ representation also.
Hence
X
The action of SU(dA ) × SU(dB ) maps basis elements of the form FA ? FB to other elements of that form. Hence span(FAi ? FBj ) is a proper subspace of FAB left invariant under the action of SU(dA )×SU(dB ). The representation ¯ dA dB Γ |SU(dA )×SU(dB ) is reducible and decomposes as:
¯ dA (UA ))ik (Γ ¯ dB (UB ))j (FAk ⊗ FBl ) γij (Γ l
ijkl
¯ A (UA ) ⊗ Γ ¯ B (UB )FAB . =Γ
(B14)
c.
Representation theoretic constraint imposed by the compositional structure
An immediate consequence of the above lemmas is the following: b.
Holistic systems
Lemma 4. For a locally holistic bi-partite system with ¯ dA dB the restriction of Γ ¯ dA dB to SU(dA ) × representation Γ SU(dB ) is: M d ¯ dA dB ¯ dA Γ ¯ dB ⊕ Γ Γi A Γdi B , (B15) |SU(dA )×SU(dB ) = Γ i
¯ dA and Γ ¯ dB contain a ¯ dA dB , Γ where the representations Γ trivial representation. This is not necessarily the case for Γdi A and Γdi B . Proof. In holistic systems a basis for FAB is {FAi ? k FBj , FAB }. FAB =
X
LT γij (FAi ?FBj )+
ij
X
k LT H γkH FAB = FAB +FAB (B16)
k
We consider the action of a SU(dA ) × SU(dB ) subgroup i,j=Kd ,Kd on span({FAi ? FBj }i,j=1,1A B ). FAi
?
FBj
◦ (UA ⊗ UB ) =
(FAi
◦ UA ) ?
(FBj
◦ UB )
(B17)
Lemma 5. A necessary condition for a representation ¯ dA dB to correspond to measurement postulates Fd d Γ A B of a bipartite system with subsystems with postulates ¯ dA dB to SU(dA ) × FdA and FdB is that the restriction of Γ ¯ ¯ dB . SU(dB ) contains a subrepresentation ΓdA Γ We observe that a representation which does not have this feature cannot correspond to a bipartite system. In such a representation one cannot describe joint probabilities. This imposes further restrictions on the representations classified in [6]. 3.
Representation theoretic criterion for local tomography
Consider a bi-partite system with alternative measurement postulates FdA dB , FdA and FdB . If the theory is locally tomographic then necessarily: ¯ dA dB ¯ ¯ Γ |SU(dA )×SU(dB ) = ΓA ΓB
(B20)
¯ A and Γ ¯ B both contain a unique trivial repreWhere Γ sentation. By contraposition, any system which has a
13 ¯ dA dB which does not have this form when representation Γ AB restricted SU(dA ) × SU(dB ) cannot obey local tomography. This allows us to establish the following test for violation of local tomography: Lemma 6. Given a bi-partite system with measurement postulates FdA dB , FdA and FdB . The associated represen¯d , Γ ¯ dA and Γ ¯ dB . If Γ ¯ dA dB tation are Γ AB A B |SU(dA )×SU(dB ) is not of the form (B20) then the system is holistic.
a.
¯ dA
Affine representation and existence of trivial times trivial as a criterion
a.
Young diagram
A Young diagram is a collection of boxes in leftjustified rows, where each row-length is in non-increasing order. The number of boxes in each row determine a partition of the total number of boxes n. This partition denoted λ is associated to the Young diagram which is said to be of shape λ. We write |λ| = n for the total number of boxes. A Young diagram with m rows (labelled 1 to m) where row i has ni boxes (by definition n1 ≥ n2 ≥ ... ≥ nm ) is written λ = (n1 , n2 , ..., nm ). By definition: X
Γ and Γ contain a unique trivial subrepresentation, hence they can be decomposed as; ¯ dA = 1dA ⊕ Γ ˇ dA , Γ ¯ dB = 1dB ⊕ Γ ˇ dB , Γ
(B21)
(C1)
λ1 =
λ2 =
,
,
λ3 =
.
