arXiv:1412.8323v8 [quant-ph] 13 Dec 2017 ainabout mation ihtersmere stm evolution time Höhn: Andres as Philipp symmetries their together respectively, with rebit, spaces and state sys- qubit as a disc elementary for and ball single Bloch the the a for tem: of solutions case two admit simplest rules four the xsec fcmlmnayinformation; (3) complementary of existence ue il eesbetm vlto and quantifying evolution measure, quadratic time a reversible The a arises. yield rebits rules and arbi- qubits of many structure trarily correlation the how gener- en- ally, more of and, non-locality monogamy tanglement, and theory, entanglement quantum how in measurements jective rsre nbtenitroain;and, (4) interrogations; in-between preserved ueof struc- ture compatibility and complementarity independence, a ensuing the permits how approach elucidating This constructive possible. evolution such be consistent to every and tions interroga- in-between time in continuously oe,asrig()almto h amount the to available on information limit im- of a are (1) asserting rules Four posed, un- all assumptions. highlight to are derlying attempt framework we and the simple of tools mathematical eiee nttt o hoeia hsc,3 aoieSt 2017 Caroline 14, 31 December Physics, Theoretical for Institute Perimeter Höhn Andres Philipp acquisition from information on theory rules quantum reconstructing for Toolbox cetdin Accepted omto custo.Tefcsle nan on lies observer focus The in- acquisition. on formation rules of elementary from theory systems quantum qubit the reconstructing for u cec n ehooy nvriyo ina Boltz- Quan- Vienna, Austria for Vienna, of 1090 Center University 3, Technology, Vienna manngasse and and Science Sciences; tum Aus- of Information, Academy Quantum & trian Optics Quantum for Institute O iayqetosand questions binary s‘aaou fkolde about knowledge’ of ‘catalogue ’s edvlpa prtoa approach operational an develop We O O s‘aaou fkolde ochange to knowledge’ of ‘catalogue ’s sttlaon fifraint be to information of amount total ’s O Q sqetosmthsta fpro- of that matches questions ’s O u a eiaino unu theory, quantum of derivation nergtn system a interrogating n t u m S ial,i ssonthat shown is it Finally, . [email protected] 071-7 lc il overify to title click 2017-11-27, S ssaei ae as taken is state ’s O 2 h mere the (2) ; rsn address: Present , O sinfor- ’s S S The . with etNrh aelo nai,Cnd 2 2Y5 N2L Canada Ontario, Waterloo, North, reet ayqbt scmltdi companion a [ in paper completed is arbitrarily qubits for many reconstruction The groups. fifraintertctos[ tools applications theoretic for information arena of computa- physical natural and the information is tion quantum of field tire informa- thus [ and perspectives tional entropic to adheres naturally ytm fabtaiymn uisfo elemen- from for qubits many theory arbitrarily quantum of of systems structure the develop and deriving to formalism for is framework informational ambition example novel Our a concrete theory. the quantum of of means address by shall question we ap- article, this this the In of theory. structure propriate the with be intertwined should deeply communication, or acquisition infor- mation e.g. activities, informational restric- certain or of rules tions elementary the that is The idea about overriding theories. namely of some- architecture and physics, content us physical about tell also deeper can thing situations, con- physical to applications crete useful their con- from theoretic apart cepts, information whether question the oio nrpe,cnb nesodi terms in [ understood entropy entanglement be examples, of can of space-time entropies, generally number more horizon vast or entropy, a hole black Among phe- physical specific under- nomena. interpreting for in and useful proving applications standing often of physics, number growing modern ever an have theory seen information from concepts and Tools Introduction 1 of re- terms novel in the questions. theory a on) quantum of yields rely formulation and not in- interpretation does is lational approach (but This by case. spired rebit the nates hsmnsrp,b otat smtvtdby motivated is contrast, by manuscript, This 1 hr nadtoa ueelimi- rule additional an where ] 8 – 12 ,ad bv l,teen- the all, above and, ], 3 – 7 ,thermodynamics ], 13 ]. 1 tary operational postulates – a task which is com- on the acquisition of information would it make pleted in the companion paper [1]. While neither sense to speak about an absolute state of a sys- this question nor the fact that one can recon- tem within a purely operational approach (unless struct quantum theory from elementary axioms is one accepts the existence of an omniscient and new and has been extensively explored before in absolute observer as an external standard). But various contexts [14–25], we shall approach both thanks to the existence of complementarity, im- from a novel constructive perspective and with plied by ~, an observer may not access all conceiv- a stronger emphasis on the conceptual content able properties of the system at once. Further- of the theory. The ultimate goal of this work is more, the observer can choose the experimental therefore very rudimentary: to redo a well estab- setting and thereby which property of the system lished theory – albeit in a novel way which is es- she would like to reveal (although, clearly, she pecially engineered for exposing its informational cannot choose the experimental outcome). Under and logical structure, physical content and dis- our purely operational premise, we shall treat the tinctive phenomena more clearly. In other words, situation as if the system does not have any other we shall attempt to rebuild quantum theory for properties than those accessible to the observer at qubit systems from scratch. any moment of time. In particular, the system’s state is naturally interpreted as representing the In such an information based context it is nat- observer’s state of information about the system. ural to follow an operational approach, describing These ideas are in agreement with earlier propos- physics from the perspective of an observer. Ac- als in the literature [26,27] and, most specifically, cordingly, we shall work under the premise that with the relational interpretation of quantum me- we may only speak about the information an ob- chanics [28, 29]. server has access to in an experiment. Our ap- proach will thus be purely operational and epis- Of course, in order for different observers who temic (i.e. knowledge based) by construction and may communicate (by physical interaction) to shall survive without ontic statements (i.e. ref- have a basis for agreeing on the description of erences to ‘reality’). We shall thus say nothing a system, some of its attributes must be observer about whether or not ‘hidden variables’ could give independent such as its state space, the set of rise to the experiences of the observer. Under possible measurements on it and possibly a limit these circumstances we adopt an ‘inside view of on its information content. But without adhering physics’, holding properties of systems as being to an external standard against which measure- relationally, rather than absolutely defined. ment outcomes and states could be defined, it is as meaningless to assert a system’s physical state Indeed, more generally the replacement of ab- to be independent of its relations to other sys- solute by relational concepts goes in hand with tems as it is to relate a system’s dynamics to an the establishment of universal (i.e. observer in- absolute Newtonian background time. dependent) limits. For instance, the crucial step from Galilean to special relativity is the realiza- It is worthwhile to investigate what we can tion that the speed of light c constitutes a univer- learn about physics from such an operational sal limit for information communication among and informational approach. For this endeavour observers. The fact that all observers agree on we shall adopt the general conviction, which has this limit is the origin of the relativity of space been voiced in many different (even conflicting) and time. Similarly, the crucial step from classi- ways before in the literature [26, 28, 30–41], that cal to quantum mechanics is the recognition that quantum theory is best understood as an opera- the Planck constant ~ establishes a universal limit tional framework governing an observer’s acquisi- on how much simultaneous information is acces- tion of information about a system. While most sible to an observer. While less explicit than in earlier works take quantum theory as given and the case of special relativity, this simple observa- attempt to characterize and interpret its physical tion suggests a relational character of a system’s content with an emphasis on information infer- quantum properties. More precisely, the process ence, here and in [1] we take a step back and show of information acquisition through measurement that one can actually derive quantum theory from establishes an informational relation between the this perspective. This will require a focus on the observer and system. Only if there was no limit informational relation between an observer and a

Accepted in Quantum 2017-11-27, click title to verify 2 system and the rules governing the observer’s ac- an externally given transformation group between quisition of information. More precisely, our ap- different reference frames – one can also derive proach will be formulated in terms of the observer the Lorentz group as the minimal group trans- interrogating a system with elementary questions. lating between different observer’s descriptions While the present work has been inspired by re- of physics from their informational relations, es- lational ideas, we emphasize that the sequel does tablished by communication with quantum sys- not actually rely on them so this should not dis- tems [42]. courage a reader unsympathetic with relational More fundamentally, relational ideas are actu- interpretations of quantum theory. Our approach ally required and commonly employed in the con- will reconstruct and produce a novel formulation text of background independent quantum gravity of quantum theory, but will clearly not single out approaches where the notion of coordinates dis- the relational interpretation as ‘the right one’. appears together with a classical notion of space- However, it will support a partial interpretation, time within which a dynamics could be defined. namely that quantum theory is a law book gov- Instead, one has to resort to dynamical degrees erning an observer’s acquisition of information of freedom to define physical reference frames about physical systems. (i.e., dynamical ‘rods’ and ‘clocks’) relative to This is clearly not the only physical situation which a meaningful dynamics can be formulated to which such a relational approach applies; it in the first place. This constitutes the relational likewise opens up a novel perspective on elemen- paradigm of dynamics [43–51] but goes beyond tary space-time structure which is encoded in the a purely informational and operational approach informational relations among different observers and thus clearly beyond the scope of this work. and can be exposed by a communication game. For instance, without presupposing a particular The remainder of this manuscript is organized space-time structure – and thus without assuming as follows.

Contents

1 Introduction 1

2 Why a(nother) reconstruction of quantum theory? 5

3 Landscape of inference theories 5 3.1 The standard landscape of generalized probabilistic theories ...... 6 3.2 A novel landscape of inference theories ...... 7 3.2.1 Questions and answers ...... 7 3.2.2 From information to probabilities: the state of S relative to O ...... 9 3.2.3 ‘Belief’ updating and ‘collapse’ of the state ...... 10 3.2.4 Elementary structure on Σ and Q ...... 11 3.2.5 Logical compositions of questions and rules of inference ...... 14 3.2.6 Parametrization of S’s state and tomography ...... 16 3.2.7 Compositesystems ...... 18 3.2.8 Time evolution of S’sstate ...... 18 3.2.9 The theory landscape L ...... 20

4 Principles for the quantum theory of qubits as rules on information acquisition 21 4.1 Therules ...... 21 4.2 Strategy for building the necessary tools and proving theclaim ...... 25

5 Question structure and correlations 26 5.1 Asinglegbit ...... 26 5.2 Twogbits ...... 26 5.2.1 Logical connectives of single gbit questions ...... 26

Accepted in Quantum 2017-11-27, click title to verify 3 5.2.2 Independence, complementarity and entanglement ...... 28 5.2.3 A logical argument for the dimensionality of the Bloch-sphere ...... 31 5.2.4 An informationally complete set for N = 2 qubits...... 32 5.2.5 An informationally complete set for N = 2 rebits ...... 33 5.2.6 Entanglement and non-local tomography for rebits ...... 34 5.2.7 A Bell scenario with questions: ruling out local hidden variables ...... 36 5.3 Threegbits ...... 38 5.3.1 Threequbits ...... 39 5.3.2 Independence and compatibility for three qubits ...... 40 5.3.3 An informationally complete set for three qubits ...... 42 5.3.4 Entanglement of three qubits and monogamy ...... 43 5.3.5 Maximal entanglement for three qubits ...... 44 5.3.6 Threerebits...... 45 5.3.7 Independence and compatibility for three rebits ...... 45 5.3.8 An informationally complete set for three rebits ...... 47 5.3.9 Monogamy and maximal entanglement for three rebits ...... 48 5.4 Correlation structure for N = 2 gbits...... 48 5.4.1 The logical mirror image of an inference theory ...... 49 5.4.2 Collecting the results: odd and even correlation structure for N = 2 ...... 52 5.5 The general case of N > 3 gbits...... 54 5.5.1 An informationally complete set and entanglement for N > 3 qubits ...... 54 5.5.2 An informationally complete set and entanglement for N > 3 rebits ...... 58

6 Information measure, time evolution and state space 59 6.1 Why the Shannon entropy does not apply here ...... 60 6.2 Elementary conditions on the measure ...... 60 6.3 The state of no information is an interior state ...... 61 6.4 Time evolution of the ‘Bloch vector’ ...... 62 6.5 Time evolution is injective ...... 63 6.6 Time evolution is also reversible ...... 64 6.7 Time evolution defines a group ...... 65 6.8 The squared length of the Bloch vector as information measure ...... 65 6.9 The set of all time evolutions is a subgroup of SO(DN) ...... 67 6.10 Pureandmixedstates ...... 67

7 The N = 1 case and the Bloch ball 68 7.1 A single qubit and the Bloch ball ...... 69 7.2 A single rebit and the Bloch disc ...... 70

8 Discussion and outlook 71 8.1 An operational alternative to the wave function of the universe ...... 73

Acknowledgements 74

References 74

The first part of the article up to and including ditional postulate eliminates rebits (two-level sys- section 4 includes a substantial amount of con- tems over real Hilbert spaces) in favour of qubit ceptual elaborations. The second part, by con- quantum theory. The rebit case is considered sep- trast, will become more technical upon putting arately in [2]. the novel postulates to use in sections 5-7. The reconstruction for arbitrarily many qubits is per- formed in the companion article [1] where an ad-

Accepted in Quantum 2017-11-27, click title to verify 4 2 Why a(nother) reconstruction of fairly neutral as far as an interpretation is con- quantum theory? cerned. While most of them emphasize the op- erational character of the theory, a particular in- Given that we have a beautifully working theory, terpretation of quantum theory is not strongly one may wonder why one should bother to re- suggested. construct quantum theory from operational state- The language and concepts of the present re- ments. There are various motivations for this en- construction are different. It will emphasize and deavour: concretize the view (or partial interpretation) that quantum theory is a framework governing 1. To equip the standard, physically obscure an observer’s acquisition of information about the textbook axioms for quantum theory with observed system [26, 28, 30–41]. The new postu- an operational sense. In particular, in ad- lates are simple and conceptually comprehensible, dition to its empirical success, a derivation concerning only the relation between an observer from operational statements can conceptu- and the system. Due to the simplicity, the ensu- ally justify the formulation of the theory in ing derivation is mathematically quite elementary terms of Hilbert spaces, complex numbers, and, in contrast to previous derivations, yields tensor product rule for composite systems, the formalism, state spaces and time evolution etc. groups explicitly and in a more constructive man- ner. The disadvantage, compared to other recon- 2. To better understand quantum theory within structions, is that a large number of detailed steps a larger context. By singling out quan- is required. The advantage, on the other hand, is tum theory with operational statements one the simplicity of the principles and mathematical can answer the question “what makes quan- tools, and the fact that the reconstruction affords tum theory special?”, thereby establishing a natural explanations for many quantum phenom- bird’s-eye perspective on the formalism and ena, including entanglement and monogamy, and conceivable alternatives. elucidates the origin of the unitary group.

3. It may help to understand why or why not quantum theory in its present form should 3 Landscape of inference theories be fundamental and thus why it should or should not be modified in view of attempt- The ambition of a (re-)construction of quantum ing to construct fundamental theories. By theory is to derive its formalism, state spaces, dropping or modifying some of its defining time evolution groups and permissible operations physical principles, one obtains a handle for from physical principles – in some rough analogy systematic generalizations of quantum the- to the construction of relativity theory from the ory. This may also be interesting in view of principle of relativity and the equivalence prin- quantum gravity phenomenology (away from ciple. But in order to formulate physical postu- the deep quantum regime). More fundamen- lates, we clearly have to presuppose some mathe- tally, the question arises whether an infor- matical structure within which a precise meaning mational perspective could be beneficial for can be given to them. (For example, also the quantum gravity in general. construction of special and general relativity cer- tainly presupposed a substantial amount of me- 4. The hope has been voiced that a clear inter- chanical structure.) pretation of the theory may finally emerge The procedure is thus to firstly define some from a successful reconstruction, in analogy landscape of theories, which hopefully contains to how the interpretation of special and gen- quantum theory and classical information theory, eral relativity follows naturally from its un- but within which theories are generally not for- derlying principles [28, 31, 36]. mulated in terms of the usual complex Hilbert spaces, tensor product rules, etc. The mathe- It is fair to say that the hope alluded to un- matical formulation of the landscape must there- der point 4 has not been realized thus far because fore be more elementary and, in particular, op- the existing successful reconstructions [14–25] are erational. That is, for the time being, we have

Accepted in Quantum 2017-11-27, click title to verify 5 to forget about the usual – mathematically crisp are necessarily required to be convex to permit but physically rather obscure – textbook axioms convex mixtures of states. (2) A transformation of quantum theory. While different theories will device can perform physical operations on the have the mathematical and physical structure of prepared systems (e.g., a rotation) which may the landscape in common, they may have oth- change the state of the system (e.g., by some erwise very different physical and informational group action on the state vector), but must al- properties; e.g., they may admit much stronger low it to continue its journey to (3), a measure- correlations than quantum theory [52], or weaker ment device, which detects certain experimental correlations as classical probability theory, they outcomes. The measurement devices are math- may allow exotic communication and information ematically described by so-called ‘effects’ which processing tasks to be accomplished [53, 54], and are assumed to be dual to, i.e. linear functionals so on. Secondly, given the language of this land- on, the states. (For the interested reader we note, scape, one can attempt to formulate comprehen- however, that Holevo has shown how the math- sible physical statements which single out quan- ematical incarnation of measurements as linear tum theory from within it. Going to a larger the- functionals on states follows from other simple ory landscape and beyond the languagePSfrag of replacements Hilbert operational assumptions [62].) spaces is precisely what allows us to ask the ques- preparation transformation measurement tion “what makes quantum theory special?” and, ultimately, to find an operational and physical justification for the usual textbook axioms and the standard Hilbert space formulations. systems The goal of this section is precisely to build ‘effects’ such an appropriate landscape of inference theo- dual to states convex state spaces ries both conceptually and mathematically from scratch within which we shall subsequently for- mulate those elementary rules, governing an ob- cbit square bit rebit qubit server’s acquisition of information about a sys- tem, that single out qubit quantum theory. This Figure 1: The standard operational setup of generalized novel landscape of inference theories employs a probabilistic theories with examples of allowed convex different language and is conceptually distinct state spaces of elementary two-level systems. from the by now standard landscape of general- ized probabilistic theories (GPTs) which are com- monly employed for characterizations (or gener- Measurements and states are at the heart of alizations) of quantum theory. For contrast and GPTs; ‘effects’ directly determine the outcome to put the new tools into a larger perspective, we probabilities of measurements and thereby, when begin with a brief synopsis of the GPT language a complete set is engaged, reveal the probabilistic before we establish the landscape of inference the- state of a system. An observer assumes a support- ories underlying this manuscript. ing role, giving intuitive meaning to the notion of preparation, transformation and measurements 3.1 The standard landscape of generalized of systems. The reconstructions of quantum the- probabilistic theories ory within the GPT formalism depart from rather global operational axioms, restricting the sets of It has become a standard in the literature to possible preparations (i.e. state spaces), transfor- employ the formalism of generalized probabilistic mations and measurements. The concrete acqui- theories (GPTs) for operational characterizations sition of information of the observer about the or derivations of quantum theory [14–23, 55–61]. systems is otherwise not accentuated. In par- The setup of GPTs is exclusively operational and ticular, since the outcome probabilities are the one considers three kinds of operation devices (see primary concept of GPTs, these axioms say lit- figure 1): (1) a preparation device which can spit tle about what an observer experiences in indi- out systems in some set of states defined by a vidual experimental runs, but instead focus on vector of probabilities for the outcome of fiducial the totality of a large number of experimental measurements. The state spaces of the systems runs. This has led to a whole wave of successful

Accepted in Quantum 2017-11-27, click title to verify 6 quantum theory reconstructions, employing GPT other systems carrying the information in ques- concepts in one way or another [14–23], most of tion), or possibly the question is not a senseful which, however, are fairly abstract on account of one in the first place. We shall call a question Q the rather global axioms. physically implementable on S if O can acquire a It is perhaps one of the great strengths of GPTs ‘meaningful’ answer from S to Q. An elementary that they constitute a functional and purely op- restriction on Q is that any question Qi from this erational framework which is interpretationally set be implementable on S and that, whenever O fairly neutral. It is not the ambition of this frame- asks Qi to S, S will give an answer to O. Clearly, work to elucidate the measurement problem, to Q depends on S. clarify what happens to a state during a mea- But how does O know whether S will give an surement, what probabilities are or, ultimately, answer and how can he judge whether the latter how to interpret quantum mechanics (except that is ‘meaningful’? This requires O, like any experi- it highlights its operational character). As such, menter, to have developed, from previous experi- this framework is compatible with most interpre- ences, a theoretical model by means of which he tations of quantum theory. describes and interprets his interactions with S. An answer can only be ‘meaningful’ in the con- 3.2 A novel landscape of inference theories text of this model such that our notion of imple- mentability is actually dependent on O’s model. The success of GPTs notwithstanding, we shall We shall come back to this model frequently. now change semantics and perspective to define a new landscape of theories and a novel framework The central ingredients of this framework will for (re)constructing and understanding quantum thus be questions and answers – and O’s informa- theory. Henceforth, we shall fully engage the tion about answer outcomes and their relations. observer and give primacy to his acquisition of This will lead to a novel question calculus from fundamentally limited information from observed which many crucial quantum properties will be systems. This will include being more explicit derived. That is, rather than focusing mostly about what general sort of information will be on probabilistic properties as in GPTs, this novel available to the observer even in individual exper- framework will connect more directly to what can imental runs. Probabilities, on the other hand, be measured in experiments through its empha- can be viewed as secondary and as a consequence sis on questions and their relations. As a conse- of the limited information available to the ob- quence, this framework will also produce opera- server – although, clearly, probabilities will as- tionally more compelling explanations of typical sume a pivotal role too (after all we want to re- quantum phenomena. construct quantum theory). In the sequel, we shall solely speak about the information O has about S and, correspondingly, 3.2.1 Questions and answers about the state that O assigns to S based on this information. Such a state of S is then de- As schematically depicted in figure 2, we shall fined relative to O (whether or not some hidden consider an observer O who can only interact with variables give rise to this state is a question we a system S through interrogation via questions shall not address). The act of information ac- Qi from some set of questions Q which we shall quisition establishes a relation between O and S further constrain below. (At this stage we make and this will be the center of our attention. (A no assumption about whether Q is continuous or priori, a different observer O′ may establish a discrete.) The only information which we allow different relation with S.) Although this frame- O to acquire about S is by asking questions from work will also not give rise to a unique interpre- this set Q. tation, it will support the partial interpretation In principle, of course, O could conceive of and that quantum theory is a law book governing an ask S all kinds of questions, but S could not al- observer’s acquisition of information about phys- ways give a meaningful answer; S may simply ical systems and might therefore be considered not have the desired properties or carry the in- interpretationally less neutral than GPTs. While formation O is inquiring about (e.g., S may not this connects with general ideas underlying, e.g., be complex enough, or S does not interact with the relational [28,29], Brukner-Zeilinger informa-

Accepted in Quantum 2017-11-27, click title to verify 7 Qi?

PSfrag replacements O S

Preparation Interrogation Figure 2: Schematic representation of an observer O interrogating a system S.

tional [31–35], or QBist [36–39] interpretations of we shall permit O to ask S only contains binary quantum mechanics, we emphasize that this re- questions Qi. Any Qi ∈ Q is a non-trivial ques- construction does not rely on them and should tion such that S’s answer (‘yes’ or ‘no’) is not thus not discourage readers uncomfortable with independent of its preparation. Furthermore, any these specific interpretations. Qi ∈ Q is repeatable such that O, by asking the We shall not explicitly deal with transforma- same S the same Qi m times in succession will tion and measurement (‘effect’) devices as in receive m times the same answer. GPTs; instead, these will be replaced more gener- The restriction to elementary ‘yes-no’- ally by time evolution and questions, respectively. questions greatly simplifies the discussion and, The set of all possible (information preserving) ultimately, will give rise to the quantum theory operations that O could perform on S can later of qubit systems. For instance, in quantum be identified with the set of all possible time evo- theory, a binary question could be ‘is the spin of lutions of S. However, in analogy to GPTs, we the qubit up in x-direction?’ However, it will not will assume that O has access to some method of be too difficult to generalize Q to also consist of preparing S in different ways such that S’s an- ternary, quaternary, quinary, etc. questions, but swers to O’s questions depend on the precise way we shall not attempt to do so here. Since trivial of preparation. (O could either control himself a questions are not considered, we already see preparation device or have a distinct observer O′ that not even all implementable questions will prepare systems for him.) be taken into account. We shall impose further When constructing the new landscape L of the- restrictions on Q such that, ultimately, it will be ories describing O’s acquisition of information a strict subset of all possible binary questions about S within which we shall later, in section 4, which O could, in principle, ask S. formulate our postulates, we will make a number Since this is an operational approach, it is fair of restrictions and assumptions. In order to fa- to assume that O can record the answers to his cilitate future generalizations and improvements questions asked to any system (e.g., by writing of the present construction of quantum theory – them on a piece of paper) and that he can do which constitutes a proof of principle – we shall statistics over the outcomes (e.g., by counting the attempt to be as clear as possible about the as- frequency of outcomes). We shall require that sumptions made throughout this work. every possible way of preparing S will give rise As a starter, we would like to keep Q as sim- to a particular statistics over the answers to all ple as possible, while still having non-trivial ques- Q ∈ Q; O could test these statistics by interro- tions. In particular, we do not wish to consider i gating a large number n of identically prepared trivial propositions which are always true or al- systems1 S , a = 1,...,n, sufficiently often with ways false. We shall therefore assume the follow- a 1 ing. Given the present structure, two systems S1 and S2 could only be considered as distinct in nature if either the Assumption 1. The set of questions Q which (maximal) set of questions Q which O can ask the sys-

Accepted in Quantum 2017-11-27, click title to verify 8 (at least ideally) all Qi ∈ Q. In fact, this is pre- lently as the set of all possible answer statistics cisely how O will operationally distinguish differ- or as the set of all (distinct) preparations. ent preparations of systems. Based on this notion, we shall now identify By having interrogated, in this manner, the probabilities as degrees of belief. The ‘knowledge’ n always identically prepared Sa for all possible of what Σ is for a given S will permit O to assign ways of preparation, we shall assume O to have probabilities to the outcomes of his questions. It gained a ‘stable’ knowledge of the set Σ of all pos- is very natural for O to assign probabilities to sible answer statistics for S over Q, i.e., the an- questions because he deals with statistical fluctu- swer statistics for all Q ∈ Q and all possible ways ations and furthermore, as we shall see later, with of preparing a system S.2 While ideally n →∞ is systems about which he always has incomplete in- necessary, ‘practically’ n should be large enough formation in the sense that the corresponding Σ for O to develop a theoretical model of both Q is such that he can never know the answers to all and Σ up to some accuracy which agrees with Qi ∈ Q at the same time. his observations. It is not our ambition to clarify More precisely, for a specific S and any Qi ∈ Q further what n is nor how precisely O has de- that he may ask the system next, O can assign a veloped his theoretical model, instead, we shall probability yi that the answer will be ‘yes’ (or a henceforth just assume that O has puzzled out probability ni that the answer will be ‘no’), ac- the pair (Q, Σ). As any experimenter in an ac- cording to tual laboratory, O shall interpret the outcomes of his interrogations by means of his model for (i) O’s knowledge of Σ, and (Q, Σ) and he can decide whether a given ques- (ii) any prior information that O may have about tion is contained in the set Q or not. Henceforth, the specific S. we assume that O only asks questions from Q. Our task will be to establish what this model is, Given our setup, the only prior information that based on the ensuing assumptions and postulates. O may have about the particular S (apart from what the associated Σ and Q are) must result ei- 3.2.2 From information to probabilities: the state ther from having interrogated some ensemble of of S relative to O systems identically prepared to S with some sub- set of Q beforehand3 and from the corresponding The previous subsection did not refer to the no- accumulated statistics of the asked Qj ∈ Q (e.g., tion of probabilities. But it defined two prepa- that O may have recorded on a piece of paper), rations as being identical if they give identical or from any other prior information about the statistics. This permits us to regard Σ equiva- method of preparation. In particular, this previ- ous ensemble could have been empty. tems or the totality of answer statistics for all possible For instance, if every time that O asked the preparations were distinct for S1 and S2 (if always the specific Qi to any of the identically prepared sys- same two S1,S2 are interrogated and thereafter freshly prepared again). If O can not distinguish S1 and S2 in this tems gave a ‘yes’ answer before, he will assign way for sufficiently many trials (ideally infinitely many), the prior probability yi = 1 to Qi and to the next we shall call them identical. Let S1 and S2 be identi- identically prepared S that he will interrogate. If, cal and O prepare both systems with the same procedure on the other hand, the number of ‘yes’ and ‘no’ (for instance, the setting of the preparation device is the same for both systems). If the answer statistics (frequen- answers to some other Qj was equal for the previ- cies) for S1,S2 for all Qi ∈Q become indistinguishable af- ously identically prepared systems, O will assign, ter sufficiently many trials (of preparing and interrogating 1 as a best guess, the prior probability yj = 2 to the same systems with the same procedure), we consider this Qj and to the next S. Similarly, for any these systems as ‘identically prepared’. (After completion of this work, the author was made aware that this notion other answer statistics, O would assign yi to the is similar to the definition of ‘operational equivalence’ put next S depending on the recorded frequencies of forward in [63].) ‘yes’ answers. But, thanks to his knowledge of 2We assume the preparation method to be ideal in the Σ and therefore of any possible relations in the sense that it accounts for any possible answer statistics which S can admit in O’s world for questions in Q. That 3If O asks more than one question to any S, the order- is, there do not exist other methods which can prepare S ing of the questions may matter. But then O could ask in ways that O’s method does not encompass. the questions for any S always in the same order.