(B22)
where 1d is the trivial representation of SU(d). We can decompose the tensor product action: ¯ dA Γ ¯ dB = (1dA ⊕ Γ ˇ dA ) (1dB ⊕ Γ ˇ dB ) Γ ˇ dB ) ⊕ (Γ ˇ dA 1dB ) ⊕ (Γ ˇ dA Γ ˇ dB ) . = (1dA 1dB ) ⊕ (1dA Γ (B23) The first term occurs due to the trivial component in ¯ dA dB = 1dA dB ⊕ Γ ˇ dA dB . Hence, for a locally tomographic Γ AB AB system with measurement postulates FdA dB and represen¯ dA dB , the restriction of Γ ˇ AB to the local subgroup tation Γ SU(dA ) × SU(dB ) , has the following form: dA dB ˇ dA dB ˇ dB ˇ dA ˇ dA ˇ dB Γ |SU(dA )×SU(dB ) = (1 Γ ) ⊕ (Γ 1 ) ⊕ (Γ Γ ) . (B24) This shows that for a locally tomographic theory the repˇ dA dB resentations Γ AB|SU(dA )×SU(dB ) cannot contain any terms dA dB 1 1 . By contraposition we establish:
Lemma 7. Given measurement postulates FdA dB of the composite of two subsystems with postulates FdA and FdB ˇ dA dB has a subrepthen if the associated representation Γ AB dA dB resentation 1 1 upon restriction to SU(dA )×SU(dB ) then the composite system is holistic.
Appendix C: Background representation theory 1.
ni = n
i
¯ dB
Background
ˇ notation when referring to In this section we drop the Γ representations of SU(d) without the trivial subrepresentation. We use the notation Γ to refer to representations without this trivial component.
(C2) The above Young diagrams are λ1 = (2, 1), λ2 = (3, 3, 2, 1) and λ3 = (5, 2, 1, 1). In the case where a diagram has m multiple rows of the same length l we write lm instead of writing out “l” m times.. For instance λ2 = (32 , 2, 1)
b.
Representations of SU(d)
The special unitary group SU(d) is the Lie group of d × d unitary matrices of determinant 1. A finitedimensional representation of SU(d) is a smooth homomorphism SU(d) → GL(V ), with V a finite-dimensional (complex) vector space. Irreducible representations of SU(d) are labelled by Young diagrams with at most d rows. There is no limit on the total number of boxes. A column of d boxes can be removed from a Young diagram of a representation of SU(d) without changing the representation labelled by the diagram. Hence an irreducible representation of SU(d) corresponds to an equivalence class of Young diagrams up to columns of size d. Typically the canonical representative member of the equivalence class is the Young diagram where all columns of length d have been removed. With this choice of representative element of each class the Young diagram of representations of SU(d) have at most d − 1 rows. For example the equivalence class of rectangular Young diagrams of columns of length d are all mapped to the empty diagram (since all columns are removed).
λ1 =
,
λ2 =
,
λ3 =
. (C3)
14 These three Young diagrams are equivalent up to columns of length 4. They all label the same representation of SU(4). The canonical representative of the equivalence class is λ3 . In the following the expression “representation of SU(d) labelled by the Young diagram λ” is used to mean “representation of SU(d) labelled by the equivalence class of Young diagrams with representative member λ”. The young diagram labels the fundamental representation of SU(d). The empty diagram labels the trivial representation. Given a canonical representative λ we call λf the diagram in the same equivalence class which has f boxes (in other words to which a number of columns of length d have been added such that the total number of boxes is f ). For example λ2 = λ38 . Γdλ is the representation of SU(d) associated to the Young diagram λ. The index d will be dropped when the context makes it obvious. c.
Representations of the symmetric group
The symmetric group on n symbols Sn has as group elements all permutation operations on n distinct symbols. The conjugacy classes of Sn are labelled by partitions of n. Hence the number of (inequivalent) irreducible representations of Sn is the number of partitions of n. Moreover these irreducible representations can be parametrised by partitions of n. These are labelled by Young diagrams with n boxes [34]. ∆nλ denotes the representation of Sn labelled by the Young diagram λ. The index n will be dropped when the context makes it obvious. 2.
Branching rule SU(mn) → SU(m) × SU(n) a.
Definition
Consider an irreducible representation Γmn of SU(mn) λ and restrict it to a SU(m) × SU(n) subgroup: Γmn λ (U1 ⊗ U2 ), ∀U1 ∈ SU(m) , ∀U2 ∈ SU(n) .
(C4)
In general this will yield a reducible representation of SU(m) × SU(n). This representation will be built of irn reducible representations Γm µ Γν n m n (Γm µ Γν )(U1 , U2 ) = Γµ (U1 ) ⊗ Γν (U2 ) ,
(C5)
mn We write Γmn to a λ|SU(m)×SU(n) for the restriction of Γλ SU(m) × SU(n) subgroup. In general: M n Γmn Γm (C6) µ Γν , λ|SU(m)×SU(n) = µ,ν
Where there can be repeated copies for a given µ, ν. In n general finding which representations Γm µ Γν occur in this restriction is a hard problem. In the following we
outline a method which allows us to determine the muln mn tiplicity of Γm µ Γν in Γλ|SU(m)×SU(n) . λ , µ and µ will m n refer to the Young diagrams of Γmn λ , Γµ and Γν respectively. b.