Accepted in Quantum 2017-11-27, click title to verify 9 answer statistics, O can also assign prior proba- Σ of all possible answer statistics on Q which S bilities yk to questions Qk that he did not ask the admits is the state space of S. previous set of identically prepared systems. For example, Σ may be such that whenever S gives Of course, ultimately not all yi will be inde- pendent such that the full collection of probabili- a ‘yes’ answer to Qi, it will give a ‘no’ answer to ties will yield a redundant parametrization of the an immediately following Qk. Accordingly, if O state. However, this is not important for the mo- assigns a prior probability yi = 1 to Qi as above, ment and we shall come back to this shortly. he will also assign a prior yk = 0 to Qk without This definition of the state of a system S ex- previously having asked Qk. But other relations between questions will be permitted too. In par- plicitly identifies it with the ‘state of informa- ticular, it may be that the information gained tion’ that O has acquired about S; O assigns from the questions he previously asked the iden- this state to S according to his information about tically prepared systems and the structure of Σ the Qi ∈ Q. As such, the state of system S is epistemic (i.e. a ‘state of knowledge’) and a make it equally likely that the answer to Qk asked to the next S will be ‘yes’ or ‘no’. In this case, priori only meaningful relative to the observer 1 O. The interpretation of the quantum state as O will assign yk = 2 to Qk that he may ask the next S. This will become more precise along the a ‘state of information’ is certainly not new and way. has been proposed in various ways before (see We therefore take a broadly4 Bayesian perspec- also, e.g., [36–38, 40]). However, the above def- tive on probabilities: O assigns probabilities to inition is closest in spirit to the ideas underly- questions according to his ‘degree of belief’ about ing Relational Quantum Mechanics [28, 29] and the Zeilinger-Brukner interpretation [15, 31–34] S. These probabilities yi are thereby relative to the observer O. A different observer O′ may have and thereby generalizes them to the landscape of different information about S and thereby assign theories describing O’s acquisition of information different probabilities to the various outcomes of which we are in the process to establish. questions posed to S (for a discussion, within While the state of S is thus a priori only mean- quantum theory, of the consistency of different ingful relative to O, we emphasize that both the observers having different information about a set of questions Q which O may ask and the state system, see [28, 29, 64, 65]). E.g., O′ could be the space Σ are to be intrinsic to the system S. Oth- one preparing S. She could ‘know’ the statistics erwise, it would be difficult for two observers to for the specific preparation setting (from previous agree on the description of a given S. tests) and then send O the specifically prepared S without informing him about her knowledge. 3.2.3 ‘Belief’ updating and ‘collapse’ of the state For consistency, we tacitly assume the set Σ of At this stage it is important to distinguish single all possible answer statistics to coincide with the from multiple shot interrogations. Ina set of possible ‘beliefs’. That is, to every equiva- lence class of preparations of S (with ‘identically single shot interrogation O interrogates a prepared’ defining the equivalence relation) there single system S, in some prior state, with a number of questions from Q without in- shall correspond a unique ‘belief’ {yi}Qi∈Q and vice versa. Since the only way for O to acquire in- termediate re-preparations of S. The defi- formation about S is by interrogation with ques- nite answers to these questions give O def- tions in Q, the prior probabilities yi that O as- inite information about this specific S after signs to every Qi ∈ Q encode the entire informa- the interrogation. Furthermore, his knowl- tion that O has about S. Hence, we shall make edge of Σ and any prior knowledge of S the following identification. (acquired through previous interrogations of identically prepared systems) give him statis- Definition 3.1. (State of S relative to O) tical information about any questions he did The collection of all probabilities y ∀ Q ∈ Q is i i not ask S. In conjunction, the new answers the state of S relative to O. Accordingly, the set and his prior knowledge thus determine the 4We add the qualifier ‘broadly’ here since we also allow state of S after the interrogation. This will for the typical laboratory situation of ensembles of systems constitute a posterior state update rule, and (which may or may not contain more than one element). thereby a ‘belief’ update form a prior to a

Accepted in Quantum 2017-11-27, click title to verify 10 posterior state for a single system which we ensemble state such that his assignments of shall turn to shortly (and which clearly de- the yi will fluctuate less the larger the num- pends on the specific way O interrogates S). ber of interrogated systems. This process This posterior state of S will reflect O’s defi- gives rise to an ensemble state updating. In- nite information about every asked question dependent of this updating, the prior state Qi by featuring either yi = 0 or yi = 1 due to that O assigns to any individual system may repeatability, depending on whether the an- experience a ‘collapse’ during the interroga- swer was ‘no’ or ‘yes’, respectively (assuming tion of that specific system because his infor- for now, of course, that Σ is such that the mation about the specific system may have answers to the selection of questions that O changed. Accordingly, O will have to distin- asked can be known simultaneously). guish the ensemble state from the posterior state of any system in the ensemble. But the If this posterior state does not coincide with collection of posterior states determines the the prior state that O assigned to S before ensemble state. the interrogation, based on his prior infor- mation about S, then S’s state has ‘col- lapsed’ relative to O during the interroga- 3.2.4 Elementary structure on Σ and Q tion. Hence, a state ‘collapse’ only occurs if O’s posterior information about S does The present structure is still too rudimentary for not coincide with his prior information about a quantum (re)construction. We therefore need S, i.e. if O experienced an information gain more. about S via the interrogation. We shall Firstly, as in GPTs we will need O to be able to therefore view a state ‘collapse’ as O’s infor- assign a single prior state to any pair of identical mation gain about this specific S rather than systems whenever he flips a biased coin in order a ‘disturbance’5 of S (we refer the reader also to decide which of the two systems he will in- to [26, 60, 66, 67] for a related discussion). terrogate. (Equivalently, the preparation method could involve a biased coin toss, the outcome of multiple shot interrogation O interrogates which determines the preparation setting.) That an ensemble of identically prepared systems is, O will be permitted to build convex combina- Sa, a = 1,...,n, where the interrogation tions of states. of every Sa is a single shot interrogation. O will carry out such a multiple shot Assumption 2. The state space Σ of S is a interrogation to do state tomography, i.e. closed convex set. to estimate the state of the ensemble {Sa} Closure of Σ is assumed because points lying in for the specific setting of preparation or, in the boundary of Σ can be arbitrarily well approx- other words, the state of any of the systems imated by states in the interior. Perfect prepa- prior to being interrogated by O. ration and arbitrarily good preparation are op- This will also be a ‘belief’ updating, how- erationally indistinguishable and so adding the ever, a prior (or ensemble) state updating. boundary does not change the physical predic- After every interrogation of a system in the tions [16]. ensemble, O will assign probabilities yi to the Next, we need to define some elementary struc- Qi in the manner described above. This will ture on Q in order to meaningfully speak about then define the prior state of the next sys- relations among questions (and answers). To this tem in the ensemble to be interrogated. By end, we shall establish additional structure on interrogating more and more systems, O will the pair (Q, Σ). We declared in assumption 1 gain more and more information about the that there are to be no trivial questions in Q the answers to which would be independent of S’s 5 A ‘disturbance’ of the system S is only meaningful if preparation. We also insisted before that O will there was an underlying ontic state (i.e. ‘state of reality’) to which, however, O would have no access. Here we shall distinguish the different ways of preparing a sys- merely speak about the information that O has access to tem S – and thus the different states he can assign and therefore not make any ontic statements, regarding – by the particular answer statistics. We shall them as excess baggage for our purposes. now strengthen these requirements, by asserting

Accepted in Quantum 2017-11-27, click title to verify 11 that there exists a distinguished state of ‘no infor- Qi should satisfy mation’, corresponding to the situation that O’s 0 bit ≤ αi ≤ 1 bit. (1) prior knowledge about S makes it equally likely for him that the answer to any Q ∈ Q is ‘yes’ or We shall not propose an explicit information mea- 6 ‘no’. We note that this assumption is a restric- sure αi (as a function on Σ) here because this tion on O’s model for the pair (Q, Σ).7 must follow from the rules on information acqui- sition postulated below and we shall indeed derive Assumption 3. There exists a special state in it therefrom later in section 6.8. Until then it will Σ, called the state of no information, which is be sufficient to work with this implicit notion of 1 given by yi = 2 , ∀ Qi ∈ Q. quantifying O’s information about any Qi. But the questions O can ask, and the informa- This state shall be the prior state that O as- tion that the corresponding answers define, may signs to S (or an ensemble thereof) in a belief not be independent. However, a priori the no- updating whenever he has ‘no prior information’. tion of independence of questions, in the sense of For example, a distinct observer O′ may prepare stochastic independence, is state dependent. For a system S and send it to O in such a way that example, for a pair of qubits in quantum theory the latter knows only the associated Σ but not the questions Q1, ‘is the spin of qubit 1 up in x- the preparation setting. In this case, as a prior, direction?’, and Q2, ‘is the spin of qubit 2 up in O will assign the state of no information to S, x-direction?’, are stochastically independent rel- 1 i.e. yi = 2 , ∀ Qi ∈ Q. Similarly, there will exist ative to the completely mixed state, but fully de- a special preparation setting which is such that a pendent relative to an entangled state (with cor- multiple shot interrogation on an ensemble pre- relation in x-direction). As a result of the state pared in this setting will give totally random an- dependence, the independence of questions may swers to O such that he will assign the state of no also be viewed as observer dependent. For in- information to the ensemble. But note that the stance, in quantum theory an observer O′ could state of any individual system after the interroga- send O an entangled pure state (with correlation tion of that system will not be the state of no in- in x-direction) and refuse to tell O which state it ′ formation because, through the interrogation, O is. Relative to O , Q1 and Q2 will be dependent, will have acquired information about that specific but they will be independent relative to O be- system (see the discussion of the state ‘collapse’ cause the latter will assign the completely mixed above). state to the pair prior to measurement. Given the state of no information, we shall pre- Since the notion of independence of questions liminarily quantify the amount of information αi is state dependent, we need a distinguished state that the definite answer to any Qi ∈ Q defines as in order to unambiguously define it – this is the 1 one bit. Similarly, whenever yi = 2 , as in the second purpose of assumption 3 and brings us in state of no information, we shall say that O has contact with a state update rule. Indeed, suppose αi = 0 bits of information about Qi. In general, O acquires S in the state of no information and under the premise that information can neither poses the question Qi ∈ Q. By assumption 1, be negative nor complex, O’s information about any Qi ∈ Q is repeatable such that, upon receiv- ing either the answer ‘yes’ or ‘no’, O will assign 6 We emphasize that GPTs are more general by, in prin- yi = 1 or yi = 0, respectively, as the probability ciple, permitting state spaces which do not contain such for S giving the answer ‘yes’ if posing Q again. a distinguished state. However, most operationally inter- i esting GPTs do possess such a state. The posterior state update rule, which enables O to update his information about a specific S in 7For instance, we note that on account of the exis- tence of binary POVMs with an inherent bias, such as compliance with the given answers, must respect (E0 = 2/3 · 1,E1 = 1/3 · 1), the pair given by Q = this repeatability. It depends on the details of this {binary POVMs} and Σ = {unit trace density matrices} update rule what the probabilities yj for all other cannot satisfy this condition because no unit trace Qj ∈ Q is after having asked only Qi. Rather density matrix exists which yields probability 1/2 for than fully specifying at this stage what this rule (E0,E1). This pair will therefore ultimately not be the solution of this reconstruction. However, a subset of is, we shall simply assume that O employs one {binary POVMs} together with the full quantum state which is consistent. Whatever this posterior state space will be the solution. update rule, we shall refer to Qi,Qj ∈ Q as

Accepted in Quantum 2017-11-27, click title to verify 12 independent if, after having asked Qi to S in gives O partial information about the answer the state of no information, the probability to Qj. Again, this relation is required to be 1 yj = 2 . That is, if the answer to Qi relative symmetric. This is equivalent to saying that to the state of no information tells O ‘noth- the joint probabilities p(Qi,Qj) relative to ing’ about the answer to Qj. We shall require the state of no information do not factorize this relation to be symmetric, i.e., Qi is in- and the answer to one question does not fully dependent of Qj if and only if Qj is indepen- imply the answer to the other. 8 dent of Qi. This is equivalent to saying that Qi,Qj are stochastically independent with We shall henceforth assume that Σ is such that respect to the state of no information, i.e. the two questions Qi,Qj which are fully or partially joint probabilities factorize relative to the dependent relative to the state of no information ∗ ∗ 1 1 1 are also fully or partially dependent relative to latter, p(Qi,Qj)= yi · yj = 2 · 2 = 4 . Here p(Qi,Qj) = p(Qj,Qi) denotes the probabil- any other state in Σ. These definitions of (in- ity that Qi and Qj give ‘yes’ answers if asked )dependence depend a priori on the update rule in sequence to the same S arriving in the through the joint probabilities. For our purposes ∗ ∗ it will turn out not to be necessary to fully specify state of no information and yi ,yj are the individual ‘yes’-probabilities in this distin- this update rule. guished state.9 Next, we need a notion of compatibility and complementarity. Qi,Qj are dependent if, after having asked Qi to S in the state of no information, the probability maximally compatible if O may know the an- yj = 0, 1. That is, if the answer to Qi rel- swers to both Qi,Qj simultaneously, i.e. if ative to the state of no information implies there exists a state in Σ such that yi,yj can also the answer to Qj. Again, we require this be simultaneously 0 or 1. relation to be symmetric. This is equivalent maximally complementary if maximal infor- to saying that, relative to the state of no in- mation about the answer to Qi forbids O to formation, Qi,Qj are stochastically fully de- ∗ ∗ have any information about the answer to Qj pendent as either p(Qi,Qj) = yi = yj = 1 ∗ ∗ at the same time (and vice versa). That is, 2 = p(¬Qi, ¬Qj) or p(Qi, ¬Qj)= yi = yj = 1 every state in Σ which features yi = 0, 1 will = p(¬Qi,Qj), where ¬Q is the negation of 1 2 necessarily have yj = (and vice versa). Q.10 2 Consequently, complementary questions are in- partially dependent if, after having asked Qi dependent, but independent questions are not to S in the state of no information, the prob- 1 necessarily complementary. Finally, Qi,Qj are ability yj 6= 0, 2 , 1. That is, if the answer partially compatible (or complementary) if max- to Qi relative to the state of no information imal information about one precludes maximal,

8 but permits non-maximal information about the It should be noted that, while this is true for projec- other. tive measurements in quantum theory, it does not hold for generalized measurements. I thank Tobias Fritz for This permits us to further constrain the up- pointing this out. date rule. Firstly, maximal complementarity has 9We emphasize that this is a definition of maximal inde- an obvious consequence for an update rule. Sec- pendence. For example, for a qubit two linearly indepen- ondly, we shall assume the following. dent directions ~n1, ~n2 in the Bloch sphere with ~n1 · ~n2 6= 0 would define two spin observables ~n1 · ~σ and ~n2 · ~σ whose Assumption 4. If Qi,Qj are maximally com- corresponding projectors would not be maximally inde- patible and independent then asking either shall pendent according to this definition. The corresponding questions would be partially dependent (see below) be- not change O’s information about the other, re- cause whenever the observer knows the answer to one gardless of S’s state. That is, asking Qi must question, the probability for a ‘yes’ answer to the other leave yj invariant – and vice versa for i, j inter- would be distinct from 1 . 2 changed. 10The most trivial example of a pair of dependent ques- tions are clearly Q and ¬Q. But there exist less trivial This is to prevent O from losing or gaining in- ones. E.g., see theorem 5.7 below. formation about some question by asking another

Accepted in Quantum 2017-11-27, click title to verify 13 question which is compatible with but indepen- the same?’. Do we allow Qij to also be imple- dent of the first. These constraints on the update mentable on S? Clearly, if Qi,Qj are maximally rule turn out to be sufficient for our reconstruc- compatible, then Qij is implementable because O tion. can always find the answer to the latter by asking Assumption 4 leads to a first result which we Qi,Qj. In this case, since Qi,Qj are simultane- shall use at times: it implies what sometimes is ously defined relative to O, we can also write referred to as ‘Specker’s principle’ [68] (see also [69–72]). We shall call Q1,...,Qn ∈ Q mutually Qij := Qi ↔ Qj, (2) maximally compatible if there exists a state of S where ↔ is the logical biconditional or where the answers to all of Q1,...,Qn are known 11 simultaneously to O. (XNOR). Qij will then automatically be maxi- mally compatible with both Qi,Qj. Theorem 3.2. (‘Specker’s principle’) If Let us now contrast this with the situation in Q1,...,Qn ∈ Q are pairwise maximally compat- which Qi,Qj are (partially or maximally) comple- ible and pairwise independent then they are also mentary. The structure introduced thus far does mutually maximally compatible. not preclude the ‘correlation question’ Qij to also be implementable and, ultimately, contained in Q Proof. Let Q1,...,Qn ∈ Q be pairwise indepen- even if Qi,Qj are complementary. In fact, it is not dent and pairwise maximally compatible. On ac- possible to decide without a theoretical model for count of repeatability, after asking S the ques- describing and interpreting (Q, Σ) whether Qij is tion Q1, O will assign the probability y1 to be implementable on S in the case that Qi,Qj are at either 0 or 1, depending on the answer. O may least partially complementary. What, however, is subsequently ask S the question Q2 upon which unambiguously clear is that for all practical pur- the probability y2 will likewise be either 0 or 1. poses Qij will not be implementable in this case. According to assumption 4, this will not change Namely, Qij would be a statement about the cor- y1 because, by assumption, both are maximally relation of two complementary questions Qi,Qj compatible and independent. By the same ar- and O could say ‘the answers are the same’, but gument, upon next asking and receiving an an- he can never directly test them individually and swer to Q3, O will assign either y3 = 1 or see that they are actually ‘the same’ at the same y3 = 0, while both y1,y2 are unchanged. Re- time. Indeed, Qi,Qj,Qij would need to form peating this argument recursively, it is clear that a mutually (partially or maximally) complemen- O can thereby generate a state of S in which all tary set. Thus, from a purely operational per- yi, i = 1,...,n, are either 0 or 1 which represents spective alone, O could never tell, even in princi- a state of S where O knows the answers to all of ple, that Qij was implementable; he simply can- Q1,...,Qn. not get an answer to Qij which he could inter- Notice that this property is satisfied for classi- pret, based on operationally accessible informa- cal bit theory and in quantum theory for projec- tion alone, as a correlation of the complementary tive, however, not for generalized measurements. Qi,Qj. It thus depends on the model for (Q, Σ) whether Q is implementable when Q ,Q are 3.2.5 Logical compositions of questions and rules ij i j complementary. Since it is impossible for O to of inference settle this question operationally, it would re- Given multiple questions, nothing, in principle, quire a model which employs hidden and opera- stops O from composing them via logical con- tionally inaccessible ontic information and which nectives to all kinds of propositions which them- is devoid of complementarity at an ontic level selves constitute questions. The issue is, however, to conclude that Qij ∈ Q in this case. For ex- whether he will be able to get an answer from S to ample, Spekkens’ elegant toy model [40] and the such a question, i.e. whether it is implementable ‘black boxes’ of [54] employ ontic states, satisfy on S and thus whether it could be contained in the structure established thus far (at least at the Q (see section 3.2.1). For example, given any two 11 questions Qi,Qj ∈ Q, O could consider the cor- That is Qij = ‘yes’ if Qi = Qj = ‘yes’ or ‘no’ and Qij = ‘no’ otherwise. relation question Qij, ‘are the answers to Qi,Qj

Accepted in Quantum 2017-11-27, click title to verify 14 epistemic level and modulo restrictions on the no- We recall from section 3.2.1 that O’s theoretical tion of convexity), and explicitly feature such a model for Q only contains implementable ques- triple of questions.12 However, such a model is tions (but not necessarily all implementable ques- in conflict with our premise of following a purely tions). operational approach which only speaks about in- Since ∗ is only applied to connect maximally formation that O has access to via direct inter- compatible questions which have also opera- rogation. The model by means of which O in- tionally simultaneous truth values relative to O, terprets the answers that he gets from S – and, we shall allow ∗ to be any of the 16 binary connec- hence, the information he can acquire about S – tives (or binary Boolean functions) ¬, ∨, ∧, ↔,... shall be based entirely on operational statements of classical Boolean or propositional logic. that O can, in principle, check through interroga- We stress that assumption 5 is a restriction on tion, and not on propositions that require hidden O’s model for Q; this assumption clearly does and inaccessible ontic information. Accordingly, not rule out that there may exist other (namely, O’s theoretical model for Q should not contain ontic) models which also yield a consistent de- (a question which is logically equivalent to) Qij scription of O’s experiences with the systems he whenever Qi,Qj are at least partially complemen- is interrogating, yet which ascribe Q ∗ Q to be tary. i j (logically equivalent to a question) contained in Of course, the ‘correlation’ (represented by the Q regardless of whether Q ,Q are compatible XNOR connective ↔) is only one of many possi- i j or not. However, we shall not worry about such ble logical connectives. More generally, we shall models here. require that O’s model for Q does not contain any logical connectives of complementary questions. We note that assumption 5 does not only ap- He can only logically connect questions which are ply to logical connectives of two questions, but maximally compatible – and thus are simultane- arbitrarily many. For example, if Qi ∗ Qj is im- ously defined with respect to him – such that he plementable (and contained in Q), according to could meaningfully write down a truth table for O’s model, then (Qi ∗ Qj) ∗ Qk is implementable the questions to be connected and the connective too iff Qi ∗ Qj and Qk ∈ Q are compatible, and question. If he cannot connect questions, he can so on. It is also important to note that assump- also not ask for the connective. tion 5 is a statement about which questions can be directly connected logically. It does not entail Assumption 5. Let Qi,Qj ∈ Q and ∗ be a logi- that O, according to this model, can only form cal connective. According to O’s theoretical model meaningful logical expressions containing exclu- for Q, Qi ∗ Qj is implementable if and only if sively questions which are mutually maximally Qi,Qj are maximally compatible. compatible. Namely, questions which are com-

12 patible might be logically further decomposable The reader familiar with Spekkens’ toy model [40] will recall the simplest (epistemic) 1-bit system which has four into other questions for which different compat- ontic states ‘1’, ‘2’, ‘3’ and ‘4’. An epistemic restriction ibility relations hold. For instance, it might be forbids an observer to know the ontic state. Instead, the that Qk in the expression (Qi ∗ Qj) ∗ Qk is only epistemic states of maximal knowledge correspond to ei- compatible with Q ∗Q , yet not with Q or Q in- ther of the following three questions (and their negations) i j i j dividually. In this case, there is no harm in never-

Q1 : “1 ∨ 2” ,Q2 : “2 ∨ 3” , theless forming the expression (Qi ∗Qj)∗Qk even though, say, Qj and Qk might be complemen- Q3 : “2 ∨ 4” , tary such that O cannot write down a truth table where ∨ is to be read as ‘or’. Q1,Q2,Q3 are mutually with all three Qi,Qj,Qk separately. However, it complementary and it can be easily checked that Q coin- 3 is clear that in this case there is a hierarchy in cides with the ‘correlation’ Q12 of Q1 and Q2: it gives ‘yes’ when the (ontic) answers to Q1,Q2 are equal and ‘no’ oth- which the logical connectives are to be executed. erwise. This relation is cyclic: Q1 is also the ‘correlation’ More specifically, the connective ∗ in (Qi∗Qj)∗Qk Q of Q ,Q and Q is the ‘correlation’ Q of Q ,Q . 23 2 3 2 13 1 3 cannot be associative since Qi ∗ (Qj ∗ Qk) is not Accordingly, there are three complementary questions for this 1-bit system, in principle the correct number for a implementable according to O’s model because qubit. Later in section 5.2.3, we will develop a different Qj,Qk are, by assumption, complementary and approach, without ontic states, in order to reason for the can therefore not be directly connected. That is three-dimensionality of the Bloch-sphere. to say, the left ∗ in (Qi∗Qj)∗Qk must be executed

Accepted in Quantum 2017-11-27, click title to verify 15 first and Qk can only be logically connected with appropriate in such cases and we shall demand the result. Thus, assumption 5 does admit logical more precisely the following. expressions containing also mutually complemen- tary questions as long as the latter are not directly Assumption 6. O’s model for Q allows him to connected through a logical connective. What as- apply exclusively classical rules of inference, ac- sumption 5 entails is that the ordering of the exe- cording to the rules of Boolean logic, to (trans- cution of the various logical connectives in a com- form or evaluate) any logical expression (or position has to respect a hierarchy which ensures subexpression of a larger expression) which is that at every step two compatible subexpressions composed purely of mutually maximally compati- are connected. This will become important later ble questions. This holds regardless of whether in the reconstruction where we shall encounter the mutually compatible questions can be logi- concrete such examples. cally decomposed into other questions which fea- ture different compatibility relations. In all other This brings us to the rules of inference by cases, no classical rules of inference (and thus no means of which O may transform or evaluate classical logical identities) are valid. logical expressions of questions and derive logi- cal identities, e.g. to establish possible logical re- In consequence, O can take any set of mutually lations among various question outcomes. The compatible questions and treat them, with clas- state update rules need, in particular, respect sical logic, as a Boolean algebra. The assump- these rules of inference. Again, the rules of in- tion states what is possible for compositions of ference which O may employ depend on his the- compatible questions and what is not possible for oretical model for (Q, Σ). For instance, in ontic compositions involving also complementary ques- models all questions will typically have simultane- tions. This will become useful later in section ous values, in contrast to purely operational ones, 5.2.7 where we shall also see what is possible for which will require different rules of inference. compositions of complementary questions. In line with our operational premise, we shall We emphasize that assumption 6 by itself does require that the rules of inference of O’s model not severely constrain the nature of any conceiv- must not only be consistent with O’s experiences able ‘hidden variable model’ which could also con- with the systems he is interrogating, but also sistently describe O’s world. However, in con- testable. In particular, any logical identities de- junction with two of the subsequent quantum rived from these rules must be testable through principles, it will rule out local ‘hidden variables’ interrogations. It is thus appropriate to call them in section 5.2.7. operational rules of inference. 3.2.6 Parametrization of S’s state and tomogra- Classical rules of inference require that the phy questions or propositions in a logical expression have truth values simultaneously. In O’s model Now that we have a notion of independence on Q any truth value must be operational such that we can say more about the parametrization and only mutually maximally compatible questions thus representation of S’s state relative to O. Not have simultaneous meaning. Nothing stops O all Qi ∈ Q will be necessary to describe the state; from applying classical rules to such kind of ques- the pairwise independent questions shall be the tions. We shall thus require that it is appropri- fundamental building blocks of the landscape L ate for O to employ classical rules of inference, of inference theories. according to Boolean logic, – and only those – Suppose there is a maximal set QM = for any logical expression or subexpression which {Q1,...,QD ∈ Q} of D pairwise independent only contains questions that are mutually maxi- (but not necessarily compatible) questions, such mally compatible. In all other cases (which still that no further question Q ∈ Q\QM exists which must abide by assumption 5), any possible rule of is pairwise independent from all members of QM inference transforming the composition of ques- too. Then, every other Q ∈ Q\QM is either tions will directly involve mutually complemen- (i) dependent on exactly one Qj ∈ QM and in- tary questions whose truth values, however, have dependent of all other QM ∋ Ql 6= Qj (if Q no simultaneous operational meaning to O. Ac- was dependent on Qj ∈ QM and partially de- cordingly, classical rules will be operationally in- pendent on QM ∋ Ql 6= Qj, then Qj,Ql could

Accepted in Quantum 2017-11-27, click title to verify 16 not be independent), (ii) partially dependent on Proof. Consider any QM . On account of pairwise some and independent of the other questions in independence, for each Qi ∈ QM there must exist QM , or (iii) partially dependent on all Qj ∈ QM . a state such that O only knows the answer to While O will not be able to infer maximal infor- this Qi with certainty, but nothing about any mation about the answers to questions of cases other Qj6=i ∈ QM , i.e. yi = 0 or 1 and all other (ii) and (iii) from his information about individ- yj6=i = 1/2. (Namely, O could have asked Qi to ual (or even subsets of) members of QM alone, S prepared in the state of no information.) But the question is whether his information about the since also in the state of no information yj6=i = full set QM will be sufficient to do so. 1/2, yi cannot in general be computed from the yj6=i. Hence, given an informationally complete Definition 3.3. (Informational Complete- D set QM , all associated probabilities {yi}i=1 are ness) A maximal set QM = {Q1,...,QD ∈ Q} necessary to parametrize the full state space Σ. of pairwise independent questions is said to be in- Now Σ is a convex set (see assumption 2). formationally complete if O’s information about Given the lack of redundancy in {yi}, a given the questions in Q determines his information M informationally complete QM with D elements about all other Q ∈ Q\QM in such a way therefore encodes Σ as a D-dimensional convex that the probabilities yi which O assigns to every RD ′ subregion of . Suppose there is a second QM Qi ∈ QM are sufficient in order for him to com- ′ ′ with D elements. In terms of QM , Σ would be pute the probabilities yj ∀ Qj ∈ Q for all prepa- described as a D′-dimensional convex subregion ′ rations of S. In this case, the set of probabilities of RD . But clearly, these two descriptions of Σ D {yi}i=1 of the Qi ∈ QM parametrizes the state must be isomorphic which, for D finite, is only that O assigns to S and thereby yields a com- possible if D ≡ D′. plete description of the state space Σ. We shall call D the dimension of Σ. Finiteness of D will later be established in sec- tion 5 with the help of the principles. We emphasize that the dimension D of Σ will ultimately not be the Hilbert space dimension in Any such QM establishes a question reference quantum theory, but the dimension of the set of frame on Q (see also [73]) and thereby also a ‘co- density matrices. ordinate system’ on Σ. In particular, in order to do state tomography with a multiple shot in- If QM was not informationally complete, O terrogation, as outlined in section 3.2.3, it will would require further questions that are partially be sufficient for O to interrogate an ensemble of dependent on at least some of the elements in identically prepared S with the questions within QM in order to fully describe the system S and its state. This situation cannot be precluded, a given QM only. Given a specific QM , there given the structure we have devised so far. How- are now three equivalent ways for O to describe S’s state: he could represent it by either the D- ever, we deem it undesirable, given that we would dimensional yes- or no-vector like to employ pairwise independent questions as building blocks for system descriptions. We shall y1 n1 therefore require that no more independent infor- y2 n2 mation about S can be learned from any question ~y :=   , ~n :=   , (3) O→S . O→S . in addition to a maximal set Q .  .   .  M      yD   nD      Assumption 7. Every maximal set QM of pair-     wise independent questions is informationally of probabilities yi and ni, i = 1,...,D, that the complete. answers to question Qi ∈ QM are ‘yes’ and ‘no’, respectively. That is, There may exist (even continuously) many such informationally complete sets of questions on Q ~yO→S + ~nO→S = p~1, (4) which, at this stage, may still be either discrete or continuous. However, their dimensions are equal. where ~1 is a D-dimensional vector with a 1 in each of its entries. Here we have introduced a new pa- Lemma 3.4. The dimensions of all maximal sets rameter: p is the probability that S is present QM on Q are equal (if finite). at all. (For example, the method of preparation

Accepted in Quantum 2017-11-27, click title to verify 17 could be such that O cannot always tell with cer- which can be posed to the composite system con- tainty when a system is spit out by the prepara- tains all questions about the individual subsys- tion device.) This probability will rescale all yi tems and that the remaining questions are lit- and ni simultaneously. However, it only becomes erally composed of these individual questions or relevant in subsection 3.2.8 and otherwise is usu- iteratively composed of compositions of them.13 ally taken to be p = 1. Evidently, the assignment Definition 3.5. (Composite System) Let of which answer to Qi is ‘yes’ and which is ‘no’ is arbitrary, but any consistent such assignment is QA, QB denote the full question sets associated fine for us. to SA,SB, respectively. Two systems SA,SB are But he could also represent the state redun- said to form a composite system SAB if any Q ∈ Q is maximally compatible with and in- dantly as a 2D-dimensional vector a A dependent of any Qb ∈ QB and if ~y ˜ P~O→S = (5) QAB = QA ∪ QB ∪ QAB, (7) ~n ! where Q˜AB only contains composite ques- which will turn out to be convenient especially tions which are iterative compositions, Qa ∗1 when p < 1. We shall write a state with a sub- Qb,Qa ∗2 (Qa′ ∗3 Qb), (Qa ∗4 Qb) ∗5 Qb′ , (Qa ∗6 script O → S to emphasize that it is the state of Qb) ∗7 (Qa′ ∗8 Qb′ ),..., via some logical con- O’s information about S. nectives ∗1, ∗2, ∗3, · · · , of individual questions Lastly, this structure also puts us into the po- Qa,Qa′ ,... ∈ QA about SA and Qb,Qb′ ,... ∈ QB sition to specify O’s total amount of informa- about SB. For an N-partite system, we use this tion about S. Clearly, the total amount of in- definition recursively. formation must be a function of the state. Let Of course, thanks to assumption 5, O can only QM = {Q1,...,QD} be an informationally com- plete set of questions in Q. Given that these ques- directly compose questions with a logical connec- tions carry the entire information O may know tive ∗ if they are compatible. There are further repercussions: let QM , QM about S, we define the total information IO→S as A B be informationally complete sets for S ,S , re- the sum of O’s information about the Qi ∈ QM , A B spectively. Then an informationally complete set as measured by the αi (1): QMAB for a composite SAB can be formed itera- D tively by joining (in a set union sense) QMA , QMB IO→S (~yO→S) := αi. (6) and adding to it a maximal pairwise independent Xi=1 set of questions which are the logical connectives (We imagine O as an agent who can write down of members of QMA with elements of QMB and/or results on a piece of paper and add these up.) The iterative compositions of such compositions and possibly questions from QM , QM . In section specific relation between αi and ~yO→S will be de- A B rived later in section 6.8; it will not be the Shan- 5.2.1 below, we shall determine which logical con- non entropy which, as discussed in section 6.1, nectives ∗ we may employ to build up QMAB from describes average information gains in repeated QMA , QMB . experiments rather than the information content in the state ~yO→S. 3.2.8 Time evolution of S’s state We shall assume that O has access to a clock and 3.2.7 Composite systems begin with a definition:

For later purpose we need to clarify the notion static states: A state P~O→S which is constant of a composite system. Since we are pursuing a in time – corresponding to the situation that purely operational approach, the notion of a com- 13 position of systems must be defined in terms of For the GPT specialists, we emphasize that this defi- nition has nothing to do with the usual requirement of local the information accessible to O through interro- tomography [15–18,20,55,59] in GPTs. To give a concrete gation. O should, in principle, be able to tell example, we shall see later that two-level systems over real a composite system apart into its constituents. Hilbert spaces (rebits) satisfy this definition while violat- Accordingly, we require that the set of questions ing local tomography.