Inner product of representations of the symmetric group
We consider two representations ∆µ and ∆ν of Sf . We construct the Kronecker product of the two matrices ∆µ (s) and ∆ν (s) for all s ∈ Sf . This creates a representation which we call the tensor product ∆µ ⊗ ∆ν (sometimes called the inner product). In general this is a reducible representation: M ∆µ ⊗ ∆ν = g(µ, ν, λ)∆λ (C7) λ
Here we abuse notation slightly to mean that g(µ, ν, λ) is the multiplicity of ∆λ in ∆µ ⊗ ∆ν . These g(µ, ν, λ) are known as the Clebsch-Gordon coefficients of the symmetric group, and understanding them remains one of the main open problems in classical representation theory. These coefficients are also relevant in quantum information theory, as they are related to the spectra of statistical operators [35]. c.
Recipe
n What is the multiplicity of Γm µ Γν in the restriction of to SU(m)×SU(n)? We adopt the approach from [36] to answer this question. Let f = |λ| be the number of boxes in the Young diagram λ. As shown above λ also labels a representation of the symmetric group on f objects Sf . This representation is ∆fλ . Take the Young diagram µ (ν) and add columns of m (n) boxes to the left until it has f boxes. The tableau obtained which we call µf (νf ) labels a representation of Sf denoted ∆fµf (∆fνf ). We remember that adding columns to the left of length m (n) keeps µ (ν) within the equivalence class of Young n diagrams labelling the representation Γm µ (Γν ). Hence µf (νf ) labels the same representation of SU(m) (SU(n)) as µ (ν). Hence the diagrams λ , µf and νf refer both to reprem sentations of the special unitary group Γmn , Γm µ (= Γµf ) λ
Γmn λ
and Γnν (= Γnνf ) as well as representations of Sf : ∆fλ , ∆fµf and ∆fνf . n Theorem 1. Γm µ Γν occurs as many times in the restriction of Γmn to SU(m) × SU(n) as ∆fλ occurs in λ ∆fµf ⊗ ∆fνf , where f = |λ| [36].
3.
Inductive lemma
m n mn Lemma 8. Consider representations Γmn ¯ , Γµ ¯ , Γν ¯ , Γλ , λ m n mn m n 0 ¯ Γµ , Γν , Γλ0 , Γµ0 and Γν 0 where λ = λ + λ , µ ¯ = µ+
15 0
0
0
0
|λ|−|ν| |λ |−|µ | | µ0 ν¯ = ν + ν 0 and |λ|−|µ| , m and |λ |−|ν m , n n mn m are integers. If Γλ|SU(m)×SU(n) contains a term Γµ m n Γnν and Γmn λ0 |SU(m)×SU(n) contains a term Γµ0 Γν 0 then n Γmn contains a term Γm ¯ µ ¯ Γν ¯. λ|SU(m)×SU(n) m n Proof. Γmn λ|SU(m)×SU(n) containing a term Γµ Γν implies
that ∆fλ occurs in ∆fµf ⊗ ∆fνf (by Theorem 1). Here µf is f −|ν| the tableau µ (ν) to which f −|µ| m ( n ) columns of length m (n) has been added so that the total number of boxes |µf | = f (|νf | = f ).
In the following we establish that Γd|SU(d)×SU(d) contains terms 1dA ΓdB ⊕ ΓdA 1dB ⊕ ΓdA ΓdB for representations Γd = Djd . We remind the reader that we have dropped the ˇ notation in this section, and that the representations Γ do not contain the trivial representation. Lemma 9. If the representation Djd contains a term DjdA 1dB and D2d contains a term D2dA 1dB when red stricted to SU(dA ) × SU(dB ) then Dj+2 contains a term dA dB Dj+2 1 upon this restriction. Proof. Let
f − |µ| m ) ), m f − |ν| n νf = ν + (( ) ). n µf = µ + ((
(C8)
Γdλ0 = D2d , ΓdµA0 = D2dA , ΓdνB0 = 1dB (C9)
m Here we recall that (( f −|µ| m ) ) indicates m rows of length f −|µ| (( m ). This implies that g(λ, µf , νf ) > 0. Similarly g(λ0 , µ0f 0 , νf0 0 ) > 0. We now show that µ ¯ = µ + µ0 and ν¯ = ν + ν 0 implies 0 that µ ¯f¯ = µf + µf 0 and ν¯f¯ = νf + νf0 0 .