Accepted in Quantum 2017-11-27, click title to verify 18 O will always assign the same probability become clear shortly, it is convenient to consider to each of his question outcomes – is called the states P~O→S with 0 ≤ p ≤ 1 for now. We static. shall return to the yes-vector later in section 6. We then have for all t If all the states O would assign to any system ~ ~ ~ S were static – according to his information – PO→S12 (t)= λ PO→S1 (t) + (1 − λ) PO→S2 (t). (8) O’s world would be a rather boring place. In or- der not to let O die of boredom, we allow the This convex combination is the state of an ‘effec- tive’ system S12 (identical to S1,S2) describing probability vector P~O→S to change in time. (But we require that Q and Σ are time independent.) O’s information about the coin flip scenario. Note We shall now make use of operational reasoning, that S12 is not a composite system according to definition 3.5 because Q12 = Q1 = Q2. to briefly consider how P~O→S evolves under time evolution. The following argument bears some Denote by Tk the time evolution map of the analogy to an argument typically employed in state of system Sk, k = 1, 2, 12, from t = t1 to t = t . Under the assumption that P~ (t) the GPT framework [14–18, 55] concerning con- 2 O→S1,2 vex mixtures and measurements, however, is dis- evolve independently of each other, given that S1 tinct in nature as we instead consider states and and S2 do not interact, equation (8) implies their mixtures under time evolution. T12[P~O→S (t1)] = T12[λ P~O→S (t1) 14 12 1 Let O have access to two identical non- ~ + (1 − λ) PO→S2 (t1)] interacting systems S1 and S2. O’s information = λ T1[P~O→S (t1)] (9) about S1 and S2 can be different such that they 1 ~ are allowed to be in distinct states ~yO→S1 and + (1 − λ) T2[PO→S2 (t1)]. ~yO→S relative to O. Now let O perform a (bi- 2 All three states P~ , k = 1, 2, 12, are elements ased) coin flip which yields ‘heads’ with probabil- O→Sk of the same Σ. As such, we assume the following. ity λ and ‘tails’ with probability (1 − λ).15 Given that S1 and S2 are identical, O can ask the same Assumption 8. Every time evolution map can questions to both systems. If the coin flip yields act on any state in Σ. ‘heads’, O will interrogate S , if it yields ‘tails’ O PSfrag replacements 1 will interrogate S2. Next, we restrict to the situation where O ex- poses both S ,S to evolve under the same time S1 S2 1 2 evolution T := T1 = T2. Thanks to assumption λ 1 − λ 8, (9) becomes in this case ~ ~ O T [PO→S12 (t1)] = λ T [PO→S1 (t1)]

+ (1 − λ) T [P~O→S (t1)] In particular, before tossing the coin, say at 2 time t1, the probability he assigns to receiving a and T is convex linear. This is clear since O may, ‘yes’ answer to question Qi (asked to either S1 in particular, prepare S1,S2 in identical states ~ ~ ~ or S2 depending on the outcome of the coin flip) PO→S1 (t1)= PO→S2 (t1)= PO→S12 (t1). This spe- 12 1 is simply the convex sum yi (t1) = λyi (t1) + cial case, when inserted into (9), yields T12 = T . 2 (1 − λ) yi (t1). (O is allowed to build this con- This scenario can be easily generalized to ar- vex combination thanks to assumption 2.) But bitrarily many identical systems. Remarkably, it this holds at any time t before O tosses the coin can be shown that convex linearity of operations (which O can also choose never to do).16 As will on probability vectors, in fact, implies full lin- earity of the operation [14, 55]. Hence, the time 14 That is, both S1 and S2 carry the same Q and Σ. evolution of the state must be linear: 15 Equivalently, O may use another system to which he ~ ~ has assigned a stable state vector by repeated interroga- PO→S(t2)= T [PO→S(t1)] tions on identically prepared systems. Given an arbitrary = A(t1,t2) P~O→S(t1)+ V~ (t1,t2), elementary question Q, he can use the probability that the answer is ‘yes’ as λ and the probability that the outcome where A(t1,t2) is a 2D × 2D matrix and V~ some is ‘no’ as (1 − λ) and thus use Q as the ‘coin’. 2D-dimensional vector. Given our assumption 16We assume λ to be constant in time. above that O’s world is not static, A(t1,t2) will

Accepted in Quantum 2017-11-27, click title to verify 19 generally be a non-trivial matrix. Requiring that all ∆t, (ii) that the evolution is actually continu- the last relation also holds at t1 = t2 for all ous, nor (iii) that every composition of evolution initial states, it follows that V~ (t,t) = 0 and matrices is again an evolution matrix. We shall A(t,t) = 1. Demanding further that the spe- return to this question in section 6 with the help cial state P~O→S = ~0, corresponding to p = 0 of the set of principles for quantum theory and and the absence of a system, is invariant un- also defer the discussion of the time evolution of der time evolution (such that no system or any ~yO→S until then. information is created out of ‘nothing’), finally For now we note, however, firstly, that any 17 yields V~ ≡ ~0, ∀ t1,t2. Given that P~O→S(t) is a A(∆t) acting on P~O→S implies a unique map probability vector for all t and (4) must always T∆t(~yO→S) acting on the yes-vector thanks to (4, hold, A(t1,t2) must be a nonnegative (real) ma- 5). Secondly, a multiplicity of time evolutions trix which is stochastic in any pair of components of S is possible, depending on the physical cir- i and i + D of P~O→S. cumstances (interactions) to which O may sub- Consequently, the elementary and natural as- ject S. The set of all possible time evolutions sumption 8 rules out non-linear and state de- which O shall henceforth be able to implement pendent time evolution, such as in Weinberg’s will be denoted by T . This set need not contain non-linear extension of quantum mechanics [74], all physically possible time evolutions. Indeed, and thereby also eliminates the many operational upon imposing the quantum principles, T will be- problems that arise with it (e.g., superluminal come the set of unitaries (rather than of arbitrary signaling) [75–77].18 completely positive maps). Clearly, T is part of For later purpose, we impose a further condi- O’s model for describing S; his model is thus a tion on the evolution of S’s state. triple (Q, Σ, T ).

Assumption 9. (Temporal Translation In- 3.2.9 The theory landscape L variance) There exist no distinguished instants of time in O’s world such that O is free to set The landscape L of theories describing O’s acqui- any instant he desires as the instant t = 0. Time sition of information about a physical system S evolution, as perceived by O, is therefore (tempo- is now the set of all theories which comply with the structure and assumptions established in this rally) translation invariant A(t1,t2)= A(t2 −t1). section. Since the time evolution matrix A can only de- In the sequel, we shall restrict O’s attention pend on the duration elapsed, but not on the par- solely to composite systems of N ∈ N general- ticular instant of time, we can collect the above ized bits (gbits), where a single gbit is character- results in the simple form: ized by the fact that O can maximally know the answer to a single question at once such that it P~O→S(t2)= A(∆t) P~O→S(t1), ∆t = t2 −t1. (10) can carry at most one bit of information. Ev- ery inference theory specifies for every system of We do not yet have sufficient evidence to con- N gbits a triple (QN , ΣN , TN ). We shall be con- clude that a given time evolution will be described cerned with the landscape of gbit theories Lgbit by a continuous one-parameter matrix group be- which contains all gbit theories, satisfying as- cause we neither know (i) that A is invertible for sumptions 1–9. To name concrete examples at this stage, L contains, among a continuum of 17 gbit A similar argument would not work for the evolution other theories, classical bit, rebit and qubit the- of ~yO S because, in principle, ~nO S 6= ~0 is possible even → → ory for all N ∈ N. We shall see more of this later, if ~yO→S = ~0 (where ~0 is a D-dimensional zero vector). This is the reason why here it is more convenient to work but for now we summarize their characteristics ~ with PO→S, which contains the information about both for N = 1, ~yO→S, ~nO→S, rather than ~yO→S alone. In fact, we shall see later in section 6 that the evolution of the latter does classical bit theory gives Q1 = {Q, ¬Q}, ~ ′ ~ involve an affine D-dimensional vector V 6= 0. Σ1 ≃ [0, 1] (for normalized states) with ex- 18It should be emphasized that assumption 8 is also tac- tremal points corresponding to Q = ‘yes’ and 19 itly made in the GPT framework [14–18, 55] for any kind ‘no’, respectively, and T1 ≃ Z2. of transformations on states, such that the present setup is in this regard not less potent. 19There are precisely two states of maximal information

Accepted in Quantum 2017-11-27, click title to verify 20 rebit theory (two-level systems on real Hilbert 4.1 The rules spaces) yields Q ≃ S1 and every two maxi- 1 As we are reconstructing a technically and empir- mally complementary questions are informa- ically well-established theory, we are in the fortu- tionally complete. Σ ≃ D2 (for normalized 1 nate position to avail ourselves of empirical ev- states) and T ≃ SO(2). 1 idence and earlier ideas on characterizing quan- tum theory in order to motivate a set of basic qubit theory has Q ≃ S2 and every triple 1 postulates. However, the ultimate justification of mutually maximally complementary ques- for these postulates will be their success in sin- tions are informationally complete. Σ ≃ B3 1 gling out qubit quantum theory within the infer- (for normalized states) and T1 ≃ SO(3). ence theory landscape Lgbit which will be com- pleted in [1]. As ‘coordinates’ on a theory space With all the assumptions made along the way, these principles will not be unique and one could the theory landscape devised here is perhaps not find other equivalent sets. (As usual, at least quite as general as the commonly employed land- many roads lead to Rome.) But we shall take scape of generalized probability theories [14–18, the below ones as a first working set which re- 23, 55, 60, 61]. In particular, GPTs easily handle stricts the informational relation between O and arbitrary finite dimensional systems, while this S, where S will be a composite system (c.f. def- remains to be done within the present frame- inition 3.5) of N ∈ N gbits. Each principle will work. Nevertheless, the new landscape L is gbit be first expressed as an intuitive and colloquial large enough and provides new tools for a non- statement, followed by its mathematical meaning trivial and instructive (re)construction of qubit within Lgbit. quantum theory that approaches the latter from a We take it as an empirical fact that there ex- conceptually and technically new angle. To facil- ist physical systems about which only a limited itate future generalizations of this work, we have amount of information can be known at any one also attempted to spotlight the assumptions un- moment of time. The standard quantum example derlying Lgbit as clearly as possible in this section. 1 is a spin- 2 particle about which an experimenter may only know its polarization in a given spatial direction, but nothing independent of that; the 4 Principles for the quantum theory of polarization ‘up’ or ‘down’ corresponds to one bit qubits as rules on information acquisi- of information. But there is also a typical classi- tion cal example, namely a ball which may be located in either of two identical boxes, the definite po- sition ‘left’ or ‘right’ corresponding to one bit of We shall now use the landscape Lgbit of gbit theo- ries and formulate the principles for the quantum information. Informationally, these two examples theory of qubit systems on it as rules on an ob- incarnate the most elementary of systems, a gbit, server’s information acquisition. These principles supporting maximally just one proposition at a constitute a set of ‘coordinates’ of qubit quantum time. But, clearly, there are more complicated systems supporting other limited amounts of in- theory on Lgbit. formation. We shall take this simple observation and raise the existence of an information limit to that O can assign to a classical bit the level of a principle. 1 0 More precisely, in analogy to von Weizsäcker’s P~O S = , P~O S = , → 0 → 1 ‘ur-theory’ [78–80], we shall restrict O’s world     to be a world of elementary alternatives which corresponding to ‘yes’ and ‘no’ answers to Q. Time evo- thereby consists only of systems which can be de- Z lution is described by the abelian group 2, given by composed into elementary gbits.20 All physical

1 0 0 1 20 1 = , P = , However, in contrast to [78–80], we shall be much 0 1 1 0     less ambitious here and will not attempt to deduce the dimension of space or space-time symmetry groups from where P swaps the two states. For classical bit theory, the systems of elementary alternatives. Recent developments permitted time evolution is therefore discrete. [21, 42, 59, 81], on the other hand, unravel a deep rela-

Accepted in Quantum 2017-11-27, click title to verify 21 quantities in O’s world are to be finite such that versa. (For an informational discussion of the he can record his information about them on a particle-wave duality in Young’s double slit ex- finite register. We wish to characterize the com- periment and a Mach-Zehnder interferometer, see posite systems in O’s world according to the finite also [32,35].) Similarly, in a Stern-Gerlach exper- limit of N bits of information he can maximally iment an experimenter may determine the polar- 1 inquire about them. ization of a spin- 2 particle in x-direction, but will be entirely oblivious about the polarization in y- Rule 1. (Limited Information) “The observer and z-direction. A subsequent measurement of O can acquire maximally N ∈ N independent the spin of the same particle in y-direction will bits of information about the system S at any render her previous information about the polar- moment of time.” ization in x-direction obsolete and keep her ig- There exists a maximal set Q , i = 1,...,N, of i norant about the spin in z-direction and so on. N mutually independent and maximally compat- That is, systems empirically admit many more ible questions in QN and no subset in QN can independent questions than they are able to an- contain more than N questions with that prop- swer at a time – thanks to the information limit. erty. We shall now return to the relation between O In other words, O can ask maximally N inde- and S and accordingly stipulate that complemen- pendent questions21 at a time to S. Accordingly, tarity exists in O’s world, however, we shall say this rule immediately implies that O can distin- nothing more about how much complementary in- guish maximally 2N states of S in a single shot formation may exist. interrogation because there will be 2N possible Rule 2. (Complementarity) “The observer O answers to the Q1,...,QN . Since QN , ΣN are in- bits trinsic to S, also the maximal amount of N bits can always get up to N new independent of that each S can carry must be intrinsic to it and information about the system S. But whenever thus be observer independent. In fact, this rule O asks S a new question, he experiences no net loss in his total information about S.” can be regarded as a defining property of all in- ′ There exists another maximal set Qi, i = ference theories in the gbit landscape Lgbit and can thus clearly not distinguish between classical 1,...,N, of N mutually independent and maximally compatible questions in QN such bits, rebits, qubits, etc. ′ that Qi,Qi are maximally complementary and We take another empirical fact and elevate ′ Qi,Qj6=i are maximally compatible and indepen- it to a fundamental principle: despite the lim- 22 ited information accessible to an experimenter dent. at any moment of time, there always exists ad- That is, after asking S a set of N indepen- ditional independent information that she may 22 learn about the observed system at other times. Alternatively, one could formulate the technical part of this rule as follows: In the N = 1 case of a single This is Bohr’s complementarity principle [82]. ′ gbit there exists, for every Q1 ∈ Q1, another Q1 ∈ Q1 ′ Consider, for instance, the prototypical quantum such that Q1,Q1 are maximally complementary. Defini- physics experiment: Young’s double slit exper- tion 3.5 would then immediately imply that a compos- iment. The experimenter can choose whether ite system of N gbits features two sets of questions, Qi, ′ i = 1,...,N, and Qj , j = 1,...,N, with the following to obtain which-way-information or an interfer- ′ properties: Qi,Qi are questions about the elementary gbit ence pattern, but not both; the complete knowl- labeled by i, all Qi are maximally independent and max- ′ edge of whether the particle went through the imally compatible, all Qj are maximally independent and ′ left or right slit is at the expense of total ig- maximally compatible and any pair Qi,Qj=6 i is maximally ′ norance about the information pattern and vice independent and compatible and every pair Qi,Qi corre- sponding to the same gbit is maximally complementary. This is a priori not equivalent to the current formulation tion between (a) simple conditions on operations with sys- of the technical part of rule 2 in the main text as the lat- tems carrying finite information (e.g., communication with ter does not necessarily refer to questions about individual physical systems) and (b) the dimension and symmetry subsystems of a composite system. However, the alterna- group of the ambient space or space-time. tive formulation would also be enough to get the correct 21Obviously, combinations (e.g., ‘correlations’) of the structure of the informationally complete sets for N qubits compatible Qi will define other bits of information which, and N rebits below. While the alternative formulation is however, will be dependent once the Q1,...,QN are asked simpler, we keep the other one as it is also published in (we shall return to this in detail in section 5). this form in the companion article [1].

Accepted in Quantum 2017-11-27, click title to verify 22 dent elementary propositions Qi, i = 1,...,N, interpreted as being in a ‘superposition’ of the ′ in a single shot interrogation, O can pose a new Q1 alternatives ‘yes’ and ‘no’, but in this case ′ (N + 1)th elementary question Q1 to S. Since O with ‘equal weight’ because both alternatives are can only know N independent bits of informa- equally likely according to O’s knowledge.23 We tion about S at a time and asking a new question emphasize that the necessary presence of super- does not lead to a net loss of information, the positions of elementary alternatives, as perceived single bit of his previous information about Q1 by O, is a consequence of the information limit must have become obsolete upon learning the an- and complementarity. ′ swer to Q1, while O’s total information about S We would like to stress that rules 1 and 2 is still N bits. The rule allows O to perform the are conceptually motivated by related propos- same procedure until he replaces his N old by N als which have been put forward first by Rov- new bits of information about S. As O’s infor- elli within the context of relational quantum me- mation about S has changed, the state of S rel- chanics [28] and later, independently, by Brukner ative to O will necessarily experience a ‘collapse’ and Zeilinger within attempts to understand the whenever he asks a new complementary question. structure of quantum theory via limited informa- As a result, it is the observer who decides which tion [31,32,34,35,73]. However, in order to com- information (e.g., spin in x- or y-direction) he will plete these ideas to a full reconstruction of qubit obtain about the system by asking specific ques- quantum theory, we have to impose further rules. tions. But clearly O will have no influence on Specifically, rules 1 and 2 say nothing about what the answer to these questions will be and what happens in-between interrogations. We re- any answer will come at the price of total igno- quire that O shall not gain or lose information rance about complementary questions. about an otherwise non-interacting S without asking questions. A few further explanations concerning this complementarity rule are in place. First of all, Rule 3. (Information Preservation) “The to- this rule clearly rules out classical bit theory. Sec- tal amount of information O has about (an oth- ondly, the postulate asserts the existence of max- erwise non-interacting) S is preserved in-between imally complementary questions in O’s world, interrogations.” however, makes no statement about whether par- IO→S is constant in time in-between interroga- tially independent or partially complementary tions for (an otherwise non-interacting) S. questions may exist too. Thirdly, the peculiar requirement that every question of rule 2 is com- Correspondingly, IO→S is a ‘conserved charge’ plementary to exactly one from rule 1 is chosen of time evolution; this is a simple observation such that each subsystem of a composite system which will become extremely useful later. In fact, features complementarity. For example, consider notice that rule 3 could also be viewed as defin- the N = 1 case, corresponding to a single gbit, ing the notion of non-interacting systems (as per- for which rules 1 and 2 entail a complementary ceived by O). ′ pair Q1,Q1. This complementarity per gbit gen- Finally, we come to the last rule of this eralizes to arbitrary N since every Qi in rule 1 manuscript which is about time evolution. Em- ′ and every Qj in rule 2 may correspond to one of pirical evidence suggests, at least to a good ap- N gbits. More complicated complementarity re- proximation, that the time evolution of an ex- lations arising from rules 1 and 2 will be discussed perimenter’s ‘catalogue of knowledge’ about an in section 5. observed system is continuous – in-between mea- surements. More precisely, the specific prob- Notice that the complementarity rule implies the existence of a notion of ‘superposition’ – even 23We shall later see that, even in a state of maximal in states of maximal information. For example, information of N bits, O may possibly have only partial take N = 1 and let O know the answer to Q1 or incomplete knowledge about the outcomes of any ques- tion in an informationally complete set QM of pairwise with certainty, y1 = 1, such that his informa- N independent questions (c.f. assumption 7). In this case, tion about S saturates the limit of one bit. This O’s information about all questions in QMN will be in a will leave him oblivious about the complementary general state of superposition because their correspond- ′ ′ 1 ing probabilities cannot all be 1 , otherwise it would be Q1 such that he would have to assign y1 = 2 . 2 The state of information he has about S can be the state of no information.

Accepted in Quantum 2017-11-27, click title to verify 23 abilistic statements an experimenter can make violates this postulate due to its discrete time about the outcomes of measurements, i.e. his ac- evolution group (see section 3.2.9). tual information about the systems, change con- It is now our task to verify what the triples tinuously in time. We promote this to a further of (QN , ΣN , TN ) for each N ∈ N are which obey postulate for O’s world, however, with a further rules 1–4. Remarkably, it turns out that these requirement. four rules cannot distinguish between real and We note that the state space and the time evo- complex quantum theory, i.e. between real and lutions are interdependent as every legal time evo- complex Hilbert spaces. But the state spaces and lution must map a legal state to a legal state, i.e. the orthogonal and unitary groups of time evo- T∆t(~yO→S) ∈ ΣN , ∀ T∆t ∈TN and ∀ ~yO→S ∈ ΣN . lutions of rebit and qubit quantum theory are, Accordingly, what is the set of legal states de- in fact, the only ‘solutions’ within Lgbit to these pends on what is the set of legal time evolutions rules. In the sequel of this manuscript we shall – and vice versa. We would like the pair (ΣN , TN ) employ rules 1–4 in order to develop the necessary to be as ‘big’ as compatibility with the other rules tools, within Lgbit, for eventually proving the fol- allows in order to equip the other rules with as lowing more precise claim in [1, 2]. In fact, as a general a validity as possible. But there are mul- simple example of the newly developed tools, we tiple ways of ‘maximizing’ the pair. Namely, the shall already prove the claim for N = 1 at the interdependence implies that the larger the num- end of this article. ber of states, the tighter the constraints on the set Claim. Lgbit contains only two solutions for the of time evolutions – and vice versa. We required pair (ΣN , TN ) which are compatible with rules 1– before that O’s world should not be ‘boring’ and 4: therefore feature a non-trivial time evolution of states. We shall now sharpen this requirement: it 1. rebit quantum theory [2], where ΣN co- N N is more interesting for O to live in a world which incides with the space of 2 × 2 density 24 R2 ⊗N N ‘maximizes’ the number of possible ways in which matrices over ( ) and TN is PSO(2 ). any given state of S can change in time, rather 2. qubit quantum theory [1], where ΣN co- than the number of states which it can be in rela- incides with the space of 2N × 2N density tive to O. We shall thus require that any consis- 2 ⊗N N matrices over (C ) and TN is PSU(2 ). tent, continuous time evolution of O’s ‘catalogue Furthermore, states evolve unitarily accord- of knowledge’ about S in-between interrogations ing to the von Neumann evolution equation. is physically realizable. Since different ways of evolving the state of a system correspond to dif- We remind the reader that the time evolution ferent interactions (e.g., among the members of a groups TN for density matrices in rebit and qubit composite system) we thereby maximize the set quantum theory are projective because they cor- † of possible interactions among systems. O’s world respond to ρ 7→ U ρ U , where for (1) rebits ρ is a N N N is a maximally interactive place! 2 × 2 real symmetric matrix, U ∈ SO(2 ) and † denotes matrix transpose; and (2) for qubits ρ N N N Rule 4. (Time Evolution) “O’s ‘catalogue of is a 2 × 2 hermitian matrix, U ∈ SU(2 ) and knowledge’ about S evolves continuously in time † denotes hermitian conjugation. in-between interrogations and every consistent Furthermore, using one additional rule on QN such evolution is physically realizable.” which allows O to ask S any question that “makes (probabilistic) sense”, we show in [1, 2] that TN is the maximal set of transformations T∆t on states which is continuous in ∆t and compatible 1. in the rebit case [2], QN is (isomorphic with rules 1-3 and the structure of Lgbit. to) the set of projective measurements onto the +1 eigenspaces of Pauli operators25 over N Of course, during an interrogation, ~yO→S may R2 . This is the set of rank-(2N−1)- change discontinuously, i.e. ‘collapse’, on account projectors. of complementarity. As innocent as the require- ment of continuity of time evolution in rule 4 ap- 24A real density matrix is a symmetric, positive semidef- R2 ⊗N pears, it turns out to be absolutely crucial in or- inite matrix on ( ) . N der to single out a unique information measure 25The set of Pauli operators over R2 is the set of trace- less real symmetric 2N ×2N matrices with eigenvalues ±1. αi(~yO→S). Notice also that classical bit theory

Accepted in Quantum 2017-11-27, click title to verify 24 2. in the qubit case [1], QN is (isomorphic physically sufficient. to) the set of projective measurements onto Notice that it is necessary to directly recon- the +1 eigenspaces of Pauli operators26 over struct the space of density matrices over Hilbert N C2 (i.e., the set of rank-(2N−1)-projectors) spaces from the rules rather than the underly- and the outcome probability for any Q ∈ QN ing Hilbert spaces themselves. The reason is that to be answered with ‘yes’ by S in some state the latter contain physically redundant informa- is given by the Born rule for projective mea- tion (norm and global phase), while the rules refer surements. only to information which is directly accessible to O. Since both real and complex quantum theory come out of rules 1–4, we shall impose a further rule on O’s information acquisition in [1] in order 4.2 Strategy for building the necessary tools to also eliminate rebit theory. However, we note and proving the claim that rebit theory is mathematically contained in qubit quantum theory and that rebits can ac- Our strategy and procedure for developing tools tually be produced in laboratories. Hence, one and intermediate results before proving the main might also hold rules 1–4 and the rule that O claim is best summarized as a diagram (see figure may ask S any question that “makes sense” as 3).

limited information complementarity information preservation time evolution

PSfrag replacements

independence, compatibility and cor- reversible time evolution (in sec. 6) relation structure on QN (in sec. 5) and information measure (in sec. 6.8)

time evolution

Σ1 is a ball with D = 2, 3, and T2 is either PSO(4) or PSU(4) ⇒ R 3 C 3 T1 is either SO(2) or SO(3) (in sec. 7) Σ2 convex hull of P or P (in [1, 2])

(QN , ΣN , TN ) for N > 2 (in [1, 2])

Figure 3: Strategy and steps for the reconstruction.