f − |µ| m f 0 − |µ0 | m ) ) + (( ) ) m m (f + f 0 − |µ| − |µ0 |) m =µ ¯ + (( ) ) m f¯ − |¯ µ| n =µ ¯ + (( (C10) ) )=µ ¯f¯ . m
µf + µ0f 0 = µ + µ0 + ((
And similarly for ν¯. Let us make use of a property of the Clebsch Gordon coefficients called the semi-group property. If g(λ, µf , νf ) > 0 and g(λ0 , µ0f 0 , νf0 0 ) > 0 then g(λ+λ0 , µf + µ0f 0 , νf + νf0 0 ) > 0 [37]. By the semi-group property ¯ ¯ µ we have g(λ, ¯ ¯, ν¯ ¯) > 0. This implies that ∆f¯ ocf
¯
f ¯
λ
curs in ∆fµ¯f¯ ⊗ ∆fµ¯f¯. By Theorem 1 this implies that m Γmn contains a term Γm ¯ ¯ . µ ¯ Γν λ|SU(m)×SU(n)
Appendix D: Existence of locally tomographic subspace
As shown in [6] the representations Γd corresponding to alternative measurement postulates for systems with pure states Cd (d > 2) are of the form Γ=
M
Djd ,
(D1)
j∈J
where J is a list of non-negative integers (at least one of which is not 0 or 1) and Djd are representations of SU(d) labelled by Young diagrams (2j, j, . . . , j ). | {z } d−2
Γdλ = Djd , ΓdµA = DjdA , ΓdνB = 1dB dA d , Γdν¯B = 1dB Γλ¯ = Dj+2 , Γµ¯ = Dj+2 where
λ = (2j, j d−2 ), µ = (2j, j dA −2 ), ν = 0 , λ0 = (4, 2d−2 ), µ0 = (4, 2dA −2 ), ν 0 = 0 , ¯ = (2j + 4, (j + 2)d−2 ), µ λ ¯ = (2j + 4, (j + 2)dA −2 ), ν¯ = 0 . We observe ¯ = λ + λ0 , µ λ ¯ = µ + µ0 , ν¯ = ν + ν 0 ¯ = d(j + 2) f = |λ| = jd, f 0 = |λ0 | = 2d, f¯ = |λ| Next we check that the quantities below are integer valued: |λ| − |µ| jd − jdA = = j(dB − 1) , m dA |λ| − |ν| jd − 0 = = jdA , n dB |λ0 | − |µ0 | 2d − 2dA = = 2(dB − 1) , m dA |λ0 | − |ν 0 | 2d − 0 = = 2dA . n dB From Lemma 8 it follows that if Djd contains a representation DjdA 1dB and D2d contains a representation d D2dA 1dB when restricted to SU(dA ) × SU(dB ) then Dj+2 dA dB contains a representation Dj+2 1 . Lemma 10. If the representation Djd contains a term 1dA DjdB and D2d contains a term 1dA D2dB when red stricted to SU(dA ) × SU(dB ) then Dj+2 contains a term dB dA 1 Dj+2 upon this restriction. Proof. Same as above with relabelling of µ’s for ν’s. Lemma 11. If the representation Djd contains a term DjdA DjdB and D2d contains a term D2dA D2dB when red stricted to SU(dA ) × SU(dB ) then Dj+2 contains a term dA dB Dj+2 Dj+2 upon this restriction.
16 Proof. Let Γdλ = Djd , ΓdµA = DjdB , ΓdνB = DjdB ,
We observe that if we include the trivial representation then we have that all representations Γ9trivial ⊕ Dj9 restricted to SU(3) × SU(3) have terms:
Γdλ0 = D2d , ΓdµA0 = D2dA , ΓdνB0 = D2dB , Γλ¯ =
d Dj+2 ,
Γdµ¯A
=
dB , Dj+2
Γdν¯B
=
(13 13 ) ⊕ (13 Dj3 )
dB , Dj+2
where
⊕ (Dj3 13 ) ⊕ (Dj3 Dj3 ) . This is equal to:
λ = (2j, j d−2 ), µ = (2j, j dA −2 ), ν = (2j, j dB −2 ) λ0 = (4, 2d−2 ), µ0 = (4, 2dA −2 ), ν 0 = (4, 2dB −2 ) ¯ = (2j + 4, (j + 2)d−2 ), µ λ ¯ = (2j + 4, (j + 2)dA −2 ), ν¯ = (2j + 4, (j + 2)dB −2 ) . We observe ¯ = λ + λ0 , µ λ ¯ = µ + µ0 , ν¯ = ν + ν 0 ¯ = d(j + 2) f = |λ| = jd, f 0 = |λ0 | = 2d, f¯ = |λ| Next we check that the quantities below are integer valued: |λ| − |µ| m |λ| − |ν| n |λ0 | − |µ0 | m |λ0 | − |ν 0 | n
jd − jdA dA jd − jdB = dB 2d − 2dA = dA 2d − 2dB = dB =
= j(dB − 1) ,
which is of the form Γ3 Γ3 where the representations Γ3 contain a trivial representation (as in equation (15)). 2.