That is, we shall firstly ascertain, in section 5, which, besides becoming important later, yield the independence, compatibility and correlation simple explanations for typical quantum phenom- structure on QN which is induced by rules 1 and ena such as, e.g., entanglement, non-locality and 2. This section will be particularly instructive monogamy relations. This is also where we shall and deliver many elementary technical results determine the dimensionality of Σ1 by a simple argument and, using this result, derive the di- N 26The set of Pauli operators over C2 is the set of trace- mensionalities of all other state spaces too. Next, less hermitian 2N × 2N matrices with eigenvalues ±1. in section 6, we shall use rules 3 and 4 in order

Accepted in Quantum 2017-11-27, click title to verify 25 to conclude that a specific time evolution, is re- maximal complementarity) versible and, in fact, described by a continuous one-parameter matrix group. These results will QM1 = {Q1,Q2,...,QD1 } (11) then be used in section 6.8, together with rules because rule 1 prohibits further pairwise indepen- 3 and 4, to derive an explicit information mea- dent (partially or maximally) compatible ques- sure αi(~yO→S) which is unique under elementary tions. (Recall that maximally complementary consistency conditions. In section 7, the conjunc- questions are automatically independent.) Rules tion of these outcomes will then be employed to 1 and 2 imply D1 ≥ 2. It turns out that we need show that for N = 1 one either obtains a two- to consider two gbits in order to upper bound D1. or three-dimensional Bloch ball as a state space We shall call such questions for a single gbit Σ1 and that the group of all possible time evo- individual questions. lutions T1 is accordingly either SO(2) or SO(3). The claim for N > 1 will require substantial addi- tional work and will be proven in the companion 5.2 Two gbits paper [1] for qubits and in [2] for rebits. The N = 2 case requires substantially more work. First of all, since this is a composite system (c.f. definition 3.5), the individual questions about the 5 Question structure and correlations two single gbits must be contained in an infor-

mationally complete set QM2 . We shall again In this section, we shall solely employ rules 1 and slightly change the notation, as compared to sec- 2 in order to deduce the independence, compati- tion 4.1: the maximal complementary set QM bility and correlation structure for the questions 1 (11) for gbit 1 will be denoted by Q1,...,QD1 , in an informationally complete set QMN (c.f. as- while the elements of QM1 for gbit 2 will be de- sumption 7). To this end it is instructive to lookPSfrag replacements ′ ′ noted with a prime Q1,...,QD1 . It will be conve- at the individual cases N = 1, 2, 3 in some de- nient to depict the question structure graphically. N tail before considering general N ∈ . We shall Representing individual questions henceforth as denote by DN the dimension of the state space vertices, QM2 certainly contains the following ΣN . gbit 1 gbit 2 ′ 5.1 A single gbit Q1 Q1 ′ Rules 1 and 2 taken together imply that for N = 1 Q2 Q2 there exists at least one maximally complemen- Q Q′ tary pair Q ,Q′ . Since N = 1 is fixed, it is now . 3 . . 3 1 1 . . . more convenient to count the independent ques- ′ QD Q. (12) tions by an index, rather than a prime.27 Let us 1 D1 therefore slightly change the notation and write This structure abides by rules 1 and 2 and any Qi ′ Q1 henceforth as Q2, such that the rules imply will be independent of and maximally compatible the existence of a maximally complementary pair ′ with any Qj (cf. definition 3.5). Q1,Q2. But, applied to a single gbit only, rules 1 and 2 tell us nothing more about how many 5.2.1 Logical connectives of single gbit questions questions maximally complementary to Q1 exist. There may arise another Q3 maximally comple- But a composite system may also admit compos- mentary to both Q1,Q2 etc. Notice that, for a ite questions, built with logical connectives of the single gbit, an informationally complete set of subsystem questions. We must now determine pairwise independent questions must be given by which composite questions are allowed and which a maximal set of mutually maximally comple- must be added to the individual questions in or- mentary questions (i.e., no further Q ∈ Q1 can der to form an informationally complete QM2 of be added to this set without destroying mutual pairwise independent questions. To this end, we must firstly unravel which logical connective ∗ O 27Rules 1 and 2 count compatible questions, here we can employ at all in order to construct compos- need to count maximally complementary questions. ite questions which are pairwise independent of

Accepted in Quantum 2017-11-27, click title to verify 26 the individual ones. Recall assumption 5 that choose to use the XNOR, already introduced in O, according to his theoretical model, may only (2), and operationally interpret it as a ‘correlation ′ compose compatible questions with logical con- question’; Qij := Qi ↔ Qj is to be read as ‘are ′ ′ nectives ∗ and consider Qi,Qj as an exemplary the answers to Qi and Qj the same?’. But the compatible pair. In truth tables, we shall hence- XOR could equivalently be employed.28 forth symbolize ‘yes’ by ‘1’ and ‘no’ by ‘0’. Since, according to assumption 7, O must be able to build an informationally complete set Remark. (‘=’ in logical expressions.) To for a composite system which, in particular, in- avoid confusion, we emphasize that whenever we volves pairwise independent composite questions, write an equality sign ‘=’ in a logical expression ′ we would like to conclude that Qi ↔ Qj are not in the sequel, we do not interpret it as another only implementable, but also contained in a QM2 logical connective ∗, but as an actual equality of together with the individuals. However, this is the values of the propositions on the left and right subject to a few consistency checks on pairwise ′ hand side. For example, the expression Qi = Qj independence. is not itself a proposition which takes truth val- The XNOR ↔ is a symmetric logical connec- ues 0 or 1, but is intended to mean that the truth tive, Q = Q ↔ Q′ = Q′ ↔ Q (but note that ′ ij i j j i value which Qi takes is identical to that which Q j Qij 6= Qji), and, thanks to its associativity, takes. ′ Qi ↔ Qij = Qi ↔ (Qi ↔ Qj) While permitted binary connectives, it is clear = (Q ↔ Q ) ↔ Q′ = Q′ , that, e.g., the AND and OR operations ∧ and ∨, i i j j respectively, cannot be employed alone to build 1 ′ ′ Qij ↔ Q = Qi (14) a new independent question from Qi,Qj because j | {z } Q ∧Q′ = 1 implies Q = Q′ = 1 and Q ∨Q′ = 0 i j i j i j such that ↔ defines a closed relation on implies Q = Q′ = 0 such that certain answers i j Q ,Q′ ,Q . This has the following ramifications: to these two connectives imply answers to the in- i j ij (1) For any compatible pair Q ,Q′ , the ‘correla- dividuals. But if Q ∗ Q′ is to be pairwise inde- i j i j tion’ operation ↔ gives rise to precisely one ad- pendent from Q ,Q′ , an answer to it must not i j ditional question Q , and (2) the set Q ,Q′ ,Q imply the answer to either of Q ,Q′ and, con- ij i j ij i j will indeed be pairwise independent, according to versely, the answer to only Q or only Q′ cannot i j the definition in section 3.2.4, because neither Q imply Q ∗ Q′ . As can be easily checked, it must i i j nor Q′ alone can determine Q relative to the satisfy the following truth table: j ij state of no information (not even partially), oth- erwise, together with the determined Qij, they ′ ′ Qi Qj Qi ∗ Qj would (at least partially) determine each other 0 1 a via (14) – in contradiction with the independence ′ 1 0 a a 6= b a, b ∈ {0, 1}. (13) of Qi,Qj. That is, Qij is independent of both Qi ′ 1 1 b and Qj and since we assume independence to be a symmetric relation, it must also be true the other 0 0 b way around. 2 Consequently, the D1 correlation questions The only logical connectives ∗ satisfying this Qij, i, j = 1,...,D1, are candidates for addi- truth table are the XNOR ↔ for a = 0 and b = 1 tional questions in QM2 besides the 2D1 indi- and its negation, the XOR ⊕ for a = 1 and b = 0. vidual ones of (12). Graphically, we shall repre- ′ ′ As Qi ⊕ Qj = ¬(Qi ↔ Qj), we may only choose sent Qij as the edge connecting the vertices cor- ′ one of the two binary connectives to define new responding to Qi and Qj. For example, the fol- pairwise independent questions. We henceforth lowing question graphs

28As an aside, the XNOR can be expressed in terms of

′ ′ the basic Boolean operations as Qi ↔ Qj =(¬Qi ∨ Qj ) ∧ ′ (Qi ∨ ¬Qj ).

Accepted in Quantum 2017-11-27, click title to verify 27 PSfrag replacements PSfrag replacements PSfrag replacements

gbit 1 gbit 2 gbit 1 gbit 2 gbit 1 gbit 2 Q11 ′ Q11 ′ ′ Q1 Q1 Q1 Q1 Q11Q1 Q1 Q Q31 ′ Q22 31 ′ Q31 ′ Q2 Q2 Q2 Q2 Q2 Q2 Q22 Q22 Q23Q Q′ Q Q23 Q′ Q23Q Q′ . 3 . . 3 . 3 . . 3 . 3 . . 3 QD1D1 ...... QD1D1 . . . ′ QD1D1 ′ ′ QD1 QD1 QD1 QD1 QD1 QD1

represent legal sets of questions. Note that due if and only if its corresponding vertex is a vertex to assumption 5, there can only be edges between of the edge corresponding to Qij. By symmetry, ′ gbit 1 and gbit 2 but not between the vertices of the analogous result holds for Qj. a single gbit, e.g., Q1 and Q2, because they are complementary. Proof. Qi and Qij are maximally compatible by Our next task is to analyse the independence construction. Consider therefore Qi and Qkj for ′ k 6= i and j = 1,...,D1. Clearly, Qi and Q are and compatibility structure of the Qij. j maximally compatible and pairwise independent and so are Q′ and Q . Maximal complemen- 5.2.2 Independence, complementarity and entan- j kj tarity of Qi and Qkj for k 6= i now follows from glement ′ Qk = Qkj ↔ Qj (see (14)), which is maximally We begin with a simple, but important observa- complementary to Qi, and lemma 5.1. tion which we will frequently make use of. For example, Q1 and Q12 are maximally com- Lemma 5.1. Let Q1,Q2,Q3 ∈ QN be such that patible, while Q1 and Q22 are maximally com- Q1 is maximally compatible with and pairwise in- plementary. The intuitive explanation for the dependent of both Q2,Q3 and that Q3 is maxi- incompatibility of Q1 and Q22 is as follows: if mally complementary to Q1 ↔ Q2. Then Q2,Q3 O knew the answers to both simultaneously, he are maximally complementary. would know more than one bit of information about gbit 1 because Q1 defines a full bit of in- Proof. Suppose Q ,Q were at least partially 2 3 formation about it while Q22 could be regarded compatible. In that case there must exist a state as defining half a bit about each of gbit 1 and of S in which O has maximal information α2 = 1 2. But in view of rule 1, O should never know bit about Q2 and at least partial information more than one bit about a single gbit, even in a α3 > 0 bit about Q3 (or vice versa). Let O composite system. Considering the information now ask Q1 which is maximally compatible with contained in correlation questions to equally cor- and pairwise independent of both Q2,Q3 to S respond to gbit 1 and 2, lemma 5.2 suggests that in this state. Hence, according to assumption 4, O’s information will always be equally distributed by asking Q1, O cannot change his information over the two gbits for states of maximal knowl- about either Q2,Q3. Thus, after asking Q1, O edge, i.e. O will know equally much about gbit 1 will have maximal information about Q1,Q2 and and 2.29 We shall make this more precise in [1]. at least partial information about Q3. But max- ′ We saw before that Qi,Qj,Qij are pairwise in- imal information about Q ,Q implies maximal 1 2 dependent. Lemma 5.2 implies that also Qi and information about Q ↔ Q which is maximally 1 2 Qkj for i 6= k are independent. We can make use complementary to Q3. Consequently, Q2 and Q3 of this result to show the following. must be maximally complementary too. Lemma 5.3. The Qij, i, j = 1,...,D1 are pair- This has immediate useful implications. wise independent.

Lemma 5.2. Qi is maximally compatible with Proof. Consider Qij and Qkl and suppose i 6= Qij, ∀ j = 1,...,D1 and maximally complemen- k. Then Qi and Qij are maximally compatible, tary to Qkj, ∀ k 6= i and ∀ j = 1,...,D1. That 29 is, graphically, an individual question Qi is max- Clearly, this is not true for non-maximal knowledge. E.g., let O always ask only Q . imally compatible with a correlation question Qij 1

Accepted in Quantum 2017-11-27, click title to verify 28 while Qi and Qkl are maximally complementary 5.2 it follows that Qkl is maximally complemen- ′ by lemma 5.2. Hence, when knowing Qi and Qij tary to both Qi and Qj such that the answer to O cannot know Qkl such that Qij and Qkl must Qkl will give one new independent bit of infor- ′ be independent. (Recall from section 3.2.4 that mation but renders O’s information about Qi,Qj dependence requires the answer to Qij to always obsolete. But by rule 2 O cannot experience a net imply at least partial knowledge about Qkl, i.e., loss of information by asking a new question and 1 PSfrag replacements PSfrag replacements yij = 0, 1 implies ykl 6= 2 ). The analogous argu- after asking Qkl he must still know 2 bits about ment holds for when j 6= l. gbitS. 1 Hence, after acquiring thegbit answer 1 to Qkl he gbitmust 2 still know the answer togbitQij 2such that both Recalling from subsection 5.2.1 that the set of areQ1 maximally compatible. Q1 composite questions in QM2 must be non-empty, Q2 Q2 lemmas 5.2 and 5.3 have an important corollary. QTo3 give graphical examples, QQ311 and Q21 are ′ maximallyQD1 complementary dueQ toD1 the intersection Corollary 5.4. Qi,Qj,Qij are pairwise indepen- ′ ′ in Q , while Q13 and Q21 are maximallyQ1 compati- dent for all i, j = 1,...,D1 and, thanks to as- ′ 1 ′ bleQ2 because their edges do not intersectQ2 in vertices sumption 7, will thus be part of an information- ′ ′ Q3 Q3 ally complete set QM2 . ′ ′ QD1 QD1 Q11 Q′ Next, we consider the compatibility and com- 1 Q11 plementarity structure of the correlation ques- Q21 Q21 Q Q tions Qij. 22 22 Q13 Q13 . . Lemma 5.5. Qij and Qkl are maximally com- QD1D1 . QD1D1 . patible if and only if i 6= k and j 6= l. That is, graphically, Qij and Qkl are maximally compat- PSfrag replacements ible if their corresponding edges do not intersect This questiongbit structure 1 has significant conse- in a vertex and maximally complementary if they quences: sincegbit the 2 correlation questions Qij and intersect in one vertex. Qkl are both independentQ1 and maximally com- patible for i 6= k and j 6= l, O can ask both of Proof. Suppose Q and Q with i 6= k were Q2 ij kj them simultaneously, thereby spending the maxi- at least partially compatible. Then there must Q3 mal amount of QND= 2 independent bits of infor- exist a state of S in which the answer to Qij 1 mation he may acquireQ′ according to rule 1 over is fully known to O and in which also αkj > 0 1 composite questions.′ For example, he may ask S bit (or vice versa). Let O ask Q′ to S in this Q2 j Q and Q simultaneously′ 11 22 Q3 state. Since, by lemma 5.2, this question is max- ′ QD imally compatible with both Qij,Qkj (and pair- 1 Q11 wise independent of both), according to assump- ′ Q21 tion 4, by asking Qj, O cannot change his infor- Q22 mation about both Qij,Qkj. That is, after hav- ing also asked Q′ , O must have maximal informa- Q13 j . ′ QD D . tion about Qij,Qj and partial information about 1 1 . ′ Qkj. But maximal information about Qij,Qj im- (15) plies also maximal information about Qi which, however, is maximally complementary to Qkj by upon which he must be entirely oblivious about lemma 5.2. Hence, Qij,Qkj with i 6= k must be the individual gbit properties represented by ′ maximally complementary. Qi,Qj. (Lemma 5.2 implies that no individual Consider now Qij and Qkl with i 6= k and j 6= l. question is maximally compatible with two non- ′ Let O ask S both Qi and Qj the answers to which intersecting bipartite questions at once.) That is, imply the answer to Qij according to (2). Thanks O has only composite, but no individual infor- to rule 1, this defines a state of maximal knowl- mation about the two gbits; they are maximally edge of 2 independent bits and O may not know correlated. But this is precisely entanglement. any further independent information. Next, let Indeed, the question graph (15) will ultimately O ask the same S the question Qkl. From lemma correspond to four Bell states in either of the two

Accepted in Quantum 2017-11-27, click title to verify 29 ′ ′ question bases {Q1,Q1} and {Q2,Q2}, represent- would be an interesting question to investigate ing the four possible answer configurations ‘yes- what the precise sufficient and necessary condi- yes’, ‘yes-no’, ‘no-yes’ and ‘no-no’ to the corre- tions for entanglement are in the quantum case lation questions Q11,Q22 (see [31] for a related in this informational formulation. However, we perspective within quantum theory). In fact, if shall leave this open for now. O now ‘marginalized’ over gbit 2, corresponding Since there are only N = 2 independent bits of to discarding any information about questions in- information to be gained, according to rule 1, the volving gbit 2, he would find gbit 1 in the state above sufficient condition for entanglement thus of no information relative to him because the means, in particular, that O has more compos- discarded information also contained everything ite than individual information. In general, two he knew about gbit 1. Other compatible edges gbits will be referred to as maximally entangled will correspond to Bell states in different ques- relative to O if he has only composite informa- tion bases. tion about them which is incompatible with any Inspired by the compelling proposal within individual information. By contrast, two gbits quantum theory in [33, 34], we shall define states would be in a ‘product state’ if O has maximal ~yO→S of a bipartite gbit system in Lgbit as individual knowledge of 2 bits about them. For ′ instance, if he knows that Q1 = Q1 = 1, S would entangled if the composite information satisfies be in such a product state relative to him. But D1 i,j=1 αij > 1 bit, notice that even in this case, O would have one P dependent bit of information about the correla- where αij(~yO→S) is O’s information about the tions because clearly Q = 1 too. correlation question Qij. While there will be 11 D1 Note also that O can ‘collapse’ two gbits into states with i,j=1 αij ≤ 1 bit which can indeed 30 an entangled state: as the most extreme exam- be consideredP as classically composed, in con- trast to [33,34], we emphasize that the above will ple, consider the case that O receives a ‘product only be a sufficient, but not a necessary condi- ensemble state’ of two gbits in a multiple shot tion for entanglement, as in the quantum case interrogation. O may then ask the next pair of there will exist entangled states violating it.31 It gbits the questions Q11,Q22. Upon receiving the answers, the two gbits will have ‘collapsed’ into 30 1 E.g., consider a state with y11 = 1 and yi = 2 for all a maximally entangled posterior state relative to other questions in QM2 , corresponding to O knowing with O, despite having been in a ‘product state’ prior certainty that Q11 = 1 and nothing else. The discussion D1 bit to interrogation. Entanglement is thus a property surrounding (1) implies already that i,j=1 αij = 1 , while the individual information is zero. In quantum the- of O’s information about S. ory, this state would correspond toP the separable state We emphasize that, for systems with limited in- 1 1 ρ = 4 ( + σx ⊗ σx). formation content, entanglement is a direct con- 31 D1=3 bit sequence of complementarity – at least for states For the quantum case, states with i,j=1 αij ≤ 1 where defined as classically composed in [33, 34]. This is, of maximal information. To illustrate this obser- P however, in conflict with the usual definition of entangled vation, consider two classical bits (cbits). Since states as those which are non-separable. Namely, consider the family of quantum states (on the r.h.s. expressed in 1 1 the same in z-direction, respectively. Using the quadratic z-basis) for − ≤ a ≤ 2 3 3 information measure αij = (2 yij − 1) which we derive in 1 1 subsection 6.8 and which was also proposed for quantum ρa := 1 + (σz ⊗ 1 + 1 ⊗ σz − σz ⊗ σz) 4 3 theory by the authors of [33, 34], these states give  1 3 0 0 0 3 1 2 2 2 0 3 a 0 Icomp := αij = (2 yxx − 1) +(2 yyy − 1) +(2 yzz − 1) +2a(σx ⊗ σx + σy ⊗ σy)) =   . 0 a 1 0 i,j=1 3 X   1 2  0 0 00  +0 · · · +0= + 8a bit ≤ 1 bit,   9 This clearly is a family of non-separable states for a 6= 0.   1 2 In particular, for a = this is the reduced state which and for the individual information Iindiv := (2yz1 − 1) + 3 2 bit one obtains from a W-state upon marginalizing over a (2yz2 − 1) + 0 · · · +0=2/9 . Icomp can be arbitrar- 2 1 ily close to 1/9 bit for a sufficiently small a 6= 0 so that qubit. These states yield yz1 = yz2 = 3 , yzz = 3 and 1 1 even Icomp

Accepted in Quantum 2017-11-27, click title to verify 30 PSfrag replacements

Q2 Q3

QD1

′ Q2 there is no complementarity′ in this case, there are correlations of correlation questions, e.g., what Q3 only threeQ pairwise′ independent questions: the about Q11 ↔ Q22? D1 ′ individuals Q1,Q1 about cbit 1 and cbit 2, re- spectively, and the correlation Q would form an Q 11 5.2.3 A logical argument for the dimensionality of informationally21 complete Q , graphically rep- Q MN the Bloch-sphere resented as 22 Q13 The answer to the last question, in fact, is inter- QD D cbit 1 cbit 2 1 1 twined with the dimensionality D1 of the state . Q11 ′ .Q1 Q1 space Σ1 of a single gbit and thus, ultimately, with the dimensionality of the Bloch-sphere. De- riving the dimension of the Bloch sphere has also ′ Q1,Q1,Q11 are mutually maximally compatible been a crucial step in the successful reconstruc- and this composite system satisfies rule 1 too such tions of quantum theory via the GPT framework. that O may only acquire maximally N = 2 inde- While Hardy’s pioneering reconstruction [14] did pendent bits of information about this pair of not fully settle this issue (it required an opera- cbits. Clearly, also in this classical case it is pos- tionally somewhat obscure ‘simplicity axiom’ to sible for O to acquire only composite information obtain D1 = 3), it was later explicitly solved about the pair of cbits, namely by only asking by impressive group-theoretic arguments in [59] Q11 and not bothering about the individuals. O which show that, under certain information the- would be able to find out whether identically pre- oretic constraints, qubit quantum theory is the pared pairs of cbits in an ensemble are correlated only theory with non-trivial entangling dynam- by always asking Q11 in a multiple shot interro- ics. This result was then exploited to relate the gation. There will indeed exist states such that dimension of the Bloch-sphere to the dimension ′ 1 y11 = 1 and y1 = y1 = 2 , however, the pair of of space via a communication thought experi- cbits will not be in a state of maximal informa- ment [21]. However, unfortunately, the ingredi- tion relative to O because he has only spent one of ents of these arguments neither fit mathemati- his two available bits. A classical state of max- cally nor conceptually fully into our framework imal information corresponds to O knowing the which relies on distinct structures than GPTs. ′ answers to two questions of the three Q1,Q1,Q11, On the other hand, although not yielding full but then by (2, 14) O will also know the answer quantum theory reconstructions, approaches as- to the third. That is, in a state of maximal in- serting an epistemic restriction (i.e. a restriction formation about two cbits, O will always have of knowledge) over ontic states admit very sim- maximal individual information about the pair. ple and elegant arguments for a 1-bit system to For cbits, O cannot spend the second bit also in have a state space spanned by three independent composite information. By contrast, it is a conse- epistemic states [40, 41, 54]. These arguments quence of the complementarity rule 2 that in our can be carried out by considering a single system case here, O can actually exhaust the entire infor- over a 2-bit ontic state which at the epistemic mation available to him by composite questions, level, however, is effectively a 1-bit system due thereby giving rise to entanglement. to epistemic restrictions. The arguments involve Thus far, we have only considered individual binary connectives (at the ontic level) of comple-

and correlation questions. All are part of QM2 . mentary questions which, while non-problematic But can there be more pairwise independent ques- when dealing with ‘hidden variables’, are, how-

tions in QM2 ? These would have to be obtained ever, deemed illegal in O’s model of the world ac- from the individuals and correlations via another cording to our assumption 5. We do not wish to composition with an XNOR. Composing the indi- make any ontological commitments in our purely ′ viduals Qi,Qj with the correlation questions via operational approach. Consequently, we must XNOR, e.g. Qi ↔ Qil, will yield nothing new reason differently and without ontic states to de- because questions need to be maximally compat- duce the dimensionality of Σ1. In our case, this ible in order to be logically connected by O and requires to consider two gbits rather than just for compatible pairs it already follows from (14) one and involves entanglement, in analogy to the that the individual and correlation questions are group-theoretic GPT derivation in [21, 59] which logically closed under ↔. However, what about likewise requires two gbits and entanglement.

Accepted in Quantum 2017-11-27, click title to verify 31 Theorem 5.6. D1 = 2 or 3. such that either

Qjj = Q11 ↔ Q22, or

Proof. By lemma 5.5, Qij and Qkl are indepen- Qjj = ¬(Q11 ↔ Q22), ∀ j = 3,...,D1. dent and maximally compatible if i 6= k and But then clearly Qjj, j = 3,...,D could not j 6= l. But any maximal set of pairwise maxi- 1 be pairwise independent if D > 3. The same mally compatible correlation questions then con- 1 argument can be carried out for any set of D tains precisely D questions. Graphically, this 1 1 pairwise independent and maximally compatible is easy to see: one can always find D non- 1 correlations Q , i.e. for any question graph with intersecting edges between the D vertices of gbit ij 1 D non-intersecting edges, and any choice of a 1 and the D vertices of gbit 2. For example, the 1 1 pair of maximally compatible correlations which D ‘diagonal correlations’ Q , i = 1,...,D 1 ii 1 O may ask first. We must therefore conclude that D1 ≤ 3, while rule 2 implies that D1 ≥ 2.

Q11 PSfrag replacements We shall refer to D1 = 2 as the ‘rebit case’ and

Q22 to D1 = 3 as the ‘qubit case’; for the moment this is just suggestive terminology, however, we Q33 shall see later and in [2] that the D1 = 2 case . . will, indeed, result in rebit quantum theory (two- QD1D1 level systems over real Hilbert spaces), while the D1 = 3 case will give rise to standard quantum theory of qubit systems [1].

are pairwise maximally compatible and pairwise 5.2.4 An informationally complete set for N = 2 independent (lemma 5.3) such that, by theorem qubits 3.2 (Specker’s principle), they must also be mu- tually maximally compatible. Hence, O may ac- The two cases have distinct properties as regards quire the answers to all D1 Qii at the same time. questions which ask for the correlations of bi- However, rule 1 forbids O’s information about partite questions and it is necessary to consider S to exceed the limit of N = 2 independent them separately. We begin with the simpler qubit bits. Accordingly, the D1 correlation questions case D1 = 3 for which correlations of correlation Qii cannot be mutually independent if D1 > 2 questions do not define new information for O and the answers to any pair of them must imply about S such that six individual and nine cor- the answers to all others. For instance, suppose relation questions constitute an informationally

O has asked Q11 and Q22. The pair must de- complete set QM2 . These will ultimately cor-

termine the answers to Q33,...,QD1D1 such that respond to the propositions ‘the spin is up in the latter must be Boolean functions of Q11,Q22. x−,y−, z−direction’ for the individual qubits 1 Hence, Qjj = Q11 ∗Q22, j 6= 1, 2, for some logical and 2 and ‘the spins of qubit 1 in i− and qubit 2 connective ∗ which preserves that Q11,Q22,Qjj in j− direction are the same’ where i, j = x,y,z. are pairwise independent. Table (13) implies that Note that the density matrix for two qubits has ∗ must either be the XNOR ↔ or the XOR ⊕ 15 parameters.

Theorem 5.7. (Qubits) If D1 = 3 then the correlation questions Qij are logically closed under ↔ ′ and QM2 = {Qi,Qj,Qij}i,j=1,2,3 constitutes an informationally complete set for N = 2 with D2 = 15. Furthermore, for any two permutations σ, σ′ of {1, 2, 3} either

Qσ(3)σ′ (3) = Qσ(1)σ′(1) ↔ Qσ(2)σ′ (2), or Qσ(3)σ′ (3) = ¬(Qσ(1)σ′ (1) ↔ Qσ(2)σ′ (2)) (16)

and either

Qσ(3)σ′ (3) = Qσ(1)σ′(2) ↔ Qσ(2)σ′ (1), or Qσ(3)σ′ (3) = ¬(Qσ(1)σ′ (2) ↔ Qσ(2)σ′ (1)) (17)

Accepted in Quantum 2017-11-27, click title to verify 32 such that either

Qσ(1)σ′(1) ↔ Qσ(2)σ′ (2) = Qσ(1)σ′ (2) ↔ Qσ(2)σ′ (1), or

Qσ(1)σ′(1) ↔ Qσ(2)σ′ (2) = ¬(Qσ(1)σ′ (2) ↔ Qσ(2)σ′ (1)). (18)

Proof. Statements (16, 17) are an immediate is an informationally complete set of questions consequence of the argument in the proof of the- for N = 2 and D1 = 3 such that D2 = 15. orem 5.6 which can be applied to any correla- For example, if σ, σ′ are both trivial then the tion question graph with D1 = 3 non-intersecting PSfrag replacements′ theorem appliedPSfrag to the replacements following two question edges and thus to any two permutations σ, σ of . . {1, 2, 3}. From this it directly follows that the graphs. . Q11 correlation questions Qij are logically closed un- Q11 Q12 der ↔ such that in the case D1 = 3 correlations Q22 of correlations such as Qij ↔ Qkl do not define Q12 Q22 Q21 any new independent questions. Accordingly, Q21 Q33 Q33

′ QM2 = {Qi,Qj,Qij}i,j=1,2,3 implies for D1 = 3 either

Q33 = Q11 ↔ Q22, or Q33 = ¬(Q11 ↔ Q22) (19) PSfrag replacements PSfrag replacements and either

Q33 = Q12 ↔ Q21, or Q33 = ¬(Q12 ↔ Q21) (20) Q3 Q3 such that either QD1 QD1

Q11 ↔ Q22 = Q12 ↔ Q21, or Q11 ↔ Q22 = ¬(Q12 ↔ Q21) (21) ′ ′ Q3 Q3 ′ ′ QD1 QD1 In sections 5.2.7 and 5.4 we shall determineQ11 corresponding to the two correlation questions whether negations ¬ occur in the expressions (16– graphs Q21 ′ ′′ 18). For Q,Q ,Q maximally compatible and re-Q22 lated by an XNOR, we shall henceforth distin- ′ Q11 ′ Q1 Q12 Q1 Q12Q1 Q1 guish between Q33 ˜ Q Q′ Q Q22 Q′ even correlation: if Q = Q′ ↔ Q′′, and Q33 2 Q21 2 2 2 . . . Q33 . Q˜33 odd correlation: if Q = ¬(Q′ ↔ Q′′).32 . .