Reducible representations
All representations of the form (D1) contain a locally tomographic part since each term Djd in the sum provides a locally tomographic part. However we observe that in general the direct sum of the locally tomographic parts is not a locally tomographic subspace. Consider a representation Dkd ⊕ Dld . The restriction of this representation will have components:
= j(dA − 1) ,
(1dA DkdB ) ⊕ (DkdA 1dB ) ⊕ (DkdA DkdB )
= 2(dB − 1) ,
⊕ (1dA DldB ) ⊕ (DldA 1dB ) ⊕ (DldA DldB ) .
(D2)
However we observe that this is not equal to: = 2(dA − 1) .
From Lemma 8 it follows that if Djd contains a representation DjdA DjdB and D2d contains a representation d D2dA D2dB when restricted to SU(dA )×SU(dB ) then Dj+2 dA dB contains a representation Dj+2 Dj+2 . The above three lemmas entail that: Lemma 12. If the representation D2d contains a representation (1dA D2dB ) ⊕ (D2dA 1dB ) ⊕ (D2dA D2dB ) upon restriction to SU(dA )×SU(dB ) and the representation D3d contains a representation (1dA D3dB )⊕(D3dA 1dB )⊕(D3dB D3dB ) upon restriction to SU(dA ) × SU(dB ) then all representation Djd contain a representation (1dA DjdB )⊕(DjdA 1dB ) ⊕ (DjdA DjdB ) upon restriction to SU(dA ) × SU(dB ). 1.
(13 ⊕ Dj3 ) (13 ⊕ Dj3 ) ,
Proof for d = 9
Using Sage software we can show that D29 contains a representation (13 D23 ) ⊕ (D23 13 ) ⊕ (D23 D23 ) upon restriction to SU(3) × SU(3) and the representation D39 contains a representation (13 D33 )⊕(D33 13 )⊕(D33 D33 ) upon restriction to SU(3) × SU(3). By Lemma 12 all representations Dj9 contain a representation (13 Dj3 ) ⊕ (Dj3 13 ) ⊕ (Dj3 Dj3 ) upon restriction to SU(3) × SU(3).
(1dA (DkdB ⊕ DldB )) ⊕ ((DkdA ⊕ DldA ) 1dB ) ⊕ ((DkdA ⊕ DldA ) (DkdB ⊕ DldB )) ,
(D3)
since the last term will contain cross terms not present in (D2). From the above consideration it follows that all representations of SU(9) of the form (D1) have a locally tomographic subrepresentation upon restriction to SU(3)× SU(3). Appendix E: Violation of local tomography in all alternative measurement postulates
In the following we establish that Γ9|SU(3)×SU(3) is not of the form (B24) for all representations Γ9 of SU(9) corresponding to non quantum state spaces with pure states C9 . We show explicitly that every such restriction contains a term of 13 13 , where 13 is the trivial representation of SU(3). 1.
Arbitrary dimension d
We now construct a proof by induction to show that the representations Djd are not compatible with local tomography when restricted to SU(dA ) × SU(dB ). A representation Djd will violate local tomography if it is not
17 of the form (B24) when restricted to SU(dA ) × SU(dB ). It suffices to show that there is a term 1dA 1dB in this restriction in order to show that it is not of this form. Lemma 13. If the representations Djd and D2d of SU(d) contain a term 1dA 1dB when restricted to SU(dA ) × d SU(dB ) then so does Dj+2
Where J is a list of positive integers containing at least one integer which is not 1. Since at least one (non-trivial) subrepresentation in Γ9 has a 13 13 when restricted to SU(3) × SU(3) so does the representation Γ9 .
3.
Violation of local tomography for all theories
Proof. Let Γdλ = Djd , ΓdµA = 1dA , ΓdνB = 1dB Γdλ0 = D2d , ΓdµA0 = 1dA , ΓdνB0 = 1dB d Γλ¯ = Dj+2 , Γµ¯ = 1dA , Γdν¯B = 1dB ,
where λ = (2j, j d−2 ), µ = 0, ν = 0 , λ0 = (4, 2d−2 ), µ0 = 0, ν 0 = 0 , ¯ = (2j + 4, (j + 2)d−2 ), µ λ ¯ = 0, ν¯ = 0 . We observe ¯ = λ + λ0 , µ λ ¯ = µ + µ0 , ν¯ = ν + ν 0 ¯ = d(j + 2) f = |λ| = jd, f 0 = |λ0 | = 2d, f¯ = |λ| Next we check that the quantities below are integer valued: |λ| − |µ| m |λ| − |ν| n |λ0 | − |µ0 | m 0 |λ | − |ν 0 | n
jd dA jd = dB 2d = dA 2d = dB =
Every representation Γ9 of SU(9) of the form (E1) has a representation 13 13 when restricted to SU(3)×SU(3). It follows from this that the restriction of Γ9 to SU(3) × SU(3) is not of the form required for local tomography. From this it follows that all non-quantum C9 systems which are composites of two C3 systems violate local tomography. In order to show that a theory with systems Cd (for every d > 1) violates local tomography, it is sufficient to show that one of the systems in the theory violates local tomography. Since all C9 non-quantum systems violate local tomography it follows that all non-quantum theories with systems Cd violate local tomography.
Appendix F: Violation of purification
In this appendix we show that all alternative measurement postulates violate purification (for an arbitrary choice of ancillary system dimension). Consider a system SAB = {CdA dB , ΓAB } which is the composite of two systems SA = {CdA , ΓA } and SB = {CdB , ΓB }. Here the representations Γ are of the form (D1). Let us define the following equivalence classes of pure global states:
= jdB , = jdA , = 2dB , = 2dA .
0
From Lemma 8 it follows that if Djd contains a representation 1dA 1dB and D2d contains a representation d 1dA 1dB when restricted to SU(dA ) × SU(dB ) then Dj+2 contains a representation 1dA 1dB when restricted to SU(dA ) × SU(dB ).
0
[|ψiAB ]UB = {|ψiAB ∈ CdA dB | |ψiAB = IA ⊗ UB |ψiAB } (F1) All members of the same equivalence class are necessarily mapped to the same reduced state of Alice. Otherwise, Bob could signal to Alice. Let us call the set of all these equivalence classes RB . RB := {[|ψiAB ]UB | |ψiAB ∈ CdA dB }
Hence it suffices to show that D2d and D3d contain 1dA dB 1 when restricted to SU(dA ) × SU(dB ) to show that Djd does for any j > 1. 2.
Existence of 13 13 for all non-quantum representations of SU(9)
The map from global states to reduced states can be defined on the equivalence classes [|ψiAB ]UB since two members of the same equivalence class are always mapped to the same reduced states. R : RB → SA is the map from equivalence classes to reduced states: R([|ψiAB ]UB ) = ω ¯ A (|ψiAB ) ,
D29
D39
Using Sage software we can show that and have a representation 13 13 when restricted to SU(3)×SU(3). By Lemma 13 all representations Dj9 , j > 1 have this property. An arbitrary representation corresponding to an alternative Born rule for C9 is of the form: M Γ9 = Dj9 (E1) j∈J
(F2)
(F3)
where ω ¯ A (|ψiAB ) is the reduced state obtained in the standard manner from the global state |ψiAB (as outlined in the following appendix). Next we prove that the image of R is smaller than SA for any non-quantum measurement postulates. In other words there are some (local) mixed states in SA which are not reduced states of the global pure states |ψiAB .
18 In the Schmidt decomposition a state |ψiAB is:
|ψiAB =
dA X i=1
λi |iiA |iiB , λi ∈ R ,
X
λ2i = 1 ,
1.