5.2.5 An informationally complete set for N = 2 (we add the correlation of correlations as a new rebits edge without vertices to the graph). The dif- Next, let us consider the rebit case D1 = 2. O can ference to the qubit case is that Q33, Q˜33 can ′ ′ ask the four individual questions Q1,Q2,Q1,Q2 now not be written as correlations of individ- ′ and the four correlations Q11,Q12,Q21,Q22. But ual questions Q3,Q3 since the latter do not ex- O can also define the two new correlation of cor- ist. Clearly, Q12,Q21,Q33 and Q11,Q22, Q˜33 are relations questions two pairwise independent sets. The question is whether Q33, Q˜33 are independent of each other Q33 := Q12 ↔ Q21, Q˜33 := Q11 ↔ Q22 (22) and of the individuals. The following lemma even 32We note that this also implies Q′ = ¬(Q ↔ Q′′) and asserts complementarity to the latter. (Note that Q′′ = ¬(Q ↔ Q′). this lemma holds trivially in the case D1 = 3.)

Accepted in Quantum 2017-11-27, click title to verify 33 Lemma 5.8. The questions asking for the cor- 2, O is not allowed to experience a net loss of relation of the bipartite correlations, Q12 ↔ Q21 information. Hence, Q12 (or Q21) must be max- and Q11 ↔ Q22, are maximally complementary imally compatible with Q˜33 (they are also in- ′ ′ ˜ to and independent of any Q1,Q2,Q1,Q2. dependent). That is, Q12,Q21, Q33 are mutu- ally maximally compatible according to theorem Proof. We firstly demonstrate independence of 3.2 (Specker’s principle) and O may also ask all Q1 and Q11 ↔ Q22. This follows from not- three at the same time. But then the same ar- ing that Q22 is maximally complementary to gument as in the proof of theorem 5.6 applies Q (lemma 5.2) and maximally compatible with 1 such that either Q˜33 = Q12 ↔ Q21 or Q˜33 = Q ↔ Q and arguments completely analogous 11 22 ¬(Q12 ↔ Q21). This implies either Q33 = Q˜33 or to those in the proof of lemma 5.3. Next, we es- Q33 = ¬Q˜33. Accordingly, there is only one in- tablish complementarity of Q ↔ Q and Q . 11 22 1 dependent correlation of correlations Q33. Since Suppose the contrary, namely, that Q ↔ Q 11 22 Q11,Q12,Q21,Q22 and Q33 are then, by construc- and Q were at least partially compatible such 1 tion, logically closed under the XNOR ↔ and Q33 that there must exist a state with αQ11↔Q22 = 1 is maximally complementary to all individuals bit and α1 > 0 bit. Clearly, both are maxi- ′ ′ Q1,Q2,Q1,Q2, no further pairwise independent mally compatible with and independent of Q11. question can be built from this set such that it is According to assumption 4, asking Q to this 11 informationally complete. Hence, D2 = 9. state does not change O’s information about Q ↔ Q and Q . However, maximal infor- 11 22 1 5.2.6 Entanglement and non-local tomography for mation about Q ↔ Q and Q also implies 11 22 11 rebits maximal information about Q22 which is max- imally complementary to Q1. Hence, Q1 and The rebit case D1 = 2 thus has a very spe- Q11 ↔ Q22 must be maximally complementary. cial question and correlation structure: Q33 = The arguments works analogously for the other Q12 ↔ Q21 is the only composite question which cases. is maximally complementary to all individuals, but maximally compatible with all correlations Finally, we show that (19–21) hold analogously Qij. By contrast, e.g., Q11 is maximally comple- for the rebit case D1 = 2. From this it follows mentary to Q ,Q′ ,Q ,Q and maximally com- ′ 2 2 12 21 that the four individual questions Qi,Qj, the four ′ patible with Q1,Q1,Q22 and maximally compati- correlations Qij, i, j = 1, 2, and the correlation ble with the correlation of correlations Q33. Con- of correlations Q33 form an informationally com- sequently, Q33 assumes a special role in the en- plete set Q for rebits. That is, in contrast M2 tanglement structure: once O knows the answer to the qubit case, there is now one non-trivial to this question he may no longer have any fur- correlation of correlations question which is pair- ther information about the outcomes of individ- wise independent from the individuals and corre- ual questions, but may have information about lations. correlation questions. That is, two rebits can be in a state of non-maximal information of 1 bit Theorem 5.9. (Rebits) If D1 = 2 then QM2 = ′ relative to O, corresponding to the latter only {Qi,Q ,Qij,Q33}i,j=1,2 constitutes an informa- j having maximal knowledge about the answer to tionally complete set for N = 2 with D2 = 9 and either Q33 and no information otherwise, and still be maximally entangled because any individual in- Q11 ↔ Q22 = Q12 ↔ Q21, or formation is forbidden in that situation. Notice that this is not true for pairs of qubits because Q11 ↔ Q22 = ¬(Q12 ↔ Q21). even if everything O knew about the pair was the Proof. O can begin by asking S the maximally answer to Q33, he could still acquire information ′ compatible Q11 and Q22 upon which he also about the individuals Q3,Q3 such that one could knows the answer to Q˜33. O then possesses the not consider such a state as maximally entangled. maximal amount of N = 2 independent bits Even stronger, O will always know the answer of information about S. Next, O can ask Q12 to Q33 if the two rebits are maximally entangled (or Q21) which, according to lemma 5.5, is max- and he has maximal information about S. This imally complementary to Q11,Q22. But, by rule follows from (6): for a maximally entangled state

Accepted in Quantum 2017-11-27, click title to verify 34 of maximal information (no information about in- This question structure also has severe reper- dividuals), the following must hold cussions for rebit state tomography: it must ul-

2 2 timately be non-local. For rebits, D1 = 2, the ′ probability that Q = 1 could be written as IO→S = (αi + αi) + αij + α33 33 i=1 i,j=1 X X y = p(Q = 1,Q = 1)+ p(Q = 0,Q = 0) =0 33 12 21 12 21 = α11 + α12 + α21 + α22 + α33 = 3 bits. | {z } where p(Qi,Qj) denotes here the joint probabil- The last equality follows from the fact that once ity distribution over Qi,Qj. That is, in a multiple O knows the answers to two maximally compat- shot interrogation, O could ask both Q12,Q21 to ible questions, he will also know the answer to the identically prepared rebit couples and from their correlation and since this correlation is in the statistics over the answers, he could also de- the pairwise independent question set the total termine y33. But this probability cannot be de- information defined in (6) will always yield 3 bits composed into joint probabilities over the individ- ′ for N = 2 independent bit systems in states of ual questions according to Q33 = “(Q1 ↔ Q2) ↔ 33 ′ ′ ′ maximal knowledge. If we now require that O (Q2 ↔ Q1)” because Q1,Q2 and Q1,Q2 are max- cannot know more than a single bit about max- imally complementary. Therefore, O would not imally complementary questions then, in a state be able to determine y33 by only asking individ- of maximal information of N = 2 independent ual questions to the two rebits and the statis- bits, we must have tics over these answers.34 That is, for rebits, state tomography would always require correla- α + α = α + α = 1 bit 11 12 21 22 tion questions and in this sense be non-local. ⇒ α33 = 1 bit Note that this stands in stark contrast to qubit ′ such that O must have maximal information pairs where Q33 = Q3 ↔ Q3 can be written about Q . Therefore, the correlation of corre- in terms of individual questions such that also 33 ′ ′ y33 = p(Q3 = 1,Q = 1)+ p(Q3 = 0,Q = 0) can lations Q33 can be viewed as the litmus test for 3 3 entanglement of two rebits. be determined by the statistics over the answers ′ ′ ′ to Q3,Q3 only. Qubit systems (and quantum the- We note that the individuals Q1,Q2,Q1,Q2 will ultimately correspond to projections onto the ory in general) are thus tomographically local. The requirement of tomographic locality, ac- +1 eigenspaces of σx,σz, while the Qij corre- spond to projections onto the +1 eigenspaces of cording to which the state of a composite system can be determined by doing statistics over mea- σi ⊗ σj, i, j = x, z, and Q33 corresponds to the surements on its subsystems, is a standard con- projection onto the +1 eigenspace of σy ⊗ σy, dition in the GPT framework [15–18, 20, 55, 59] where σx,σy,σz are the Pauli matrices [1,2]. Ob- servables and density matrices on a real Hilbert and thus directly rules out rebit theory. How- space correspond to real symmetric matrices. ever, in contrast to derivations within the GPT landscape, we shall not implement local tomog- This is the reason why σy is not an observable 2 raphy here because it will be interesting to see on R (it corresponds to the ‘missing’ Q3), but the differences between real and complex quan- σy ⊗ σy is a real symmetric matrix and thus an observable on R2 ⊗ R2. tum theory from the perspective of information This gives a novel and simple explanation for inference and we shall thus carry out the recon- struction of both here and in [1, 2]. the discovery that σy ⊗ σy determines the entan- glement of rebits [83]: a two-rebit density matrix 34For example, in a multiple shot interrogation O could ρ is separable if and only if Tr(ρσy ⊗ σy) = 0, ′ first ask Q1,Q2 on a set of identically prepared rebit cou- i.e. if the state has no σy ⊗ σy component. This ples to find out whether the answers are correlated. On a ′ statement means in our language that a state is second identically prepared set, O could then ask Q2,Q1 separable if and only if α33 = 0 and is consistent which are maximally complementary to the first questions with our observation above because any informa- he asked. From the statistics over the answers, O would be able to determine also the probabilities for Q12 and Q21. tion about the individuals is incompatible with But since he needed two separate interrogation runs to information about Q33 due to complementarity. determine the statistics for Q12 and Q21, he would not be able to infer from this any information whatsoever about 33 We are implicitly using here also rules 3 and 4. the statistics of answers to Q33.

Accepted in Quantum 2017-11-27, click title to verify 35 Local tomography is usually taken as the ori- instructive to illustrate these equations by means gin of the tensor product structure for composite of a question configuration analogous to a Bell systems in quantum theory [16, 17, 55, 59]. How- scenario. ever, one has to be careful with this statement (A) Suppose Q11 ↔ Q22 = Q12 ↔ Q21 was because there exist two distinct tensor products: true. Before we determine whether it is consis- there is (a) the tensor product of Hilbert spaces, tent with our background assumptions and pos- C2 ⊗N R2 ⊗N e.g., ( ) for qubits and ( ) for rebits, and tulates, we note that this is the case of classi- (b) the tensor product of unnormalized probabil- cal (or realist) logic: O can consistently interpret ity vectors or density matrices. A tensor prod- any such configuration by means of a local ‘hid- uct of type (b) defines a sufficient support only den variable’ model. For example, consider the for composite qubit systems, but not for compos- case Q11 = Q22 = 1 (which is compatible with ite rebit systems; the space of hermitian matrices the rules) such that Q ↔ Q = 1 and, conse- C2 ⊗N 11 22 over ( ) is the N-fold tensor product of her- quently, also Q ↔ Q = 1. We represent the C2 12 21 mitian matrices over , but the space of sym- last two equations by the following two graphs metric matrices over (R2)⊗N is not PSfragthe N-fold replacements PSfrag replacements tensor product of symmetric matrices over R2 (see Q12 also [83]). What local tomography implies is the Q21 Q11 Q12 tensor product (b), but as the example of rebits Q11 Q22 shows, the tensor product of type (a) also exists Q22 Q21 without it.

5.2.7 A Bell scenario with questions: ruling out where both edges solid and of equal colour means local hidden variables that the corresponding questions are correlated. O could consistently read this configuration as We shall now settle the issue of the relative nega- follows: “Since “Q = Q′ ” and “Q = Q′ ”,I tion ¬, i.e. whether 1 1 2 2 would precisely have to conclude that “Q12 = Q11 ↔ Q22 = Q12 ↔ Q21, or Q21”, i.e. Q12 ↔ Q21 = 1, if the individuals ′ ′ Q1,Q2,Q ,Q had definite values which, how- Q11 ↔ Q22 = ¬(Q12 ↔ Q21), 1 2 ever, I do not know.” For instance, Q11 = Q22 = for both rebits and qubits; one of these equations Q12 ↔ Q21 = 1 would be consistent with the four must be true (see theorems 5.7 and 5.9). It is ontic states:

PSfrag replacements PSfrag replacements PSfrag replacements PSfrag replacements 1 1 1 1 0 0 0 0 1 01 1 0 0 0 0 1 1 .

Therefore, O could join the two graphs consis- ilarly, O can interpret any other configuration tently (without knowing the truth values of the with Q11 ↔ Q22 = Q12 ↔ Q21 in terms of a individuals) local ‘hidden variable’ model which assigns defi- nite values to the individuals and, conversely, as can be easily checked, any assignment of definite ′ ′ (23) (or ontic) values to the individuals Q1,Q2,Q1,Q2 leads necessarily to Q11 ↔ Q22 = Q12 ↔ Q21. i.e., draw all four edges simultaneously which cor- We now preclude this possibility. We note that responds to all four edges having definite (al- Q11 ↔ Q22 = Q12 ↔ Q21 can be rewritten in beit unknown) values at the same time. Sim- terms of the individuals as

′ ′ ′ ′ (Q1 ↔ Q1) ↔ (Q2 ↔ Q2) = (Q1 ↔ Q2) ↔ (Q2 ↔ Q1) (24)

Accepted in Quantum 2017-11-27, click title to verify 36 which is a classical logical identity that is al- switched. In classical Boolean logic, this iden- ways true for classical Boolean logic. It states tity follows from the associativity and symme- that the brackets can be equivalently regrouped, try of the XNOR. Namely, suppose for a moment ′ ′ i.e. that the order in which the XNOR ↔ that Q1,Q2,Q1,Q2 had simultaneous truth val- is executed in the logical expression can be ues. Then Boolean logic would tell us that

′ ′ ′ ′ ′ ′ (Q1 ↔ Q1) ↔ (Q2 ↔ Q2)= (Q1 ↔ Q1) ↔ Q2 ↔ Q2 = Q1 ↔ (Q2 ↔ Q1) ↔ Q2 ′ ′ ′ ′ = Q 2 ↔ Q1 ↔ (Q2 ↔
Q1) = (Q1 ↔ Q2) ↔ (Q2
↔ Q1). (25)

′ ′ Of course, relative to O, Q1,Q2,Q1,Q2 do not is also consistent with assumption 5 since only have simultaneous truth values. Instead, assump- maximally compatible questions are directly con- tion 6 implies that, for a classical logical iden- nected. tity to hold in O’s model, either (a) all ques- It is impossible for O to interpret this situation tions in the involved logical expressions are mu- in terms of local ‘hidden variables’ which assign tually maximally compatible or (b) the identity definite values to the individuals simultaneously. follows from applying classical rules of inference Namely, consider, again, the case Q11 = Q22 = to (sub-)expressions which are entirely written in 1 such that Q11 ↔ Q22 = 1 and, consequently, terms of mutually maximally compatible ques- now Q ↔ Q = 0. This configuration may be PSfrag replacements 12 PSfrag21 replacements tions and subsequently decomposing the result graphically represented as Q logically into other questions (with possibly dis- 12 Q21 Q11 tinct compatibility relations). However, since Q12 ′ ′ Q11 Q1,Q2 and Q ,Q form maximally complemen- 1 2 Q22 Q Q tary pairs, neither is the case here. In particu- 22 21 lar, the right hand side in (24) does not follow from applying any rules of Boolean logic to the total expression Q11 ↔ Q22 or those subexpres- where one edge solid the other dashed, but same ′ ′ sions Q1 ↔ Q1 and Q2 ↔ Q2 on the left hand colour, means that the corresponding answers are side which are fully composed of maximally com- anti-correlated. One can easily check that, in con- patible questions.35 Hence, (24) violates assump- trast to (23), it is impossible to consistently join tion 6 such that we must rule out the possibility the two diagrams by drawing all four edges si- Q11 ↔ Q22 = Q12 ↔ Q21. multaneously (which would correspond to all in- dividuals and thus all edges having definite truth (B) We are thus already forced toPSfrag the conclu- replacements PSfrag replacements sion that values), as one would obtain a frustrated graph

Q11 ↔ Q22 = ¬(Q12 ↔ Q21). (26) must hold. Indeed, its equivalent representation < . (27) ′ ′ (Q1 ↔ Q1) ↔ (Q2 ↔ Q2) ′ ′ = ¬(Q1 ↔ Q2) ↔ (Q2 ↔ Q1) O must conclude that neither the four individu- does not constitute a classical logical identity als nor the four edges can have definite (but un- and is thus consistent with assumption 6. It known) values all at the same time. The same 35The only relevant rules in this case, as in (25), follow verdict holds for any other configuration with from the properties of the XNOR, namely its symmetry Q11 ↔ Q22 = ¬(Q12 ↔ Q21). and associativity in Boolean logic. However, it is clear Since either (A) or (B) must be true by theo- that, thanks to the occurring complementarity, the argu- ments of (25) no longer apply. For instance, the first step rems 5.7 and 5.9, and (A) violates assumption 6, in (25) is illegal, according to assumption 5, because Q2 while (B) is consistent with both assumptions 5 ′ and Q1 ↔ Q1 are complementary such that they may not and 6, we conclude that (26) is the correct rela- be directly connected. tion.

Accepted in Quantum 2017-11-27, click title to verify 37 Note that this argument holds for both rebits made for any four questions (i.e., not necessar- and qubits and, for qubits, also for any other pairs ily individuals) Q,Q′,Q′′,Q′′′ which are such that ′ ′ ′ ′′ ′′′ Qi,Qj6=i and Qk,Ql6=k of pairs of maximally com- Q,Q and Q ,Q are maximally complementary plementary individuals (see theorem 5.7). In fact, pairs, while Q and Q′ are each maximally com- even more generally, the same argument can be patible with both Q′′,Q′′′; also in this case one would have to conclude that

(Q ↔ Q′′) ↔ (Q′ ↔ Q′′′)= ¬ (Q ↔ Q′′′) ↔ (Q′ ↔ Q′′) . (28)

This will become relevant later on. For now we Q33 in (19, 20) and more generally in theorem 5.7 observe that exchanging the positions of the com- for other permutations of non-intersecting edges. plementary Q′′,Q′′′ from the left to the right hand This difference between rebits and qubits results side introduces a negation ¬. again from the fact that Q33 is defined as the ′ This relative negation ¬ in (28) precludes a correlation of the individuals Q3,Q3 for qubits, classical reasoning for the distribution of truth while it is a correlation of correlations for rebits. values over O’s questions. In fact, as just seen, This has a remarkable consequence: rebit theory it rules out local ‘hidden variable models’, analo- is its own ‘logical mirror image’, while qubit the- gously to the Bell arguments. We also recall that ory’s ‘logical mirror image’ is distinct from qubit assumption 6 was a statement about which rules theory. This topic will be deferred to section 5.4 of inference and logical identities are not appli- because we firstly need to understand the ques- cable to logical compositions involving mutually tion structure for the N = 3 case in order to complementary questions. By contrast, (28) is discuss the odd and even correlation structure of now a first non-classical logical identity showing two qubits further. one possible non-classical way of reasoning in the presence of complementarity. 5.3 Three gbits The correlation structure for rebits is now clear from these results; (26) implies Q33 = ¬Q˜33 for It will be both useful and instructive to explicitly the correlations of correlations defined in (22). consider the N = 3 case for rebits and qubits. This settles the fate of all possible relative nega- As a composite system, we can view three gbits, tions ¬ for rebits. However, these results do labeled by A,B,C, either as three individual sys- not fully determine the odd and even correla- tems, as three combinations of one individual and tion structure for qubits: we still have to clarify a bipartite composite system or as a tripartite whether there is an overall negation ¬ relative to system:

PSfrag replacements PSfrag replacements PSfrag replacements C C C

A B A B A B PSfrag replacements PSfrag replacements C C

A B A B (29)

According to definition 3.5 of a composite system, and any permissible logical connectives thereof. Q3 must then contain the individual questions of This results in different structures for rebits and all three gbits, any bipartite correlation questions qubits.

Accepted in Quantum 2017-11-27, click title to verify 38 5.3.1 Three qubits correlations and Q˜111 would not be pairwise in- dependent. PSfragWe replacements shall begin with the qubit case D1 = 3. From (13) we already know that the logi- Clearly, according to definition 3.5, all 3 × 3 = cal connective yielding new pairwise independent 9 individual questions, henceforth denoted as questions must either be the XNOR or the XOR. Qi ,Qj ,Qk , and all 3 × 9 = 27 bipartite A B C For consistency with the bipartite structure, we correlation questions, from now on written as continue to employ the XNOR. There is an obvi- Qi j ,Qi ,k ,Qj k , i, j, k = 1, 2, 3, will be part A B A C B C ous candidate for an independent tripartite ques- of an informationally complete set QM . As be- 3 tion, namely fore, we represent individuals and correlations graphically as vertices and edges, respectively, e.g. Qijk = QiA ↔ QjB ↔ QkC (30) A B C which thanks to the associativity and symmetry Q Q 1A 1C of ↔ can also equivalently be written as

Q1A1C Qijk = Qi ↔ Qj k = Qi j ↔ Qk Q1A3B Q A B C A B C PSfrag replacementsQ2A 1B Q2C Q1B 3C = QiAkC ↔ QjB . (31)

Q2A2B

Q2B 2C Q3A Q2B Q3C (Since the notation for this tripartite ques- Q111 tion is unambiguous from the ordering of Q333 i, j, k we drop the subscripts A,B,C in Qijk.) Q 322 Q3B This structure is also natural from the differ- ent compositions in (29). Qijk thus defined is by construction maximally compatible with

QiA ,QjB ,QkC ,QiAjB ,QiAkC , QjBkC and, for sim-

A B C ilar reasons to the independence of QiAjB from Q Qi ,Qj (see the discussion below (14)), also Q 1A1C Q A B 1A 1C pairwise independence of the latter. Note that Q1 this question does not stand in one-to-one cor- Q1A3B B respondence with the proposition “the answers Q2A Q2C PSfrag replacements to QiA ,QjB ,QkC are the same"; e.g., QiA = 1 Q2A2B Q2B 2C and QjB = QkC = 0 also gives Qijk = 1. It is Q1B 3C easier to interpret this question via (31) as ei- Q Q2B Q 3A 3C ther of the three questions “are the answers to Q111 QiA ,Qj k /QiAjB ,Qk /Qi k ,QjB the same?". Q333 B C C A C Q There are 3 × 3 × 3 = 27 such tripartite 322 Q3B questions Qijk, i, j, k = 1, 2, 3. We shall repre- depict valid question graphs. sent them graphically as triangles. For example, But in order to complete the individuals and Q111,Q322,Q333 are depicted as follows: bipartite correlations to Q we now have to M3 A B C consider logical connectives of these questions

which are pairwise independent. This will nec- Q1A Q1C essarily involve tripartite questions because the Q111 bipartite structure is already exhausted with in- Q1A1C dividuals and bipartite correlations. Clearly, we Q1A3B Q1B Q2A Q2C cannot add a question Q˜111, representing the Q1B 3C Q322 Q2 2 proposition “the answers to Q1A ,Q1B ,Q1C are the A B Q2 2 Q2B same" to the individuals and bipartite correla- B C Q3A Q3C tions because, e.g., Q˜111 = 1 would always imply Q333

Q1A1B = Q1A1C = Q1B 1C = 1. Also, Q1A1B = 0 ˜ would imply Q111 = 0 such that the bipartite Q3B

Accepted in Quantum 2017-11-27, click title to verify 39 PSfrag replacements PSfrag replacements A A B B C C

Q1A Q1A

Q2A Q2A

Q3A Q3A

Q1B Q1B

Q2B Q2B

Q3B Q3B Q Q 5.3.2 Independence and compatibility for three 1C therefore maximally complementary:1C Q Q qubits 2C 2C Q3C Q3C

We have to delve into some technical details onQ1B 1C Q1B 1C the independence and compatibility structureQ to1A3B Q1A3B explain and understand monogamy and the en-Q1B 3C Q1B 3C tanglement structure of three qubits. These re- Q2A2B Q2B 2C Q2A2B sults will also be used to prove the analogous re- Q2B 2C sults by induction in section 5.5.1 for N qubits. Q111 Q111 Individual questions from qubit A are maxi-Q333 Q333 mally compatible with the individual questionsQ322 Q322 from qubits B and C, etc. But what about the compatibility of bipartite and tripartite correla- tions? The compatibility structure of bipartite We continue with the tripartite questions. correlations of a fixed qubit pair is clear from lemma 5.5, but we have to investigate compat- Lemma 5.11. Qijk is maximally compatible with ibility of bipartite correlation questions involving QiA ,QjB ,QkC and maximally complementary to all three qubits. QlA6=iA , QmB 6=jB ,QnC 6=kC . That is, graphically, Qijk is maximally compatible with an individ-

ual QiA,B,C if the corresponding vertex is one of Lemma 5.10. QiAjB and QlBkC are maximally complementary if j 6= l. On the other hand, the vertices of the triangle representing Qijk and maximally complementary otherwise. QiAjB and QjBkC are maximally compatible and it holds PSfrag replacements Proof. Q is by construction maximally com- ijk A patible with Qi ,Qj ,Qk . On the other hand, Q ↔ Q = Q . BA B C iAjB jBkC iAkC complementarity of Q and Q is shown by C ijk lA6=iA noting that bothQ are maximally compatible with The analogous statements hold for any permuta- 1A and independentQ of QjBkC and lemma 5.1. One tion of A,B,C. That is, graphically, two bipar- 2A argues analogouslyQ for the individuals of qubits tite correlations involving three qubits are max- 3A B and C. Q imally compatible if the corresponding edges in- 1B Q2B tersect in a vertex and maximally complementary For instance, Q111 is maximally compatible Q3B otherwise. with Q1C and maximally complementary to Q2C :

Proof. Maximal complementarity of QiAjB and Q3 Q1C Ql k for j 6= l is proven by noting that both C B C Q Q111 are maximally compatible with and independent 1B 1C Q1 3 of Qi , the relation Qj = Qi ↔ Qi j and A B A B A A B Q Q lemma 5.1. 1B 3C 2C Q2A2B QiAjB and QjBkC are evidently maximally com- Q2B 2C patible since QiA ,QjB ,QkC are maximally com- patible. Moreover, Q333 Q322 QiAjB ↔ QjBkC = QiA ↔ (QjB ↔ QjB ) ↔ QkC =1 = Q iAkC | {z } This lemma also directly implies that any in- dividual and any tripartite correlation question thanks to the associativity of ↔. are pairwise independent because (1) maximally complementary questions are in particular inde-

For example, Q2A2B and Q2B 2C intersect in pendent, and (2) for maximally compatible ques-

Q2B and are thus maximally compatible, while tion pairs such as QiA ,Qijk independence argu-

Q2A2B and Q1B 1C do not share a vertex and are ments analogous to those surrounding (14) apply.

Accepted in Quantum 2017-11-27, click title to verify 40 Next we consider bipartite and tripartite cor- QmB nC ,QlAnC ,QlAmB from Qijk for l 6= i, relation questions. m 6= j and k 6= n because maximally comple- mentary questions are by definition independent Lemma 5.12. Qijk is maximally compatible and Qijk and QiAjB ,QiAkC ,QjB kC are pairwise with Qi j ,Qi k ,Qj k and, furthermore, with A B A C B C independent. Consider therefore Qijk and QmB nC , QlAnC and QlAmB for l 6= i, m 6= j QmB nC for j 6= m and k 6= n. By lemma 5.11, and k 6= n. On the other hand, Qijk is max- QkC is maximally compatible with Qijk and by imally complementary to Qi m ,Qi n ,Ql j , A B A C A B lemma 5.2 maximally complementary to Qm n . Q ,Q ,Q for l 6= i, m 6= j and B C jB nC lAkC mB kC This implies, using the arguments from the proof k 6= n. That is, graphically, Qijk is maximally of lemma 5.3, independence of Qijk,Qm n . compatible with a bipartite correlation if the edge B C The other cases follow similarly. of the latter is either an edge of the triangle cor- responding to Qijk or if the edge and triangle do Lemma 5.14. The tripartite correlation ques- not intersect. Qijk is maximally complementary tions Qijk, i, j, k = 1, 2, 3 are pairwise indepen- to a bipartite correlation question if the edge of dent. the latter and the triangle corresponding to Qijk share one common vertex. Proof. Consider Qijk and Qlmn for i 6= l. By

lemma 5.11, QiA is maximally compatible with Proof. Q is by construction maximally com- ijk Qijk and maximally complementary to Qlmn. Us- patible with QiAjB ,QiAkC ,QjBkC . Qijk = QiA ↔ ing the analogous arguments from the proof of PSfrag replacementsQjB kC and QmPSfragB nC are replacements also maximally compat- lemma 5.3, this implies that Qijk,Qlmn are inde- ible for j 6= m and k 6= n because Q is A A mB nC pendent. The same reasoning holds when j 6= m maximally compatible with Q and thanks to B iA B and k 6= n. lemma 5.5 also with Q . Complementarity of C jBkC C Qijk and Qi m for j 6= m follows from noting This has an immediate consequence: Q1 A B Q1 Athat both are maximally compatibleA with and in- Q2A Q2A Corollary 5.15. The individuals QiA ,QjB ,QkC , dependent of QkC and lemma 5.1. The reasoning Q3A Q3A the bipartite Q ,Q ,Q and the tripar- for all other cases is analogous. iAjB iAkC jBkC Q1 Q1 B B tite Qijk, i, j, k = 1, 2, 3 are pairwise independent Q2 Q2 B To give a graphical example, Q111B is maximally and thus, thanks to assumption 7, contained in Q3B Q3B compatible with Q1B 1C and Q3A3C and maxi- an informationally complete set QM3 . Q1C Q1C mally complementary to Q1A2B : Q2C Q2C Lastly, we consider the complementarity and

Q3C Q3C compatibility structure of the tripartite correla- Q111 Q111 Q1B1C tions. Q1B 1C Q1A3B Q1A3B Lemma 5.16. Qijk and Qlmn are maximally Q1A2B Q compatible if {i, j, k} and {l,m,n} overlap in one Q3A3C 1A2B or three indices and maximally complementary if Q2B 2C Q2B2C {i, j, k} and {l,m,n} overlap in zero or two in- Q3A3C dices. That is, graphically, Qijk and Qlmn are Q333 Q333 maximally compatible if their corresponding tri- Q322 Q322 angles intersect in one vertex (or coincide) and maximally complementary if the triangles share an edge or do not intersect. We still have to check pairwise independence of the bipartite and tripartite correlation questions. Proof. Compatibility for an overlap in all three indices is trivial. But also Q = Q ↔ Q Lemma 5.13. Any bipartite ijk iA jBkC and Qimn = Qi ↔ Qm n are clearly maxi- Qm n ,Ql n ,Ql m and any tripartite corre- A B C B C A C A B mally compatible for j 6= m and k 6= n because lation question Qijk are independent from one by lemma 5.5 Q ,Q are maximally com- another. jBkC mB nC patible in this case. Compatibility for the other Proof. Lemma 5.12 implies that we only cases of an overlap of Qijk and Qlmn in one index have to check pairwise independence of follows by permutation.

Accepted in Quantum 2017-11-27, click title to verify 41 The proof of the complementarity of Qijk and Theorem 5.17. (Qubits) The individuals

Qlmn for i 6= l, j 6= m and k 6= n follows from QiA ,QjB ,QkC , the bipartite QiAjB ,QiAkC ,QjB kC lemma 5.1. One may use the fact that, by lemma and the tripartite Qijk, i, j, k = 1, 2, 3 are logi- 5.12, both questions are maximally compatible cally closed under ↔ such that they form an in-

with and independent of QjBkC and that, by formationally complete set QM3 with D3 = 63 for

lemma 5.11, QiA6=lA is maximally complementary D1 = 3. to Qlmn. Similarly, one proves complementarity of Proof. For individuals and bipartite correlations

Qijk = QiA ↔ QjBkC and Qljk = QlA ↔ QjBkC of any pair of qubits the logical closure under the for i 6= l by using that both are maximally com- XNOR was already shown in section 5.2. Cor-

patible with and independent of QjBkC . Com- relation of individuals with a bipartite correla- plementarity of tripartite correlations for other tion question from another pair of qubits yields overlaps in precisely two indices follows by per- the tripartite correlation questions. Lemma 5.10 mutation. shows that also the bipartite correlation ques- tions involving distinct pairs of qubits are logi- For example, Q111 and Q212 intersect in the cally closed under ↔. We thus only have to check vertex Q and are thus maximally compatible. 1B logical closure of XNOR combinations involving By contrast, Q shares the edge Q with 111 1A1B tripartite correlation questions. Q and does not intersect at all with Q such 113 333 Lemma 5.11 shows that tripartite correlations that Q is maximally complementary to both. 111 and individuals are only maximally compatible if Q and Q intersect in the vertex Q and are 113 333 3C the individual is a vertex of the tripartite trian- therefore maximally compatible: gle. But then a combination such as

Q111 Q111 Qijk ↔ QiA = (QiA ↔ QiA ) ↔ QjB kC = QjBkC

Q212 produces another bipartite correlation, thanks to PSfrag replacements PSfrag replacements Q113 the associativity of ↔. Similarly, lemma 5.12 as- serts that tripartite and bipartite correlations are only maximally compatible if the edge of the bi- Q333 Q partite correlation is either contained in the tri- Q113 333 partite triangle or if the edge and triangle do not Q212 intersect. However, combinations of such maxi- mally compatible pairs also do not yield any new 5.3.3 An informationally complete set for three questions because, e.g., qubits

We now show that the 9 individuals, the 27 bi- Qijk ↔ QiAjB = QkC partite and the 27 tripartite correlation questions

form an informationally complete set QM3 for and, using again the associativity of XNOR and N = 3 qubits. theorem 5.7,

QσA(1)σB (1)k ↔ QσA(2)σB (2) = (QσA(1)σB (1) ↔ QσA(2)σB (2)) ↔ QkC = QσA(3)σB (3)k or ¬QσA(3)σB (3)k, Q Q = σA(3)σB (3) or ¬ σA(3)σB (3) | {z }

where σA,σB are permutations of {1, 2, 3}. Fi- sect in one vertex. But, using theorem 5.7 once nally, lemma 5.16 entails that tripartite correla- more, tions are only maximally compatible if they inter-

Accepted in Quantum 2017-11-27, click title to verify 42 QσA(1)σB (1)k ↔ QσA(2)σB (2)k = QσA(1)σB (1) ↔ QσA(2)σB (2) = QσA(3)σB (3) or ¬QσA(3)σB (3).

Permuting these examples implies that all in- that bipartite correlations of two distinct qubit dividuals, bipartite and tripartite correlations pairs are only maximally compatible if their cor- are logically closed under ↔ which by (13) is responding edges intersect in a vertex. Clearly, the only independent logical connective possibly there can then not exist any edge which is max-

yielding new independent questions. These ques- imally compatible with both Q1A1B ,Q2A2B . Sec- tions therefore form an informationally complete ondly, lemma 5.12 asserts that a bipartite correla-

set QM3 with D3 = 63. tion is only maximally compatible with a tripar- tite correlation if the corresponding edge is either 5.3.4 Entanglement of three qubits and contained in the tripartite triangle or does not monogamy intersect the triangle. This means that the only tripartite correlations maximally compatible with Let us now put all these detailed results to good Q1 1 ,Q2 2 are (a) the correlations of the lat- use and reward ourselves with the observation A B A B ter with the individuals Q1 ,Q2 ,Q3 of qubit that they naturally explain monogamy of entan- C C C C and (b) Q33k, k = 1, 2, 3. glement for the qubit case. This is best illus- For example, in the first of the following two trated with an example. Let O have asked the question graphs Q2B 2C is maximally compati- questions Q1 1 and Q2 2 to qubits A, B such A B A B ble with Q2 2 but maximally complementary to that the two are in a state of maximal informa- A B Q1 1 , while Q3 3 is maximally complementary tion of N = 2 independent bits and maximal A B B C to both. But O could ask Q222 together with entanglement relative to O. Q1A1B ,Q2A2B , as depicted in the second graph:

Q1A1B Q Q PSfrag replacements 1A1B PSfrag replacements 1A1B

Q2C Q2A2B Q222 Q2A2B Q2B 2C Q2A2B PSfrag replacements Q222

Q2C

Q3B 3C Q3B 3C Q2B 2C

Intuitively, this already suggests monogamy of

entanglement because O has spent the maximally However, asking Q1A1B ,Q2A2B and Q222 simul-

attainable amount of information over the pair taneously is equivalent to asking Q1A1B ,Q2A2B

A, B such that even if now a third qubit C enters and Q2C because Q2A2B ↔ Q222 = Q2C . the game he should no longer be able to ask any The analogous conclusion holds for any other question to the triple which gives him any fur- tripartite question maximally compatible with

ther simultaneous independent information about Q1A1B ,Q2A2B . Therefore, once O has asked the pair A, B. But any correlation of A or B the last to bipartite correlations about qubits with C would constitute additional information A, B, he can only acquire individual informa-

about A or B which would violate the informa- tion corresponding to Q1C ,Q2C ,Q3C about qubit tion bound for the subsystem of the composite C. Clearly, the same state of affairs is true for system of three qubits. any permutation of the three qubits. This is We shall now make this more precise. The monogamy of entanglement in its most extreme question is, which bipartite or tripartite correla- form: two qubits which are maximally entangled tions involving qubit C are maximally compatible can not be correlated whatsoever with any other

with Q1A1B ,Q2A2B . Firstly, lemma 5.10 states system.

Accepted in Quantum 2017-11-27, click title to verify 43 The non-extremal form of monogamy, namely is permutation invariant and measures only the the case that a qubit pair is not maximally en- information contained in the tripartite questions tangled and can thus share a bit of entanglement and thus genuine tripartite entanglement. This with a third qubit, can also be explained. To this is how one can describe the general form of end, we recall from quantum information theory monogamy of entanglement in our language. Of that monogamy is generally described with so- course, at this stage the information measure αi called monogamy inequalities, e.g., the Coffman- about the various questions Qi is only implicit, Kundu-Wootters inequality [84] but we shall derive its explicit form in section 6.8, upon which the above statements become τ ≥ τ + τ , (32) A|BC AB AC truly quantitative. These informational versions 2 where 0 ≤ τA|BC = 2(1 − Tr ρA) ≤ 1 measures of monogamy inequalities and tangles naturally the entanglement between qubit A and the qubit suggest themselves for simple generalizations to pair B,C and ρA is the marginal state of qubit A arbitrarily many qubits, thereby complementing obtained after tracing qubit B and C out of the current efforts in the quantum information liter- (not necessarily pure) tripartite state. τAB,τAC , ature (e.g., see [85]). on the other hand, measure the bipartite entan- Before we move on, we briefly emphasize that glement of the pairs A, B and A, C and are simi- monogamy is a consequence of complementarity larly obtained by tracing either C or B out of the – as is entanglement. For example, three classi- tripartite qubit state. The inequality (32) formal- cal bits also satisfy the limited information rule izes the intuition that the correlation between A 1, but due to the absence of complementarity, and the pair B,C is at least as strong as the cor- O could ask all correlations QAB,QAC ,QBC and relation of A with B and C individually. This QABC at once is the general form of monogamy. One usually also defines the so-called three-tangle [84] as the PSfrag replacementsA B difference between left and right hand side

τABC = τA|BC − τAB − τAC ∈ [0, 1] to measure the genuine tripartite entanglement C shared among all three qubits. This three-tangle which, however, is equivalent to asking the three turns out to be permutation invariant τABC = individuals QA,QB,QC . τBCA = τCAB. In our current case, a measure of entanglement 5.3.5 Maximal entanglement for three qubits sharing must be informational. At this stage, in line with our previous definition of entangle- Let us elucidate maximal tripartite entanglement ment, we simply define the analogous entangle- for three qubits, corresponding to O asking three ment measures to be the sum of O’s information mutually maximally compatible and independent about the various independent bipartite or tripar- tripartite correlation questions such that he ex- tite correlation questions hausts the information bound of N = 3 indepen- bits 3 3 3 dent with tripartite information. Lemma 5.16 asserts that tripartite questions are maxi- τ˜A|BC := αijk + αiAjB + αiAkC mally compatible if and only if they intersect in i,j,kX=1 i,jX=1 i,kX=1 3 3 exactly one vertex. For example, O could ask

τ˜AB := αiAjB , τ˜AC := αiAkC . (33) Q211,Q121,Q112 simultaneously. These are mu- i,jX=1 i,kX=1 tually independent because by (19, 20, 26)

With this definition, an informational inequality Q211 ↔ Q121 = Q3A3B or ¬Q3A3B analogous to the inequality (32) is trivial Q211 ↔ Q112 = Q3A3C or ¬Q3A3C

τ˜A|BC ≥ τ˜AB +τ ˜AC Q121 ↔ Q112 = Q3B3C or ¬Q3B3C , (34) as is the fact that the informational three-tangle i.e., their binary connectives with an XNOR do 3 not imply each other and the bipartite correla- tions are pairwise independent from the tripar- τ˜ABC :=τ ˜A|BC − τ˜AB − τ˜AC = αijk tite ones. Accordingly, asking Q ,Q ,Q i,j,kX=1 211 121 112

Accepted in Quantum 2017-11-27, click title to verify 44 will provide O with three independent bits of view the bit stemming from, say, the answer to information about the qubit triple. We note that Q2A1B 1C as being carried to one-third by each lemma 5.11 implies that once O has posed the of A,B,C.) Furthermore, if O now wanted to questions Q211,Q121,Q112, every individual ques- ‘marginalize’ over qubit C, i.e. discard any infor- tion QiA ,QjB ,QkC is maximally complementary mation involving qubit C, all that would be left of to at least one of the three tripartite correlations. his knowledge about the qubit triple would be the

That is, O cannot acquire any individual infor- answer to Q3A3B . But, as argued in section 5.2.6, mation about the three qubits and will only have this resulting ‘marginal state’ of qubits A, B rel- composite information. This is a necessary con- ative to O cannot be considered entangled. The dition for maximal entanglement. analogous results hold for ‘marginalization’ over In addition to the three implied binary corre- either A or B. As a result, the question graph lations (34), O would clearly also know the tri- (35) corresponds to genuine maximal tripartite partite correlation of all three Q211,Q121,Q112, entanglement. which amounts to 5.3.6 Three rebits Q211 ↔ Q121 ↔ Q112 = Q222 or ¬Q222. This exhausts the list of questions about We shall now repeat the same exercise for three which O would have information by asking rebits but can benefit from the results of the N = 3 qubit case. We shall thus be briefer here. The Q211,Q121,Q112. The list can be represented by the following question graph: reader exclusively interested in qubits may jump to section 5.4. According to definition 3.5, O’s questions to the three rebit system must contain the 6 indi-

viduals QiA ,QjB ,QkC , 12 bipartite correlations

QiAjB ,QiAkC ,QjBkC , i, j, k = 1, 2, and 3 bipartite

correlations of correlations Q3A3B ,Q3A3C ,Q3B 3C . To render this set informationally complete, we have to top it up with tripartite questions. There are now 8 tripartite correlations Qijk, i, j, k = 1, 2, of the kind (30, 31) and, moreover, 6 tri- (35) partite correlations of a rebit individual question with the question Q33 asking for the correlation of bipartite correlations of the other rebit pair This graph will ultimately correspond to Qi := Qi ↔ Q , the eight possible Greenberger-Horne-Zeilinger 33 A 3B 3C Q := Q ↔ Q , (GHZ) states in either of the three question 3j3 jB 3A3C Q := Q ↔ Q , i, j, k = 1, 2. bases {Q2A ,Q1B ,Q1C }, {Q1A ,Q2B ,Q1C } and 33k 3A3B kC {Q1 ,Q1 ,Q2 }, corresponding to the eight A A C Given that the individuals Q ,Q ,Q do not answer configurations ‘yes-yes-yes’, ‘yes-yes-no’, 3A 3B 3C ‘yes-no-no’, ‘yes-no-yes’, ‘no-yes-yes’, ‘no-yes-no’, exist for rebits there is now also no tripartite cor- ‘no-no-yes’ and ‘no-no-no’ to the three tripar- relation Q333. tite correlations Q211,Q121,Q112. Similarly, any other three maximally compatible tripartite cor- 5.3.7 Independence and compatibility for three relations will correspond to GHZ states in other rebits question bases.36 The independence, compatibility and comple- The correlation information in the graph (35) is mentarity structure for the questions not involv- clearly democratically distributed over the three ing an index i, j, k = 3 directly follows from the qubits A,B,C. (Intuitively, one could even qubit discussion as lemmas 5.10–5.16 also hold in 36As an aside, we note that these propositions solve the the present case for i, j, k = 1, 2. But we now little riddle in Zeilinger’s festschrift for D. Greenberger have to clarify the question structure once two [31]. indices are equal to 3 (an odd number of indices

Accepted in Quantum 2017-11-27, click title to verify 45 cannot be 3). The status of any such purely bi- Proof. One proves complementarity of QiAjB and partite relations was clarified in section 5.2 such Q3B 3C = Q1B 2C ↔ Q2B1C by employing that that here we have to consider the case that all both are maximally compatible with and inde- three rebits are involved. pendent of QiA and lemma 5.1. This also requires

invoking that QjB and Q3B3C are maximally com- Lemma 5.18. Q3B 3C is maximally complemen- plementary by lemma 5.8. tary to QiAjB , i, j = 1, 2, and maximally compat- Compatibility of Q3A3B and Q3B3C fol- ible with Q3A3B . Furthermore, lows indirectly. Namely, Q1A2B ,Q2B 1C and Q2 1 ,Q1 2 are two maximally compatible Q3 3 ↔ Q3 3 = Q3 3 . (36) A B B C A B B C A C pairs by lemma 5.10. We can then apply the The same holds for any permutations of A,B,C. reasoning around (28) to find

Q3A3B ↔ Q3B 3C = (Q1A2B ↔ Q2A1B ) ↔ (Q1B 2C ↔ Q2B 1C )

= ¬((Q1A2B ↔ Q2B 1C ) ↔ (Q1B 2C ↔ Q2A1B ))

= Q1A1C = Q2A2C

= ¬(Q| 1A1C ↔{zQ2A2C )=} Q|3A3C . {z }

The last equality holds thanks to theorem 5.9. questions in one pair are maximally compatible The same reasoning applies to any permuta- with both questions of the other. In this case tion of A,B,C. the reasoning of (28) applies and entails the cor-

relation of QiA with Q3B 3C must be maximally Next, we discuss the tripartite correlations of compatible with the correlation of QlA with QkC . individuals with the questions Q asking for the 33 Compatibility of Qi33 with QlAjB follows simi- correlation of bipartite correlations of rebit pairs. larly. Complementarity of Q and Q follows Lemma 5.19. Q is maximally compatible with i33 iAjB i33 from the fact that both are maximally compati- QiA and maximally complementary to QjB ,Qk . C ble with and independent of Qi and using argu- The same holds for any permutation of A,B,C. A ments as in previous proofs. Likewise, Qi33 and

Q3A3B follows similarly by noting that both are Proof. Compatibility of Qi33 with QiA is true maximally compatible with Q3B 3C . by construction. Complementarity of QjB and

Qi33 = QiA ↔ (Q1B 2C ↔ Q2B1C ) follows from Lemma 5.21. Any of Qi ,Q j ,Q is the observation that Q and Q are both max- 33 3 3 33k i33 jB independent from any bipartite correla- imally compatible with and independent of QiA tion question Qi j ,Qi k , Qj k and and a similar reasoning to the previous proof. A B A C B C any bipartite correlation of correlations questionQ3 3 ,Q3 3 ,Q3 3 . Qijk is also Lemma 5.20. Qi33 is maximally compatible with A B A C B C pairwise independent from the latter. Further- Q3B 3C , QjBkC , j, k = 1, 2, and QlAkC ,QlAjB for more, the Qijk and Qi33,Q3j3,Q33k are pairwise i 6= l. On the other hand, Qi33 is maximally com- independent. plementary to QiAjB ,QiAkC and Q3A3B ,Q3A3C . Furthermore, Qijk is maximally compatible with Proof. The proof is completely analogous to the Q3A3B . The same holds for any permutation of proofs of lemmas 5.3, 5.13 and 5.14. A,B,C. As before this has an important consequence. Proof. Compatibility of Qi33 with QjBkC and

Q3B 3C , as well as compatibility of Qijk with Corollary 5.22. The individuals QiA ,QjB ,QkC ,

Q3A3B is obvious. Compatibility of Qi33 with the bipartite correlations QiAjB ,QiAkC ,QjB kC ,

QlAkC for i 6= l follows indirectly by noting that the bipartite correlations of correlations

QiA ,QlA and Q3B 3C ,QkC are two pairs of max- Q3A3B ,Q3A3C ,Q3B 3C , the tripartite correla- imally complementary questions, but that the tions Qijk and the tripartite Qi33,Q3j3,Q33k,

Accepted in Quantum 2017-11-27, click title to verify 46 i, j, k = 1, 2, are pairwise independent and pendence and complementarity structure of three thus, thanks to assumption 7, part of an rebits. informationally complete set QM3 . 5.3.8 An informationally complete set for three The compatibility and complementarity struc- rebits ture of questions involving the ‘correlation of cor- The rebit questions considered thus far comprise relations’ Q33 is analogous to lemma 5.16. an informationally complete set of 35 elements.

Lemma 5.23. Qijk is maximally compatible with Theorem 5.24. (Rebits) The individu- Qi33,Q3j3,Q33k and maximally complementary als QiA ,QjB ,QkC , the bipartite correlations to Ql33,Q3m3,Q33n for i 6= l, j 6= m and k 6= n. QiAjB ,QiAkC ,QjBkC , the bipartite correlations Furthermore, Qi33 is maximally compatible with of correlations Q3A3B ,Q3A3C ,Q3B 3C , the tri- Q3j3,Q33k, but Q133 and Q233 are maximally partite correlations Qijk and the tripartite complementary. The analogous result holds for Qi33,Q3j3,Q33k, i, j, k = 1, 2, are logically closed all permutations of A,B,C. under ↔ and thus form an informationally

complete set QM3 with D3 = 35 for D1 = 2. Proof. Compatibility of Qijk with Proof. Logical closure under the XNOR for any Qi33,Q3j3,Q33k follows from the fact that pair of rebits follows from section 5.2. Combin- the constituents of the latter QiA ,Q3B 3C ,... ing individuals with bipartite questions of an- are maximally compatible with Qijk. Comple- other rebit pair produces the tripartite questions. mentarity of, e.g., Qijk and Ql33 for i 6= l can Lemma 5.10 and 5.18 assert logical closure of be shown by noting that both are maximally bipartite correlation questions and the bipartite compatible with and independent of Q3 3 and B C questions Q33 asking for the correlation of bipar- lemma 5.1. Compatibility of, say, Qi33 and tite correlations among all three rebits. We must Q3j3 can be demonstrated by using (28) and therefore only check logical closure of combina- noting that QiA ,Q3A3C and QjB ,Q3B 3C are two tions involving tripartite questions which involve pairs of maximally complementary questions indices taking the value 3 (the other cases are which are such that each question in one pair covered by theorem 5.17 for i, j, k = 1, 2). is maximally compatible with both questions Using theorem 5.9 and lemmas 5.18–5.23, the of the other pair. Finally, Q133 and Q233 are proof is mostly analogous to the proof of theorem maximally complementary because both are 5.17 such that here we will only show two non- maximally compatible with Q3B 3C and Q1A ,Q2A trivial cases which are treated differently. For are maximally complementary. example, by lemma 5.20 Q133 and Q2A2B are maximally compatible. The conjunction with ↔ This finishes our considerations of the inde- yields

Q133 ↔ Q2A2B = (Q1A ↔ Q3B3C ) ↔¬(Q1A1B ↔ Q3A3B ) (22,26)

= Q1B ↔ Q3A3C = Q313. (28,36)

Similarly, by lemma 5.23, Qi33,Q3j3 are max- imally compatible. Their XNOR conjunction gives

Qi33 ↔ Q3j3 = (QiA ↔ Q3B 3C ) ↔ (Q3A3C ↔ QjB ) = ¬(Q3A3B ↔ QiAjB ) (28)

which again coincides with some QlAmB with i 6= l and j 6= m (or the negation thereof). In

Accepted in Quantum 2017-11-27, click title to verify 47 complete analogy to these explicit examples and However, the three rebits can be similarly to the proof of theorem 5.17, one shows the log- non-monogamous in a tripartite maximally en- ical closure under the XNOR of all other cases. tangled state of maximal information. As in This gives the desired result. the qubit case (35) in section 5.3.5, one can write down a question graph corresponding to the 5.3.9 Monogamy and maximal entanglement for rebit analogues of GHZ states, representing the three rebits eight possible answers to the tripartite questions Q211,Q121,Q112 Rebits are considered as non-monogamous in the literature [86, 87]. However, this conclusion de- pends somewhat on one’s notion of monogamy and can be clarified within our language. Firstly, rebits, just as qubits, are clearly monogamous in the following sense: if two rebits A, B are maxi- mally entangled in a state of maximal information relative to O, then they cannot share any entan- PSfrag replacements

glement whatsoever with a third rebit C. This Q3A3C can be seen by repeating the argument of section Q Q 5.3.4 which holds analogously for three rebits. 3A3B 3B 3C The situation changes slightly for entangled states of non-maximal information involving ei-

ther of Q3A3B ,Q3A3C ,Q3B 3C and for tripartite maximally entangled states. As an example for (to justify this graph one has to employ the-

the former case, O could ask Q3A3B ,Q3B 3C simul- orem 5.24). Thanks to the information about

taneously which gives him two independent bits the answers to Q3A3B ,Q3A3C ,Q3B 3C , such a state of information about the rebit triple and implies would contain a non-monogamous distribution of

Q3A3C by (36) as well such that his information entanglement over A,B,C. In particular, if O could be summarized as ‘marginalized’ over rebit C, by discarding all in- formation involving C, he would be left with the

answer to Q3A3B which defines a maximally en- tangled two-rebit state of non-maximal informa- tion (see section 5.2.6) – in contrast to the qubit case of section 5.3.5.

PSfrag replacements 5.4 Correlation structure for N =2 gbits Q3 3 A C While we were able to settle the relative negation

Q3A3B Q3B 3C ¬ between the correlations of bipartite correla- tion questions for both qubits and rebits in (28) (e.g., Q11 ↔ Q22 = ¬(Q12 ↔ Q21)), we still have to clarify the odd and even correlation structure

(we depict the Q3A3B ,Q3A3C ,Q3B 3C without ver- for qubits in (19, 20) and more generally in theo- tices to emphasize the absence of the individu- rem 5.7. For example, we have to clarify whether

als Q3A ,Q3B , Q3C for rebits). This graph cor- Q33 = Q11 ↔ Q22 or Q33 = ¬(Q11 ↔ Q22), etc. responds to non-monogamously entangled rebit This will involve the notion of the ‘logical mir- states: by lemma 5.8 all individuals are maxi- ror’ of an inference theory and require the tools mally complementary to two of the three known from the previous section, covering the N = 3 questions such that O cannot acquire any indi- case. We recall that the odd and even correla- vidual information about the three rebits at the tion structure for rebits has already been settled same time. The three rebits are pairwise maxi- in section 5.2.7 through (28). This, as we shall mally entangled, albeit not in a state of maximal see shortly, is a consequence of rebit theory being information (see also section 5.2.6). its own mirror image.

Accepted in Quantum 2017-11-27, click title to verify 48 5.4.1 The logical mirror image of an inference the- O’s change of local handedness would have re- ory sulted in a switch between even and odd correla- tion structure for (19, 20). Since the handedness Consider a single qubit, described by O via an in- of the local question basis is just a convention formationally complete set Q ,Q ,Q which can 1 2 3 by O or O′, we shall allow either. Consequently, be viewed as a question basis on Q . As such, it 1 whether three correlation questions are related by defines a ‘logical handedness’ in terms of which even or odd correlation (see theorem 5.7) depends outcomes to Q ,Q ,Q O calls ‘yes’ and which 1 2 3 on the local conventions by O, O′ and both are, in ‘no’.37 Clearly, this is a convention made by fact, consistent. However, (28) will always hold. O and he can easily change the handedness by Usually, of course, one would favour the situation simply swapping the assignment ‘yes’↔’no’ of that both O, O′ make the same conventions such one question, say, Q , which is tantamount to 1 that their local question bases are equally handed Q 7→ ¬Q . This corresponds to taking the log- 1 1 (or that one observer O describes all qubits with ical mirror image of a single qubit. For a single the same handedness). Nevertheless, physically qubit this will not have any severe consequences all local conventions are fully equivalent, yet will and O is free to choose whichever handedness he lead to different representations of composite sys- desires. tems in the inference theory. However, the handedness of a single qubit ques- But there are important consistency condi- tion basis does become important when consider- tions on the distribution of odd and even cor- ing composite systems. Consider, e.g., two ob- relations. This becomes obvious when examining servers O, O′ each having one qubit and describ- three qubits A,B,C. To this end, consider the ing it with a certain handed basis. They can also trivial conjunction consider correlations Qij of their qubit pair. If O now decided to change the handedness of his local Q ↔ Q ↔ Q = 1. (37) 3A3B 3A3C 3B 3C lemma 5.10 question basis by Q1 7→ ¬Q1 this would result in From (19) we know that the correlation is either Q11,Q12,Q13 7→ ¬Q11, ¬Q12, ¬Q13, even or odd, respectively, Qij 7→ Qij, i 6= 1 Q = Q ↔ Q , or and therefore 3A3B 1A1B 2A2B

Q3A3B = ¬(Q1A1B ↔ Q2A2B ) (38) Q11 ↔ Q22 7→ ¬(Q11 ↔ Q22), and and analogously for A, C and B,C. Suppose now Q12 ↔ Q21 7→ ¬(Q12 ↔ Q21). that all three bipartite correlations in the con- Since Q33 would remain unaffected by the change junction (37) were even. Then we immediately ′ of local handedness (Q3,Q3 remain invariant), get a contradiction because, using lemma 5.10,

Q3A3B ↔ Q3A3C ↔ Q3B 3C = (Q1A1B ↔ Q2A2B ) ↔ (Q1A1C ↔ Q2A2C ) ↔ (Q1B 1C ↔ Q2B2C )

= ¬(Q1A1B ↔ Q1A1C ) ↔ (Q2A2B ↔ Q2A2C ) ↔ (Q1B 1C ↔ Q2B2C ) (28) = Q1B 1C = Q2B 2C = 0.| {z } | {z }

But this violates the identity (37) and results contradiction if one of the correlations in (37) from the relative negation between the left and was even and two were odd because then the two right hand side in (28). One would get the same negations from the odd correlation would cancel each other and one would still be left with the negation coming from (28). 37Ultimately, these three questions will define an or- thonormal Bloch vector basis in the Bloch sphere (see sec- On the other hand, everything is consistent if tion 7). The handedness or orientation of this basis will either all bipartite correlations in (38) are odd or depend on the labeling of question outcomes. one is odd and two are even because then the odd

Accepted in Quantum 2017-11-27, click title to verify 49 number of negations from the odd correlations cancels the negation coming from (28):

Q3A3B ↔ Q3A3C ↔ Q3B 3C = ¬(Q1A1B ↔ Q2A2B ) ↔¬(Q1A1C ↔ Q2A2C ) ↔¬(Q1B 1C ↔ Q2B 2C )

= ¬(Q1A1B ↔ Q1A1C ) ↔ (Q2A2B ↔ Q2A2C ) ↔¬(Q1B 1C ↔ Q2B 2C ) (28) = Q1B 1C = Q2B 2C = 1.| {z } | {z }

We can also quickly check that this is consistent and (26) (and analogously for A, C and B,C) with (20) Q1A2B ↔ Q2A1B = ¬(Q1A1B ↔ Q2A2B ).