(F4)
i
where we assume that the Schmidt coefficients are in decreasing order λi ≥ λi+1 . Two states with the same coefficients and the same basis states on Alice’s side belong to the same equivalence class [|ψiAB ]UB . Also, two Alice’s basis differing only by phases (e.g. {|iiA } and {eiθi |iiA }) give rise to the same equivalence class. Because the phases eiθi can be absorbed by Bob’s unitary. Let us count the number of parameters that are required to specify an equivalence class in RB . First, we have the dA − 1 Schmidt coefficients. Second, we note that the number of parameters to specify a basis in CdA is the same as to specify an element of U(dA ). Which is the dimension of its Lie algebra, d2A , the set of antihermitian matrices. Third, we have to subtract the dA irrelevant phases θi . The three terms together give (dA − 1) + d2A − dA = d2A − 1
(F5)
d2A
Hence − 1 parameters are needed to specify elements of RB . The set Image(R) requires the same or fewer parameters to describe as RB . This follows from the fact that every element of RB can be mapped to distinct images, or multiple elements can be mapped to the same image. Hence by requiring that Alice’s reduced states are in one-to-one correspondence with these equivalence classes, her state space must have a dimension d2A − 1. The only measurement postulates which generate a state space with this dimension are the quantum ones.
Representation of states
The set of mixed states span a real vector space. A state is a vector: F1 (ω) F2 (ω) (G1) ω ¯= , .. . FKd (ω) An arbitrary F ∈ Fd can be expressed: X F = ci Fi
¯F To each F we can associate a dual vector (or effect) E ¯ F = (c1 , ..., cK ) , E d
In the earlier appendices we gave a certain primacy to the space of OPFs, and considered the group action on the linear space spanned by the OPFs. However it is equally valid (and more common) to give primacy to the space of states and consider the group action on this space. This standard representation will be helpful for proving the consistency of the toy model. Indeed the consistency constraints required for composition translate naturally to the language of states. This section is largely without proof, as the proofs carry over naturally from the OPF formalism to the standard state formalism. We refer the reader to [33] for more detailed proofs.
(G3)
¯F · ω ¯ be effect associated By definition E ¯ = F (ω). Let u ¯·ω to the unit OPF. Since u ¯ = 1 there is a component for all states ω ¯ which is constant. That is to say we can choose F1 to be the unit OPF: 1 ω ¯= , (G4) ω ˇ The group representation Γd acting on the state space ˇ d (Γ ˇ d may be reducible or is of the form Γd = 1d + Γ irreducible). We can write ¯F · ω ˇF · ω E ¯ = c1 + E ˇ
(G5)
In the representation where the trivial component is removed (by an affine transformation) then states are ω ˇ (whose fiducial outcomes are affinely independent), the ˇ d and outcome probabilities are group representation is Γ ˇ affine functions EF of the state ω ˇ . We observe that from the uniqueness of the trivial component in Γd it follows ˇ d cannot contain any trivial subrepresentation. that Γ
2. Appendix G: Composition in GPTs
(G2)
i
Local tomography
Consider measurement postulates FdA dB , FA and FB . States for Alice and Bob can be written as: FA1 (ωA ) FA2 (ωA ) .. ω ¯A = (G6) , . K dA
FA
(ωA )
and ω ¯B =
FB1 (ωB ) FB2 (ωB ) .. . Kd FB B (ωB )
.
(G7)
19 Definition (Local tomography). A composite system SAB is locally tomographic if it has fiducial outcomes i=KdA ,j=KdB {(FAi ? FBj )}i=1,j=1 . A state of the composite system can be written: (FA1 ? FB1 )(ωAB ) (FA1 ? FB2 )(ωAB ) .. ω ¯ AB = (G8) , . Kd Kd (FA A ? FB B )(ωAB ) ¯F i ⊗ E ¯ j . Indeed any ¯ i j = E We observe that E FB FA ?FB A ¯F ?F = E ¯F ⊗ E ¯F . Product states are joint local effect E A B A B of the form ω ¯A ⊗ ω ¯B.
3.
Non-locally-tomographic theories
The states of a composite system SAB of a non-locally tomographic (or holistic) theory can be written as [32, 33]: ¯ AB λ ω ¯ AB = , (G9) ηAB where ¯ AB = λ
(FA1 ? FB1 )(ωAB ) (FA1 ? FB2 )(ωAB ) .. . Kd (FA A
?
Kd FB B )(ωAB )
,
is the locally tomographic part and 1 FAB (ωAB ) .. ηAB = , . KdA dB −KdA KdB FAB (ωAB )
(G10)
(G11)
The reduced state space
The reduced states for Alice and Bob can be computed as follows: ¯ AB ¯ dA (IA ) ⊗ u ¯ B )λ ω ¯ A = (Γ ¯ AB ¯ dB (IB ))λ ω ¯ B = (¯ uA ⊗ Γ
1.