Q3A3B = Q1A2B ↔ Q2A1B , or That is, if all correlations in (38) are odd, then

Q3A3B = ¬(Q1A2B ↔ Q2A1B ) (39) all correlations in (39) must be even. Indeed,

Q3A3B ↔ Q3A3C ↔ Q3B 3C = (Q1A2B ↔ Q2A1B ) ↔ (Q1A2C ↔ Q2A1C ) ↔ (Q1B 2C ↔ Q2B1C )

= ¬(Q1A2B ↔ Q1A2C ) ↔ (Q2A1B ↔ Q2A1C ) ↔ (Q1B 2C ↔ Q2B1C ) (28) = Q2B 2C = Q1B 1C

= Q|3B3C ↔ {zQ3B 3C =} 1 | {z }

is consistent. The tacit assumption, underlying the stan- In conclusion, if O wants to treat the bipar- dard representation of qubit quantum theory, is tite relations among all three A,B,C identically, that the handedness of each local qubit ques- then the following distribution of odd and even tion basis for A,B,C is the same, e.g., all ‘left’ correlations or ‘right’ handed. But we emphasize, that it would be equally consistent to choose one basis as ‘left’ (‘right’) and the other two as ‘right’ (‘left’) Q = Q ↔ Q 3A3B 1A2B 2A1B handed. A qubit pair with equally handed bases = ¬(Q1A1B ↔ Q2A2B ), (40) will be described by the odd and even correla- tion distribution as in (40), while a qubit pair and analogously for A, C and B,C, is the only with oppositely handed bases will be described consistent solution. We shall henceforth make the by the opposite distribution of odd and even cor- convention that the bipartite correlation structure relations. In terms of whether Q33 = Q11 ↔ Q22 among any pair of qubits be the same such that is even or odd Q33 = ¬(Q11 ↔ Q22), the three (40) holds. This turns out to be the case of qubit qubit relations yield only four consistent graphs quantum theory. for the distribution of ‘left’ and ‘right’ handedness PSfrag replacements PSfrag replacements PSfrag replacements PSfrag replacements

odd odd even even even‘left’ ‘left’ even‘right’ ‘right’ ‘right’ ‘left’ ‘left’ ‘right’ odd odd ‘left’ odd odd even odd even odd ‘right’ ‘left’ ‘right’ ‘left’ ‘right’ (41)

This gives a simple graphical explanation for framework with the opposite correlation structure the consistency observations above. The first of quantum theory, corresponding to the last two two graphs correspond to quantum theory. The graphs, is sometimes referred to as mirror quan-

Accepted in Quantum 2017-11-27, click title to verify 50 tum theory [15]. While mirror quantum theory can be carried out on a piece of paper; it is always was considered inconsistent in [15], we see here allowed. However, clearly, there cannot exist any that inconsistencies would only arise if the bi- actual physical transformation in the laboratory partite correlation structure of mirror quantum which maps states from one convention into the theory (and in particular the even correlation other. Q33 = Q11 ↔ Q22) was used for all three pairs of qubits. However, the proper formulation of mir- Within the formalism of quantum theory the ror quantum theory for three qubits corresponds transformation (b) Q 7→ ¬Q corresponds to to the last two graphs in (41) and gives a per- 1 1 the partial transpose (e.g., see [15]). It is well fectly consistent framework.38 It could be easily known that the partial transpose defines a sepa- generalized in an obvious manner to arbitrarily rability criterion for quantum states which is both many qubits.39 necessary and sufficient for a pair of qubits [88]: Obviously, the same argument can be carried a two-qubit density matrix ρ is separable if and out for any other triple of maximally compatible only if its partial transpose is positive relative to bipartite correlations appearing in theorem 5.7. a basis of standard Pauli matrix products (i.e., In conclusion, the two distinct consistent distri- represents a legal quantum state). This criterion butions of odd and even correlations, correspond- holds analogously in our language here: as seen at ing to quantum theory and mirror quantum the- the beginning of this section, the transformation ory, Q1 7→ ¬Q1 changes between odd and even corre- (a) result from different conventions of local ba- lations of bipartite correlations Qij. This would, sis handedness, and in fact, be unproblematic if O only had individ- ual information about the two qubits; it would (b) are in one-to-one correspondence through a map a classically composed state even of maxi- local relabeling ‘yes’↔‘no’ of one individual ′ mal information, say, Q1 = 1 and Q1 = 1 and question. thus Q11 = 1, to another legal classically com- posed state Q = 0, Q′ = 1 and thus Q = 0. As such, the two distinct correlation structures 1 1 11 Both states exist within both conventions. How- ultimately give rise to two distinct representa- ever, applying this transformation to a maxi- tions of the same physics and are thus fully equiv- mally entangled state with odd correlation, say, alent. The transformation (b) between the two Q = Q = 1 and thus, by (40), Q = 0 yields representations – being a translation between two 11 22 33 an even correlation Q =0= Q and Q = 1. conventions/descriptions – is a passive one and 11 33 22 The former state only exists in the quantum the- 38In fact, one could even produce the correlation struc- ory representation, while the latter exists only in ture of mirror quantum theory in the lab by using oppo- the mirror image. For other entangled states one sitely handed bases for two qubits in an entangled pair. would similarly find that Q1 7→ ¬Q1 necessar- The resulting state would be represented by the partial transpose of an entangled qubit state. This state would ily maps from one representation into the other. not be positive if represented in terms of the standard The same conclusion also holds for a transfor- Pauli matrices and thus not correspond to a legal quan- mation {Q1,Q2,Q3} 7→ {¬Q1, ¬Q2, ¬Q3}, which tum state [88]. However, the point is that mirror quantum one might call total inversion, because the odd theory would not be represented in the standard Pauli number of negations involved in the transforma- matrix basis but in the partially transposed basis which corresponds to replacing tion would likewise lead to a swap of odd and even correlations. 0 −i T 0 i σy = by σy = , i 0 −i 0     Lastly, we note that the situation is very differ- for one of the two qubits (this is a switch of the y-axis ent for rebit theory because it is its own mirror orientation). In this basis, the state would be positive. image, i.e. rebit theory and mirror rebit theory 39 Notice, however, that for more than three qubits one are identical representations. O will describe a would get more than two different consistent distributions of odd and even correlations. For example, for four qubits single rebit by a question basis Q1,Q2. Suppose there will be three cases: (1) all four have equally handed O decided to swap the ‘yes’ and ‘no’ assignments bases, (2) three have equally handed bases, (3) two have to the outcomes of Q1, such that equivalently equally handed bases. Q1 7→ ¬Q1. For a pair of rebits, this would have

Accepted in Quantum 2017-11-27, click title to verify 51 the following ramification does not hold in analogous fashion for rebits. For completely equivalent reasons, the total inver- Q11,Q12 7→ ¬Q11, ¬Q12, sion, corresponding to {Q1,Q2} 7→ {¬Q1, ¬Q2}, Q21,Q22 7→ Q21, Q22, is also a transformation which preserves the rep- resentation. and therefore

Q12 ↔ Q21 7→ ¬(Q12 ↔ Q21) ⇒ Q 7→ ¬Q . 33 33 5.4.2 Collecting the results: odd and even corre-

In contrast to the qubit case, Q33 is defined as lation structure for N = 2 a correlation of correlations Q33 := Q12 ↔ Q21 (22) and can not be written in terms of local After the many technical details it is useful to ′ questions Q3,Q3. Hence, Q33 also changes un- collect all the results concerning the compatibil- der this transformation by construction. Accord- ity, complementarity and correlation structure for ingly, Q1 7→ ¬Q1 does not lead to a swap of odd two qubits, derived in lemmas 5.2 and 5.5, theo- and even correlations for the rebit case. This rem 5.7, equation (28) and in the previous section ‘partial transpose’ therefore always maps states 5.4.1 in a graph to facilitate a visualization. We to other states within the same representation. shall henceforth abide by the convention that all For example, even a maximally entangled state bipartite relations for arbitrarily many qubits be of maximal information and even correlation, say, treated equally such that (40) must hold. For Q12 = Q21 = 1 and Q33 = 1, would be mapped to the other relations of theorem 5.7 one finds the another evenly correlated state Q12 = 0, Q21 = 1 analogous results. As can be easily verified, the and Q33 = 0. As a consequence, the Peres sep- ensuing question structure has the lattice pattern PSfrag replacementsarability criterion [88] which is valid forPSfrag qubits replacementsof figure 4, where the triangles − + Q′′ Q′′ − + QQ′ ⇔ Q = ¬(Q′ ↔ Q′′), QQ′ ⇔ Q = Q′ ↔ Q′′ (42)

denote odd and even correlation, respectively.

We recall that (26, 28) imply alternating odd and even correlation triangles for the bipartite correlation questions. However, we emphasize is also different from the graphs (23) as it in- that the Bell scenario argument of section 5.2.7 volves all six individual questions and the argu- does not require the correlation triangles involv- ment leading to (28) does not apply. Clearly, ′ ing individual questions to also admit such an given the definition Qij := Qi ↔ Qj, all such alternating odd and even pattern. For instance, triangles must be even. the following relations of Q33 The lattice structure in figure 4 contains 15 tri- ′ Q33 = Q3 ↔ Q3 = Q12 ↔ Q21 = ¬(Q11 ↔ Q22) angles and 15×3 = 45 distinct edges correspond- ing to compatibility relations. There are 60 ‘miss- PSfragare consistentreplacements with assumptions 5 and 6, despite ing edges’ corresponding to complementarity re- the absence of a. negation in Q ↔ Q′ = Q ↔ . 3 3 12 lations. Every question resides in three compat- Q21 because the latter is not a classical logical Q11 ibility triangles and is therefore maximally com- identity. The graph corresponding to the last re- Q22 patible with the six other questions in these three lation, triangles and maximally complementary to the Q12 eight remaining questions not contained in those three triangles. This means that once one ques- Q21 tion is fully known, all other information available Q33 ′ Q3 Q3 to O must be distributed over the three adjacent

Accepted in Quantum 2017-11-27, click title to verify 52 PSfrag replacements

Q1 Q2 Q3 ′ Q1 ′ Q2 identify′ identify Q3 Q13 Q23 Q12 Q21 − − +

Q31 Q33 Q32 PSfrag replacements + − + − Q22 Q11 Q13 Q23

identify

Q33 Q13 Q11 Q32 ′ + + + Q1 Q21 + ′ Q3 Q3 Q1 + ′ + Q2 Q2 ′ + Q3 Q12 ′ Q2 + + ′ Q31 Q1

Q22 Q identify 23 identify Figure 4: A lattice representation of the complete compatibility, complementarity and correlation structure of the informationally complete set QM2 for two qubits. Every vertex corresponds to one of the 15 pairwise independent questions. If two questions are connected by an edge, they are maximally compatible. If two questions are not connected by an edge, they are maximally complementary. Thanks to the logical structure of the XNOR ↔, defining a question as a correlation of two other questions, the compatibility structure results in a lattice of triangles. As clarified in (42), red triangles denote odd, while green triangles denote even correlation. Note that the two lattices represented here are connected through the nine correlation questions Qij and form a single closed lattice (which, however, is easier to represent in this disconnected manner). Every question resides in exactly three triangles and is thereby maximally compatible with six and maximally complementary to eight other questions.

triangles. In particular, if O knows the answers tion, he experience no net loss of information’ of to two of the questions in the lattice, he will also the complementarity rule 2. For instance, if O know the answer to the third sharing the same has maximal knowledge about two questions in triangle such that every triangle corresponds to the lattice, corresponding to maximal informa- a specific set of states of maximal information. tion about one triangle, it is impossible for him (Although, as we shall see shortly in section 6.8 to loose information by asking another question and, in more detail, in [1], the time evolution rule from the lattice because it will be connected to 4 implies that states of maximal information will one of the questions from the previous triangle. likewise exist such that two independent bits can As a concrete example, suppose O knew the an- be distributed differently over the lattice.) swers to Q21,Q13,Q32. He could ask Q22 next. Since Q22 is connected by an edge to Q13, he Notice that every triangle is connected by an would know the answers to both upon asking Q22 edge (adjacent to one of its three vertices) to ev- and thereby then also the answer to Q31 – no net ery other vertex in the lattice. This embodies loss of information occurs. the statement ‘whenever O asks S a new ques-

Accepted in Quantum 2017-11-27, click title to verify 53 PSfrag replacements

Q1 Q2 Q3 ′ Q1 ′ Q2 Q′ It is straightforward to check, e.g., using the identify3 identify ′ ansatz ′ Q1 Q2 Q12 Q21 |ψi = α |z−z−i + β |z+z+i + γ |z−z+i + δ|z+z−i + + + for a two qubit pure state and translating it into the various basis combinations xx,xy,yx,yy,..., Q1 Q33 Q2 that the lattice structure of figure 4 is pre- cisely the compatibility and correlation struc- + − + ture of qubit quantum theory. For instance, ′ Q11 Q22 Q11,Q22,Q33 correspond to projectors onto the Q ′ 1 Q2 +1-eigenspaces of σx ⊗ σx, σy ⊗ σy and σz ⊗ σz. The three questions sharing a red triangle means, Figure 5: A lattice representation of the complete com- patibility, complementarity and correlation structure of e.g., that Q11 = Q22 = 1 and Q33 = 0 is an al- the informationally complete set QM2 for two rebits. lowed state while Q11 = Q22 = Q33 = 1 is illegal. The explanations of the caption of figure 4 also apply Indeed, ignoring normalization, in quantum the- here. ory one finds

|x+x+i − |x−x−i = −i|y+y+i + i|y−y−i asserting that O shall not experience a net loss of = |z+z−i + |z−z+i information by asking further questions. for α = β = 0 and γ = δ = 1, corresponding to the propositions Q = 1: “the spins are corre- 11 5.5 The general case of N > 3 gbits lated in x-direction"; Q22 = 1: “the spins are correlated in y-direction"; and Q33 = 0: “the We are now prepared to investigate the indepen- spins are anti-correlated in z-direction".40 But dence, compatibility and correlation structure en- no quantum state exists such that the spins are suing from rules 1 and 2 on a general QN , i.e. of also correlated in z-direction if they are correlated O’s possible questions to a system S composed of in x- and y-direction (this would be mirror quan- N > 3 gbits. In particular, we shall exhibit an in- tum theory). Every other triangle in the lattice formationally complete set QMN for both qubits corresponds similarly to four pure quantum states and rebits. O can view the N gbits as a composite (representing the answer configurations ‘yes-yes’, system in many different ways: as N individual ‘yes-no’, ‘no-yes’, ‘no-no’ to the two independent gbits, as one individual gbit and a composite sys- questions per triangle). tem of N − 1 gbits, as a system composed of 2 Finally, we also collect the results on the com- gbits and a system composed of N − 2 gibts, and patibility, complementarity and correlation struc- so on. All these different compositions yield, of ture of two rebits, derived in lemmas 5.2, 5.5 and course, the same question structure. It is simplest 5.8, theorem 5.9 and equation (26), in a lattice to interpret the N gbits recursively as being com- structure in figure 5. There are six triangles and posed of a composite system of N − 1 gbits and a 6 × 3 = 18 edges representing compatibility rela- new individual gbit. Definition 3.5 of a composite tions. Every question resides in exactly two tri- system then implies that QN must contain (1) the angles and is thereby maximally compatible with questions of QN−1 for N − 1 gbits, (2) the set Q1 four and maximally complementary to four other of the new gbit, and (3) all logical connectives of questions. As in the qubit case in figure 4, every the maximally compatible questions of those two triangle is connected by an edge to every other sets. This entails slightly different repercussions vertex in the lattice, in conformity with rule 2 for qubits and rebits.

40 α+β Similarly, |ψi = α |z−z−i + β |z+z+i = 2 (|x+x+i + β−α α+β 5.5.1 An informationally complete set and entan- |x−x−i) + 2 (|x+x−i + |x−x+i) = 2 (|y+y−i + β−α glement for N > 3 qubits |y−y+i)+ 2 (|y+y+i+|y−y−i). But |x+x+i+|x−x−i = |y+y−i+|y−y+i and |y+y+i+|y−y−i = |x+x−i+|x−x+i. Hence, once Q33 = 1, one indeed gets Q11 ↔ Q22 = 0 A natural candidate for an informationally com- even though α, β are unspecified such that Q11,Q22 may plete question set is given by the set of all possible be unknown. XNOR conjunctions of the individual questions of

Accepted in Quantum 2017-11-27, click title to verify 54 the N gbits lemma 5.3 this implies independence of Qµ1···µN

and Qν1···νN . Lastly, suppose now µa = 0 and Qµ1µ2···µN := Qµ1 ↔ Qµ2 ↔···↔ QµN (43) νa 6= 0 (as we have µa 6= νa not both can be 0). Then Q with i 6= 0 is maximally compatible (we recall from (13) that the logical connective ia6=νa with Q and maximally complementary to yielding independent questions is either ↔ or ⊕). µ1···µN Q . By the same argument, this again im- Here we have introduced a new index notation: ν1···νN plies independence of Q and Q . µ is the question index for gbit a ∈ {1,...,N} µ1···µN ν1···νN a Consequently, the set (43) will be part of an and can take the values 0, 1, 2, 3. As before the informationally complete set. We note that a index values i = 1, 2, 3 correspond to the indi- hermitian matrix of trace equal to 1 on a 2N - viduals Q ,Q ,Q . On the other hand, the 1a 2a 3a dimensional complex Hilbert space (i.e. qubit index value µ = 0 implies that none of the three a density matrix) is described by 4N − 1 param- individual questions of gbit a appears in the con- eters. junction (43), i.e. always Q ≡ 1. For instance, 0a Next, we must elucidate the compatibility and complementarity structure of this set. Q100000···000 := Q11 , Q0003020···0 := Q34 ↔ Q26 ,

Lemma 5.26. Qµ1···µN and Qν1···νN are etc. Note that Q000···000 corresponds to no ques- tion. The set (43) thus contains all individual, maximally compatible if the index sets bipartite, tripartite, and up to N-partite cor- {µ1,...,µN } and {ν1,...,νN } differ by an relation questions for N gbits. We emphasize even number (incl. 0) of non-zero indices, that, due to the special multipartite structure (and associativity) of the XNOR, a question such maximally complementary if the index sets {µ1,...,µN } and {ν1,...,νN } differ by an as Q111···111 does not incarnate the question ‘are odd number of non-zero indices. the answers to Q11 ,Q12 , · · · ,Q1N all the same?’. For example, for N = 4, Q1 = Q1 = 0 and 1 2 For example, for N = 2, Q11 and Q22 differ Q = Q = 1 also yield Q = 1. This is 13 14 11121314 by two non-zero indices and are thus maximally important for the entanglement structure. compatible. By contrast, Q10 = Q1 and Q22 dif- We begin with an important result. fer by an odd number of non-zero indices and are N 41 thereby maximally complementary. Lemma 5.25. The 4 − 1 questions Qµ1···µN , µ = 0, 1, 2, 3, are pairwise independent. Proof. Let Qµ1···µN and Qν1···νN disagree in an odd number, call it 2n + 1, of non-zero indices.

Proof. Consider Qµ1···µN and Qν1···νN . The two We can always reshuffle the index labeling of the questions must disagree in at least one index oth- N qubits such that now a = 1,..., 2n + 1 cor- erwise they would coincide. Let the questions responds to the qubits on which Qµ1···µN and differ on the index of gbit a, i.e. µa 6= νa. Sup- Qν1···νN differ by non-zero indices, i.e. µa 6= νa pose µa,νa 6= 0. Then Qµa is maximally compat- and µa,νa 6= 0. The remaining qubits labeled ible with Qµ1···µa···µN = Qµ1 ↔ ··· ↔ Qµa ↔ by b = 2n + 2,...,N are then such that Qµ1···µN

··· ↔ QµN and maximally complementary to and Qν1···νN either agree on the non-zero index,

Qν1···νa···νN = Qµ1 ↔ ··· ↔ Qνa ↔ ··· ↔ µb = νb 6= 0 or at least one of µb,νb is 0. That

QνN since Qµa ,Qνa are maximally complemen- is, after the reshuffling the index labeling, we can tary. Using the same argument as in the proof of write the questions as

Qµ1···µN = (Qµ1 ↔···↔ Qµ2n+1 ) ↔ (Qµ2n+2 ↔···↔ QµN ) disagreement maximally compatible | {z } | {z } Qν1···νN = (Qν1 ↔···↔ Qν2n+1 ) ↔ (Qν2n+2 ↔···↔ QνN ) . (44)

41 z }| { z }| { We deduct the trivial question Q000···000. Obviously, one arrives at the same number by counting the distri- bution of individuals, bipartite,...and N-partite correla- N N k tions over N qubits as a binomial series k=1 k 3 = (3+1)N − 1. P
Accepted in Quantum 2017-11-27, click title to verify 55 The parts of the questions where the index We can now proceed by induction. Lemmas sets either agree or feature zeros coincide with 5.2, 5.5, 5.10–5.16 imply that the statement of

Qµ2n+2···µN and Qν2n+1···νN and are clearly maxi- this lemma is correct for n = 0, 1. Let the state- mally compatible (dropping here zero indices). ment therefore be true for n and consider n + 1. Then, (44) reads

Qµ1···µN =(Qµ1···µ2n+3 ) ↔ Qµ2n+4···µN = (Qµ1 ↔···↔ Qµ2n+1 ) ↔ (Qµ2n+2µ2n+3 ) ↔ Qµ2n+4···µN disagreement maximally compatible maximally complementary disagreement max. compat. | {z } | {z } | {z } | {z } | {z } Qν1···νN =(Qν1···ν2n+3 ) ↔ Qν2n+4···νN = (Qν1 ↔···↔ Qν2n+1 ) ↔ (Qν2n+2ν2n+3 ) ↔ Qν2n+4···νN . z }| { z }| { z }| { z }| { z }| {

But, by lemma 5.5, Qµ2n+2µ2n+3 and Qν2n+2ν2n+3 cause they disagree in 2n + 1 non-zero indices are maximally compatible with each other and (for which, by assumption, the statement of the therefore also with Qµ1···µN and Qν1···νN . This lemma holds). implies that both Qµ1···µN and Qν1···νN are max- imally compatible with and, thanks to lemma Finally, let Qα1···αN and Qβ1···βN disagree in

5.25, independent of, e.g., Qµ2n+2···µN . Com- an even number 2n of non-zero indices. Using plementarity of Qµ1···µN and Qν1···νN now fol- an analogous reshuffling of the index labeling as lows from lemma 5.1 and noting that Qµ1···µ2n+1 in the odd case above, one can rewrite the two and Qν1···νN are maximally complementary be- questions as

Qα1···αN = Qα1α2 ↔ Qα3α4 ↔···↔ Qα2n−1α2n ↔ Qα2n+1···αN differ differ differ maximally compatible | {z } | {z } | {z } | {z } Qβ1···βN = Qβ1β2 ↔ Qβ3β4 ↔···↔ Qβ2n−1β2n ↔ Qβ2n+1···βN . (45) z }| { z }| { z }| { z }| {

That is, one can decompose the disagreeing parts N = 2, 3. Let the statement therefore be true of the questions into bipartite correlations. But for N − 1 and consider a composite system of N thanks to lemma 5.5 two bipartite correlations of qubits. Since O can treat the N qubit system the same qubit pair are maximally compatible if in many different ways as a composite system and only if they differ in both indices. Conse- and the statement holds for N − 1, all XNOR quently, all the pairings of question components conjunctions of questions involving at least one of the upper and lower line, as written in (45), zero index will be contained in (43). We thus are maximally compatible and, hence, so must be only need to show that all ↔ conjunctions in-

Qα1···αN and Qβ1···βN . volving at least one N-partite correlation Qi1···iN , ia 6= 0 ∀ a = 1,...,N, produce questions already Fortunately, it turns out that the 4N − 1 ques- included in (43). tions (43) are logically closed under the XNOR. Consider Qi1···iN and Qν1···νN . Lemma 5.26 im- plies that these two questions are maximally com- Theorem 5.27. (Qubits) The 4N −1 questions patible – and can thus be connected by ↔ – if and

Qµ1···µN , µ = 0, 1, 2, 3, are logically closed under only if {i1,...,iN } and {ν1,...,νN } disagree in ↔ and thus form an informationally complete set an even number of non-zero indices. There are N QMN with DN = 4 − 1 for the case D1 = 3. now two cases that we must consider:

(a) Suppose Qi1···iN and Qν1···νN disagree in Proof. We shall prove the statement by induc- an even number of non-zero indices and, further- tion. The statement is trivially true for N = 1 more, agree on at least one index ia = νa. Thanks and, by theorems 5.7 and 5.17, holds also for to Qia ↔ Qia = 1, the conjunction then yields

Accepted in Quantum 2017-11-27, click title to verify 56 Qi1···ia···iN ↔ Qν1···ia···νN = Qi1··· 0 ···iN ↔ Qν1··· 0 ···νN . position a position a

Hence, the two questions on the right hand side (b) Suppose Qi1···iN and Qν1···νN disagree in an both contain less than N non-zero indices such even number 2n of non-zero indices and do not that the result must lie in the set (43) because agree on any non-zero index. Reshuffling the in- the statement is true up to N − 1 by assumption. dex labelings as in the proof of lemma 5.26, one can then write

Qi1···iN = Qi1i2 ↔ Qi3i4 ↔···↔ Qi2n−1i2n ↔ Qi2n+1···iN differ differ differ | {z } | {z } | {z } Qν1···νN = Qν1ν2 ↔ Qν3ν4 ↔···↔ Qν2n−1ν2n ↔ Q02n+1···0N . z }| { z }| { z }| {

(We obtain here Q02n+1···0N because Qi1···iN and compatible and by theorem 5.7 their XNOR con-

Qν1···νN do not agree on any common index and junction will yield another bipartite correlation of

Qi1···iN does not feature zero-indices.) By lemma the same qubit pair. For example, Qi1i2 ↔ Qν1ν2

5.5, the pairs of bipartite correlations differing in equals either Qj1j2 or ¬Qj1j2 for j1 6= i1,ν1 and two indices, e.g., Qi1i2 and Qν1ν2 , are maximally j2 6= i2,ν2. Accordingly, up to negation, one finds (ja 6= ia,νa, a = 1,..., 2n)

Qi1···iN ↔ Qν1···νN = Qj1j2 ↔ Qj3j4 ↔···↔ Qj2n−1j2n ↔ Qi2n+1···iN = Qj1···j2ni2n+1···iN

which is another N-partite correlation contained For example, for N = 2, Q11,Q22 correspond to in the set (43). two non-intersecting edges, i.e. 1-simplices; they are maximally compatible because they disagree As in the cases N ≤ 3, we could represent on their 1-simplices, both involving qubit 1 and 2. the compatibility and complementarity relations On the other hand, for N = 3, Q1A1B ,Q2B 2C also for N > 3 qubits geometrically by a simplicial correspond to two non-intersecting 1-simplices question graph where a D-partite question corre- but are maximally complementary because the sponds to a (D − 1)-dimensional simplex within two edges involve different qubits – A, B for the the graph (D ≤ N). The compatibility and com- first and B,C for the second edge. plementarity relations of lemma 5.26 then trans- This also helps us to understand entanglement late into abstract geometric relations according ′ for arbitrarily many qubits. Specifically, maxi- to whether a D-simplex and a D -simplex share mal entanglement will correspond to O spending or disagree on subsimplices. In particular, the the N independent bits he is allowed to acquire criterion that two distinct questions in Q are MN about the system S of N qubits on N-partite cor- maximally compatible if and only if they disagree relation questions. Lemma 5.26 guarantees that on an even number of non-zero indices means ge- for every N there will exist N maximally com- ometrically that the two questions are maximally patible N-partite questions. For instance, there compatible if and only if the two subsimplices in N are 2 ways of having N − 2 of the indices take the two question simplices which correspond to value 1 and 2 indices take the value 2. Any two the qubits they share in common either
N of the 2 corresponding questions will disagree (a) do not overlap and are odd-dimensional, or in two or
four non-zero indices (and agree on the rest) and will thus be maximally compatible (even (b) coincide. mutually according to theorem 3.2). These will

Accepted in Quantum 2017-11-27, click title to verify 57 N correspond to a set of 2 maximally compati- tributed over the various simplices and subsim- ble (N − 1)-simplices in the
question graph such plices in the question graph. However, we abstain that any two of them either disagree on an edge from analyzing such relations here further. or a tetrahedron. (There will exist even more maximally compatible N-partite questions.) For 5.5.2 An informationally complete set and entan- N N ≥ 3 it also holds that 2 ≥ N. glement for N > 3 rebits It is definitely possible to
choose N such maxi- mally compatible N-partite correlation questions We briefly repeat the same procedure for N N rebits. In analogy to (43), the natural candidate out of the 2 many such that these N questions set for an informationally complete QMN will con- do not all agree
on a single index. For similar rea- sons to the N = 2, 3 cases, this choice will con- tain stitute a mutually independent set such that ev- Qµ1µ2···µN := Qµ1 ↔ Qµ2 ↔···↔ QµN , ery individual question Qi1 ,...,QiN will be max- imally complementary to at least one of these N µa = 0, 1, 2, a = 1,...,N, (46) N-partite questions. (As a consequence of rule 1, where the notation should be clear from sec- once the answers to these N N-partite questions tion 5.5.1. However, being a composite sys- are known, they will also imply the answers to the N tem, by definition 3.5 we must permit the cor- 2 − N remaining ones by the same reasoning relation of correlations Q3a3b (22) for all a, b ∈ as
in section 5.2.3.) Accordingly, O can exhaust {1,...,N} because clearly O is allowed to ask the information limit with these N-partite corre- Q1a ,Q2a ,Q1b ,Q2b . Furthermore, thanks to (36) lation questions, while not being able to have any we have, e.g., information whatsoever about the individuals – a necessary condition for maximal entanglement. Q31323334 = Q3132 ↔ Q3334 = Q3133 ↔ Q3234 There will exist many different ways of having = Q3134 ↔ Q3233 such multipartite entanglement for arbitrary N. One could describe such different ways of entan- such that no confusion can arise about the mean- glement by generalizing the correlation measures ing of Q31323334 although there are no individu-

(33) and informational monogamy inequalities re- als Q3a into which the question could be decom- sulting therefrom. These monogamy inequalities posed. The same holds similarly for any other could also be considered as simplicial relations: even number of indices taking the value 3. Con- they restrict the way in which the available (inde- sequently, the candidate set for QMN can be writ- pendent and dependent) information can be dis- ten as

˜ QMN := Qµ1···µN , µa = 0, 1, 2, 3, a = 1,...,N only even number of indices taking value 3 

with an evident meaning of each such question. 3’s over the N indices. There are precisely Let us count the number of elements within ˜ N N−1 N N−3 QMN . ON := 3 + 3 1 ! 3 ! N−1 N Lemma 5.28. Q˜M contains 2 (2 + 1) − 1 N N N−(2n+1) non-trivial questions. + · · · + 3 2n + 1! Proof. If arbitrary distributions of the values such ways, where 2n+1 is the largest odd number µa = 0, 1, 2, 3 over the a = 1,...,N were per- smaller or equal to N. Similarly, the number of mitted, we would obtain 4N − 1 non-trivial ques- ways an even number of 3’s can be distributed tions upon subtracting Q0102···0N as in the qubit over N indices is given by case. In order to obtain the number of questions within Q˜M we thus still have to subtract all the N N N−2 N N−2m N EN := 3 + 3 + · · · + 3 , possible ways of distributing an odd number of 2 ! 2m!