Joint effects
Given effects FˆA and FˆB the joint effect on a global state |ψihψ|⊗2 AB is obtained by projecting the state onto the locally tomographic subspace and applying FˆA ⊗ FˆB . The projector onto the locally tomographic subspace is SA SB . This will not preserve the global norm S˜AB , hence we need to add in the component of the trace along AA AA (= SAB − SA SB which has been chopped off by the projection). Hence the locally tomographic part of a state is: ¯ AB = SA SB |ψihψ|⊗2 SA SB + tr(|ψihψ|⊗2 AA AB )S˜A S˜B λ AB AB (H1) ¯ is always 1, and that it has supWe see that the trace of λ port just on the locally tomographic subspace (including the trivial part). Joint effects are computed following equation (G12): (FA ? FB )(|ψiAB ) = ⊗2 ˜ ˜ tr FˆA FˆB SA SB |ψihψ|⊗2 AB SA SB + tr(|ψihψ|AB AA AB )SA SB (H2)
2.
formula
? product
Lemma 14. The ? product is bilinear, associative and satisfies C3. Proof. We first prove the bilinearity of the ? product
(G12)
Joint local effects are computed using the locally tomographic part of the state space only.
4.
In this appendix we prove that the toy model is consistent with the compositional constraints C1., C2. C3. C4. and C5.
Computing the joint effect using the ⊗2 Tr(F\ A ? FB |ψihψ|AB ) gives the same formula.
is the holistic part. The probability for a joint outcome FA ? FB is given by: ¯ AB ¯F ⊗ E ¯F ) · λ (FA ? FB )(ωAB ) = (E A B
Appendix H: Toy model
(G13)
trFˆB tr(cFˆA ) ˆ ˆ AA ⊗ AB (cF\ A ) ? FB = (cFA ) ⊗ FB + trSA trSB = c(F\ A ? FB ) trFˆA tr(cFˆB ) FA\ ? (cFB ) = (cFˆA ) ⊗ (cFˆB ) + AA ⊗ AB trSA trSB = c(F\ A ? FB ) (H3) We prove uA ? uB = uAB
(G14) S\ A ? SB = SA ⊗ SB + AA ⊗ AB = SAB
Reduced states are determined using only the locally tomographic part of the global state.
We prove (FA ? FB )(ψA ⊗ φB ) = FA (ψA )FB (φB ) :
(H4)
20
(FA ? FB )(ψA ⊗ φB ) tr(FˆA FˆB ) ⊗2 ˆ ˆ A A ) = tr |ψihψ|⊗2 |φihφ| ( F F + A B A B A B tr(SA SB ) ⊗2 ˆ ˆ = tr |ψihψ|⊗2 A |φihφ|B FA FB = FA (ψA )FB (φB )
(H5)
In the penultimate line we have used the fact that the ⊗2 overlap of product states |ψihψ|⊗2 A |φihφ|B and AA AB is 0.
3.
Reduced states
¯ AB applying the The reduced state is obtained from λ unit effect on one of the subsystems as in equation (G13): ω ¯A =
trB (SA SB |ψihψ|⊗2 AB )
+
˜ tr(|ψihψ|⊗2 AB AA AB )SA
(H6)
Lemma 15. The reduced state ω ¯ A can be written as: ω ¯ A = SA (ρA ⊗ ρA )SA + (1 − tr(SA (ρA ⊗ ρA )SA ))S˜A , (H7) where ρA = trB (|ψihψ|AB ). Proof. By Lemma 19 trB (SA SB |ψihψ|⊗2 AB ) = SA (ρA ⊗ ρA )SA
(H8)
Hence, ˜ ω ¯ A = SA (ρA ⊗ ρA )SA + (1 − tr(|ψihψ|⊗2 AB SA SB ))SA = SA (ρA ⊗ ρA )SA + (1 − tr(SA (ρA ⊗ ρA )SA ))S˜A . (H9)
Lemma 16. The reduced states ω ¯ A belong to the convex hull of the local pure states |ψihψ|⊗2 . Proof. By Lemma 15 the reduced state can be written as: ω ¯ A = SA (ρA ⊗ρA )SA +(1−tr(SA (ρA ⊗ρA )SA ))S˜A , (H10) where ρA = trB (|ψihψ|AB ). In the following we drop the A label. ρ=
X
αi |iihi| ,
(H11)
ω ¯=
X
αi2 |i, iihi, i| +
i
X
αi αj |Φij ihΦij | +
i 0. Consider the (not necessarily normalised) matrix X X S(ρ ⊗ ρ)S = αi2 |i, iihi, i| + αi αj |Φij ihΦij | , (H12) i