Accepted in Quantum 2017-11-27, click title to verify 58 where 2m is the largest even number smaller or Proof. Thanks to theorems 5.9, 5.24 and lemma equal to N. We then have 5.30, the proof of theorem 5.27 also applies here, N N except that only an even number of indices can EN + ON = (3 + 1) = 4 , N N take the value 3 and Q3a3b cannot be decomposed EN − ON = (3 − 1) = 2 into individuals Q3a ,Q3b . and thus 1 N N We close with the observation that simi- ON = (4 − 2 ) 2 larly to the N qubit case in section 5.5.1, one which yields could represent the compatibility and comple- mentarity structure via a simplicial question 4N − 1 − O = 2N−1(2N + 1) − 1 N graph. Maximally entangled states (of maximal ˜ non-trivial questions in QMN . information) will correspond to O spending all N available bits over a mutually inde- We note that the number of parameters in a N pendent set of N N-partite questions which symmetric matrix with trace equal to 1 on a 2 - is maximally complementary to every indi- dimensional real Hilbert space (i.e. rebit density vidual question. In fact, the prescription for matrix) is precisely 1 2N (2N + 1) − 1. 2 constructing a maximally N-partite entangled Next, we assert pairwise independence as re- qubit state provided at the end of section 5.5.1 quired for an informationally complete set. also applies to N rebits since only indices with Lemma 5.29. The 2N−1(2N + 1) − 1 non-trivial values 1, 2 were employed. Furthermore, for ˜ rebits one can similarly generate maximally questions in QMN are pairwise independent. entangled states of non-maximal information; Proof. The proof is entirely analogous to the e.g., O could ask only the N − 1 questions proofs of lemmas 5.3, 5.13, 5.14 and 5.25. Q3132000···,Q0323300···,Q0033340···,...,Q0···03N−13N . If O has maximal information about each such Likewise, the complementarity and compatibil- question, every rebit pair will be maximally ity structure of Q˜ is analogous to the qubit MN entangled, while O has still not reached the case. information limit (see also section 5.2.6). ˜ Lemma 5.30. Qµ1···µN ,Qν1···νN ∈ QMN are maximally compatible if the index sets 6 Information measure, time evolution {µ1,...,µN } and {ν1,...,νN } differ by an and state space even number (incl. 0) of non-zero indices, and Thus far we have only applied rules 1 and 2 to maximally complementary if the index sets derive the basic question structure on QN . We shall now slightly switch topic and return to the {µ1,...,µN } and {ν1,...,νN } differ by an odd number of non-zero indices. problems of time evolution, begun in subsection 3.2.8, and of explicitly quantifying the informa- Proof. Thanks to lemmas 5.8, 5.18–5.20 and tion which O has acquired about S by means of 5.23, the proof of lemma 5.26 also applies to interrogations with questions. The information the rebit case with the sole difference that only measure and time evolution of S’s state in be- an even number of indices in the questions can tween interrogations are intertwined through rule take the value 3 and that correlations of correla- 3 of information preservation and rule 4 of max- tions Q3a3b cannot be decomposed into individu- imality of time evolution such that we have to als Q3a ,Q3b . discuss these topics together. In (6) we have implicitly defined the total Finally, Q˜ is indeed logically closed. MN amount of O’s information about S – once the ˜ latter is in a state ~yO→S – as Theorem 5.31. (Rebits) QMN is logically closed under ↔ and is thus an informationally D N−1 N N complete set Q˜M = QM with DN = 2 (2 + N N IO→S(~yO→S)= αi(~yO→S), 1) − 1 for the case D1 = 2. Xi=1

Accepted in Quantum 2017-11-27, click title to verify 59 where αi quantifies O’s information about the gain. This update takes the form of a ‘collapse’ of outcome of question Qi ∈ QMN and satisfies the the prior into a posterior state of s single system bounds (1). We shall begin in subsection 6.1 by in a single shot interrogation, while it yields an clarifying that the information measure we seek update of the ensemble state in a multiple shot to derive here is conceptually distinct from the interrogation (see subsection 3.2.3). The proba- standard Shannon entropy, before imposing el- bility distribution represented by the state is thus ementary consistency conditions on the relation a prior distribution for the next interrogation and αi(~yO→S) in subsection 6.2. Subsequently, we use only valid until the next answer – unless the lat- these conditions and implement rules 3 and 4 to ter yields no information gain. We thus seek to establish that the set of possible time evolutions quantify the information content in the state of defines a group and, finally, to derive the explicit S relative to O. This is not equivalent to the sum functional form of IO→S in subsection 6.8. This of the average information gains, relative to that discussion will also unravel properties of state same state used as a fixed prior probability dis- spaces and lead to the notion of pure states. tribution, which are provided by specific answers to the questions in an informationally complete 6.1 Why the Shannon entropy does not apply set over repeated interrogations. here The reader should thus not be surprised to find that the end result of the below derivation will Consider a random variable experiment with dis- not yield the Shannon entropy. We emphasize, tinct outcomes each of which is described by a however, that this clearly does not invalidate the particular probability. Outcomes with a smaller use of the Shannon entropy (in the more general likelihood are more informative (relative to the form of the von Neumann entropy) in quantum prior information) than those with a larger like- theory as long as one employs it what it is de- lihood. The Shannon information quantifies the signed for: to describe average information gains information an observer gains, on average, via a in repeated experiments on identically prepared specific outcome if she repeated the experiment systems – relative to a prior state which is as- many times. (Equivalently, it quantifies the un- signed before the repeated experiments are car- certainty of the observer before an outcome of ried out and which represents the ‘known’ prob- the experiment occurs.) The probability distri- ability distribution. bution over the different outcomes is assumed to be known. In particular, none of the outcomes in any run of the experiment will lead to an update 6.2 Elementary conditions on the measure of the probability distribution. There are a few natural requirements on the re- Here, by contrast, we are not interested in how lation between αi and ~yO→S: informative one specific question outcome is rel- ative to another and how much information O (i) The Qi ∈ QMN are pairwise independent. would gain, on average, with a specific answer if Accordingly, αi should not depend on the he repeated the interrogation with a fixed ques- ‘yes’-probabilities yj6=i of other questions tion many times on identically prepared systems. Qj6=i such that αi = αi(yi). Instead, we wish to quantify O’s prior informa- tion (e.g., gained from previous interrogations) (ii) All Qi ∈ QMN are informationally of equiv- about the possible question outcomes on only alent status. The functional relation be- the next system to be interrogated. The role of tween yi and αi should be the same for all i: the probability distribution over the various out- αi = α(yi), i = 1,...,DN . comes (to all questions) is here assumed by the state which O ascribes to S. This state is not as- (iii) If O has no information about the outcome sumed to be ‘known’ absolutely, instead, as dis- of Qi, i.e. yi = 1/2, then αi = 0 bit. cussed in section 3.2, it is defined only relative to the observer and represents O’s ‘catalogue of (iv) If O has maximal information about the out- knowledge’ or ‘degree of belief’ about S. In par- come of Qi, i.e. yi = 1 or yi = 0, then αi = 1 ticular, this state is updated after any interro- bit; both possible answers give 1 bit of in- gation if the outcome has led to an information formation.

Accepted in Quantum 2017-11-27, click title to verify 60 D 42 (v) The assignment of which answer to Qi is contains a basis of R N so that it is a DN - ‘yes’ and which is ‘no’ is arbitrary and the dimensional closed convex subset of RDN . Fur- functional relation between αi and yi (or ni) thermore, since yi ∈ [0, 1] for all yi in ~yO→S, ΣN should not depend on this choice. Hence, is clearly bounded and thus, by the Heine-Borel- α(yi) = α(ni) must be symmetric around theorem, compact. As a compact convex set with yi = 1/2 (see also (iv)); αi quantifies the non-empty interior, ΣN is thus a convex body amount of information about Qi, but does and, in particular, has volume [89]. not encode what the answer to Qi is. Next, we focus on the state of no information 1 ~yO→S = ~1. Given strict convexity of IO→S on (vi) On the interval y ∈ (1/2, 1], the relation 2 i ΣN , the conditions of the previous subsection en- between α and yi should be monotonically tail that the state of no information is the global increasing such that O’s information about 1 ~ minimum of IO→S with IO→S( 2 1) = 0 bits – the answer to Qi is quantified as higher, the in agreement with its uniqueness. It is also an higher the assigned probability for a ‘yes’ interior state. outcome. Likewise, on [0, 1/2), α(yi) must be monotonically decreasing. In particular, Lemma 6.1. The state of no information lies in α shall be continuous and strictly convex; the interior of ΣN . 1 2 i.e., for all yi ,yi ∈ [0, 1] and every λ ∈ [0, 1] Proof. The Qi ∈ QMN are pairwise indepen- 1 2 1 2 dent so for each such Qi there exist two states α(λyi + (1 − λ) yi ) ≤ λ α(yi ) + (1 − λ) α(yi ) yi 1 1 1 1 ni ~y O→S = ( 2 , · · · , 2 , 1, 2 , · · · , 2 ) and ~y O→S = ( 1 , · · · , 1 , 0, 1 , · · · , 1 ) corresponding to O having and the inequality shall be strict whenever 2 2 2 2 1 2 1 only asked Qi to S in the state of no information yi 6= yi and 0 <λ< 1. Hence, yi = is the 2 and having received the answers Qi = ‘yes’ and global minimum of α on [0, 1]. Qi = ‘no’, respectively. Since ΣN is convex, it Since now also contains all convex mixtures of these special yi ni states. The DN ~y O→S, as well as the DN ~y O→S D N each define a basis of RDN such that the set of all IO→S(~yO→S)= α(yi) (47) their convex mixtures define a DN -dimensional i=1 D X convex polytope contained in ΣN ⊂ R N . In and each summand is strictly convex, we also 1 yi 1 ni 1 ~ particular, given that 2 ~y O→S + 2 ~y O→S = 2 1 have that IO→S is a strictly convex function on yields the state of no information for each i, it ΣN . Accordingly, in the coin flip scenario of sub- is clear that it lies in the interior of the convex section 3.2.8, O’s information about the mixed polytope. state ~yO→S12 = λ ~yO→S1 +(1−λ) ~yO→S2 is smaller than his maximal information about any of its Given that time evolution preserves the total information by rule 3, we shall also be interested constituents ~yO→S1 , ~yO→S2 – unless the outcome of the coin flip was certain, or S1,S2 are in the in the level sets of IO→S. same state. This is consistent with the fact that Lemma 6.2. Let IO→S fulfill conditions (i)–(vi) O’s information about the outcome of any ques- of section 6.2. For sufficiently small ε > 0, the tion (asked to either S or S , depending on the 1 2 level set outcome of the coin flip) is clouded by the random coin flip outcome. Lε := {~yO→S ∈ ΣN |IO→S(~yO→S)= ε} (48)

lies in the interior of ΣN and is homeomorphic 6.3 The state of no information is an interior D −1 to S N . In this case, Lε constitutes the full state boundary of the fat level set LI≤ε := {~yO→S ∈ In order to discuss the action of time evolution ΣN |IO→S(~yO→S) ≤ ε} which contains the state on the state space ΣN , we have to derive a few 42 Each of the following DN -dimensional vectors further properties of the latter, some of which 1 1 1 1 1 1 1 1 (1, 2 , 2 , · · · , 2 ), ( 2 , 1, 2 , · · · , 2 ), · · · , ( 2 , · · · , 2 , 1) rep- depend on the properties of IO→S. resents a legal state ~yO→S, corresponding to O only Firstly, since DN is finite for finite N, ΣN knowing the answer to precisely one question from QMN with certainty and ‘nothing else’. is finite dimensional in this case. In fact, ΣN

Accepted in Quantum 2017-11-27, click title to verify 61 of no information and is homeomorphic to the to the closed unit ball and its boundary is home- closed unit ball in RDN . omorphic to SDN −1.

Proof. The state of no information is the global These results will become important for estab- 1 ~ lishing that the set of time evolutions is a group. minimum of IO→S with IO→S( 2 1) = 0 and, thanks to lemma 6.1, lies in the interior of ΣN . Furthermore, condition (vi) of subsection 6.2 re- 6.4 Time evolution of the ‘Bloch vector’ quires that α(yi) increases monotonically away 1 In subsection 3.2.8, we have only considered the from yi = 2 for each i so that IO→S in (47) must likewise increase monotonically in each direction time evolution of the redundantly parametrized 1 ~ RDN 2DN -dimensional state P~O→S. According to (10), away from ~yO→S = 2 1 in . Owing to the continuity of IO→S, we can therefore find a suffi- this time evolution is linear, described by a ma- ciently small ε > 0 such that any state attained trix A(∆t), and only depends on the interval by moving away from the state of no information ∆t = t2 − t1, where t1,t2 are arbitrary instants of in any direction until reaching IO→S = ε still lies time in between O’s interrogations on a given sys- in the interior of ΣN . Accordingly, for such ε> 0, tem. However, because of (4), it clearly suffices Lε must lie in the interior of ΣN and so must LI≤ε to consider the yes-vector ~yO→S (or no-vector with the state of no information in its own inte- ~nO→S) alone to describe the state of S relative rior. Given that IO→S :ΣN → R is continuous to O. Let us therefore determine, how ~yO→S and and ΣN is closed, IO→S is a closed convex func- ~nO→S evolve under time evolution. To this end, ~ tion. Closed convex functions have closed convex we decompose A(∆t) and PO→S in (10), fat level sets [89] so that LI≤ε is a closed convex ~yO→S(∆t) a(∆t) b(∆t) ~yO→S(0) subset in the interior of ΣN . Clearly, LI≤ε is DN - = , dimensional since we moved away in all directions ~nO→S(∆t) ! c(∆t) d(∆t) ! ~nO→S(0) ! 1 ~ from ~yO→S = 2 1 to reach Lε and the states with IO→S = ε define the limit points of LI≤ε in each where a(∆t), b(∆t), c(∆t), d(∆t) are nonnegative direction so that Lε constitutes its full bound- (real) DN × DN matrices. Using the normal- ary. Any DN -dimensional closed convex subset ization (4) (with p = 1), one finds that both DN of R with non-empty interior is homeomorphic ~yO→S, ~nO→S evolve affinely,

~yO→S(∆t) = [a(∆t) − b(∆t)] ~yO→S(0) + b(∆t)~1,

~nO→S(∆t) = [d(∆t) − c(∆t)] ~nO→S(0) + c(∆t)~1. (49)

We shall now determine relations among the four must hold for all initial states and all times. matrices a, b, c, d. Firstly, the normalization (4) Hence, for all ∆t ∈ R,

~1 = ~yO→S(∆t)+ ~nO→S(∆t)

= [a(∆t) − b(∆t)] ~yO→S(0) + [d(∆t) − c(∆t)] ~nO→S(0) + [b(∆t)+ c(∆t)] ~1

= [a(∆t) − b(∆t)+ c(∆t) − d(∆t)] ~yO→S(0) + [b(∆t)+ d(∆t)] ~1. (4)

Since the set of all possible initial states ~yO→S(0) Next, we note that, by assumption 3, the state of contains a basis of RDN , we must conclude no information is unique and, by rule 3, clearly preserved under time evolution. Inserting the a(∆t) − b(∆t)= d(∆t) − c(∆t), 1 ~ state of no information, e.g., as ~nO→S(∆t)= 2 1 [b(∆t)+ d(∆t)] ~1= ~1. (50)

Accepted in Quantum 2017-11-27, click title to verify 62 into (49), this entails 6.5 Time evolution is injective We continue by demonstrating that time evolu- [c(∆t)+ d(∆t)] ~1= ~1. (51) tion of S’s states must be injective on ΣN . In subsection 6.6, we will use this result to prove 43 In conjunction, (50, 51) therefore imply that time evolution is also surjective on ΣN and thus, in fact, reversible. ~yO→S(∆t) = [a(∆t) − b(∆t)] ~yO→S(0) + b(∆t)~1, Lemma 6.3. Let T (∆t):Σ → Σ be a time ~n (∆t) = [a(∆t) − b(∆t)] ~n (0) + b(∆t)~1. N N O→S O→S evolution as given in (52) for any ∆t ∈ R. If IO→S is strictly convex, rule 3 entails that T (∆t) Consequently, defining the DN × DN time evolu- is injective. tion matrix Proof. Assume T (∆t) was not injective. Then ′ ′′ T (∆t) := a(∆t) − b(∆t), (52) there would exist states ~y O→S(t1) 6= ~y O→S(t1) such that ′ yields a linear time evolution of what we shall ~yO→S(t2) := T (∆t)~y O→S(t1) henceforth call the generalized Bloch vector ′′ = T (∆t)~y O→S(t1). (55) 2 ~yO→S − ~1: Now consider the coin flip scenario of section

2 ~yO→S(∆t) − ~1= ~yO→S(∆t) − ~nO→S(∆t) 3.2.8. O can prepare S1 in the state ~yO→S1 (t1) := ′ ~y O→S(t1) and S2 in the state ~yO→S2 (t1) := = T (∆t) 2 ~yO→S(0) − ~1 . (53) ′′ ~y O→S(t1) at time t1 before tossing the coin. By   (8), Notice that T (∆t) need neither be nonnegative ~y (t )= λ ~y (t ) + (1 − λ) ~y (t ), nor stochastic in any pair of its components (in O→S12 1 O→S1 1 O→S2 1 contrast to A(∆t)). Henceforth, we shall only and, on account of (55) and rule 3, consider the T (∆t) governing the time evolution I (~y (t )) = I (~y (t )) of the Bloch vector O→S1 O→S1 1 O→S2 O→S2 1 = IO→S1 (~yO→S1 (t2)) = I (~y (t )). (56) ~rO→S := 2 ~yO→S − ~1 (54) O→S2 O→S2 2 Thus, using (56) and strict convexity of IO→S, and often employ the latter to parametrize states. this yields for a coin flip with 0 <λ< 1

IO→S12 (~yO→S12 (t1))

On the other hand, (55) implies that, at time t2, such that O would find IO→S12 (~yO→S12 (t2)) = IO→S1 (~yO→S1 (t2))

= IO→S2 (~yO→S2 (t2)). (58) ~yO→S (t2)= ~yO→S (t2)= ~yO→S (t2) 12 1 2 Hence, (57, 58) entail that O’s information about S12 has increased between t1 and t2, despite not 43As an aside, note that A(∆t) can thus be replaced by having tossed the coin and asked any questions. This is in contradiction with rule 3. We conclude a(∆t) b(∆t) that T (∆t) must be injective. A(∆t)= , b(∆t) a(∆t)   D Since ΣN contains a basis of R N (see subsec- so that time evolution is symmetric under a swap of the tion 6.3) it is clear that T (∆t) is also injective DN ‘yes/no’-labeling for all Q ∈ QMN at once, i.e. under on R . For finite dimensional square matrices, ~yO→S ↔ ~nO→S. T (∆t) being injective is equivalent to it being

Accepted in Quantum 2017-11-27, click title to verify 63 bijective on RDN . Hence, to every T (∆t) there T (∆t) is also closed. must exist an inverse matrix T −1(∆t) such that T (∆t)T −1(∆t)= T −1(∆t)T (∆t)= 1. Next, we employ this result to show that time However, is this T −1(∆t) also a legal time evo- evolution is surjective on interior level sets. 44 lution? This would be the case if T (∆t) was Lemma 6.5. Any allowed time evolution also surjective and thereby reversible on ΣN ⊂ is surjective on the interior level sets, i.e. RDN . Indeed, we shall demonstrate this property [T (∆t)](Lε)= Lε and thus also [T (∆t)](LI≤ε)= next. LI≤ε. Proof. Lemma 6.4 entails that T (∆t) defines a 6.6 Time evolution is also reversible continuous, open and closed map from Lε to it- To establish reversibility of time evolution we self. It therefore maps all open sets to open sets shall resort to tools from topology and convex and all closed sets to closed sets in Lε. In par- sets. ticular, it must map all sets that are both open and closed in Lε to other sets which are both D −1 Lemma 6.4. Restricting any time evolution open and closed. Since Lε ≃ S N is con- T (∆t):ΣN → ΣN to an interior level set Lε nected, the empty set and the full Lε are the (48) defines a continuous, open and closed map only sets in Lε which are both closed and open. from Lε to itself. Now, by lemma 6.3, T (∆t) is injective so that [T (∆t)](Lε)= Lε. Proof. As an injective square matrix T (∆t) de- fines a continuous invertible map from RDN to it- The last result is sufficient to finally establish self and so the pre-image of any open set in RDN that time evolution is surjective also on the state space. is another open set. Rule 3 implies [T (∆t)](Lε) ⊆ Lε and that no state from outside Lε can evolve Theorem 6.6. Any allowed time evolution is into Lε. The open sets of Lε in the induced topol- surjective on ΣN , i.e. [T (∆t)](ΣN )=ΣN , and ogy are the intersections of Lε with open sets of has | det T (∆t)| = 1. D R N . Consider any such open set V ⊂ Lε and DN any open set U ⊂ R such that V = U ∩ Lε. Proof. Given that an interior fat level set LI≤ε The pre-image [T −1(∆t)](U) is again open as is is, by lemma 6.2, homeomorphic to the closed −1 D therefore [T (∆t)](U)∩Lε which, thanks to rule unit ball in R N it is a convex body and has 3, must be non-empty if V is non-empty. Hence, volume [89]. Lemma 6.5 yields [T (∆t)](LI≤ε) = T (∆t) defines a continuous map from Lε to it- LI≤ε and so its volume is left invariant by self. For similar reasons, this map is also open. T (∆t) which, according to (53), acts linearly on Finally, the closed map lemma states that a con- states ~rO→S in the Bloch vector parametriza- tinuous map from a compact to a Hausdorff space tion. Theorem 6.2.14 in [89] then implies that is also closed. Thanks to lemma 6.2, we have | det T (∆t)| = 1 and therefore also that the vol- D −1 Lε ≃ S N which is both compact and Haus- ume of [T (∆t)](ΣN ) is equal to the volume of ΣN . dorff and so the map from Lε to itself defined by Consider now the volume difference

δ [T (∆t)](ΣN ), ΣN := vol [T(∆t)](ΣN) ∪ ΣN − vol [T(∆t)](ΣN) ∩ ΣN .      

−1 Clearly, [T (∆t)](ΣN ) ⊆ ΣN , and Hence, T (∆t) maps all legal states to legal so δ [T (∆t)](ΣN ), ΣN = vol (ΣN) − states and must, by rule 4, be legal also. vol [T(∆t)](ΣN) = 0. Exercise 6.2.2 in [89]
Corollary 6.7. Any time evolution permitted by then shows that δ [T (∆t)](ΣN ), ΣN = 0 is only
the rules is reversible. possible if [T (∆t)](Σ N )=ΣN .

44I thank a referee for pointing out that addressing this question was overlooked in a previous draft version.

Accepted in Quantum 2017-11-27, click title to verify 64 6.7 Time evolution defines a group

Given that any time interval can be decomposed into two time intervals, ∆t = ∆t1 +∆t2, and the evolution of ~rO→S = 2 ~yO→S − ~1 is continuous by rule 4, O must find

T (∆t2) T (∆t1) ~rO→S(0) = T (∆t2) ~rO→S(∆t1)= ~rO→S(∆t)= T (∆t) ~rO→S(0),

and thus that multiplication is abelian, by a single evolution generator and single param- eter ∆t parametrizing the duration. We note, T (∆t1 +∆t2)= T (∆t2) T (∆t1)= T (∆t1) T (∆t2). however, that a multiplicity of time evolutions of S is possible, depending on the physical circum- (The last equality follows from time translation stances (interactions) to which O may subject S. invariance.) From the last equation and T (0) = Different time evolutions will be generated by dif- 1, using ∆t − ∆t = 0, we can also infer that ferent generators, but each time evolution will T −1(∆t) = T (−∆t). If we permit O to consider form a one-parameter group as discussed above. the time evolution of S for any duration ∆t ∈ R, In fact, the full set of time evolutions TN which it follows that the product of any two time evolu- O is able to implement must likewise be a group. tion matrices is again a time evolution matrix. In Namely, if T (∆t), T ′(∆t′) correspond to two dis- summary, we therefore gather that a given time tinct interactions, then rule 4 implies that also evolution, as perceived by O and under the cir- T (∆t) · T ′(∆t′) is a legal time evolution for any cumstances to which he has subjected S, is such state and since both T, T ′ are invertible their full that: set must be a group. This implies, in particular, that TN is a group. We shall return to this further (i) T (0) = 1, below and in [1].

(ii) to every T (∆t) there exists an inverse 6.8 The squared length of the Bloch vector as T −1(∆t)= T (−∆t), information measure

(iii) the multiplication of any two time evolution We now have sufficient structure in our hand matrices is again a time evolution matrix, to determine the functional relation between αi and and yi. Given that the generalized Bloch vector 2 ~yO→S −~1 transforms nicely under time evolution (iv) matrix multiplication is obviously associa- (53), it is useful to parametrize αi by 2yi − 1, i.e. tive. αi = α(2yi − 1). Rule 3 entails that O’s total in- formation about (an otherwise non-interacting) S In conclusion, under the assumptions of section is a ‘conserved charge’ of time evolution 3.2 and rules 3 and 4, a given time evolution de- IO→S(~yO→S(∆t)) = IO→S(~yO→S(0)) fines therefore an abelian, one-parameter matrix group. Hence, a given time evolution is described which translates into the condition

DN DN I T (∆t) 2 ~y (0) − ~1 = α T (∆t) (2y (0) − 1) O→S O→S  ij j     Xi=1 jX=1 DN   = α (2yi(0) − 1) = IO→S(2 ~yO→S(0) − ~1). (59) Xi=1

If T (∆t) was a permutation matrix, (59) would ple, N classical bits are governed by the evolu- hold for any function α(2yi − 1). For exam- tion group Z2 ×···× Z2. However, permutations

Accepted in Quantum 2017-11-27, click title to verify 65 form a discrete group, while in our present case (vi) of subsection 6.2, enforces the quadratic re- 2 {T (∆t), ∆t ∈ R} constitutes a continuous one- lation αi = (2yi − 1) . To this end, we once more parameter group. This is where continuity of time invoke the coin flip scenario.45 evolution, as asserted by rule 4, becomes crucial. Given our parametrization in terms of the Under a reasonable assumption on the informa- Bloch vector 2 ~yO→S − ~1, O’s information about tion measure, we shall now show that continuity the outcomes of his questions in the coin flip sce- of time evolution, together with conditions (i)– nario can be written as follows

~ ~ ~ IO→S12 2 (λ ~yO→S1 + (1 − λ) ~yO→S2 ) − 1 = IO→S12 λ (2 ~yO→S1 − 1) + (1 − λ)(2 ~yO→S2 − 1)

  DN   1 2 = α λ (2yi − 1) + (1 − λ)(2yi − 1) . Xi=1  

It is instructive to consider the case in which O rescaled is state independent. For that reason, is entirely oblivious about S2 such that the lat- the relative information loss should likewise de- 1 ~ ter is in the state of no information ~yO→S2 = 2 1 pend only on the coin flip, quantified by λ, and relative to him, but that O has some informa- not on the state ~yO→S1 . For instance, if we also ~ tion about S1. In this case, 2 ~yO→S12 − 1 = considered the case that the coin flip was certain, ~ λ(2 ~yO→S1 −1) and (assuming the outcome of the i.e. λ = 0, 1, then clearly for λ = 1 we must have coin flip is not certain) strict convexity of IO→S f = 1 ∀ ~yO→S1 ∈ ΣN and for λ = 0 it must hold

(see subsection 6.2) implies f = 0 ∀ ~yO→S1 ∈ ΣN . We shall make this into a requirement on the information measure for all IO S (λ (2 ~yO S − ~1)) <λIO S (2 ~yO S − ~1) → 12 → 1 → 1 → 1 values of λ: or, equivalently, Requirement 1. The relative information loss I (λ (2 ~y − ~1)) (60) O→S12 O→S1 factor f in (61) is a state independent (contin- ~ = f · IO→S1 (2 ~yO→S1 − 1), uous) function of the coin flip probability λ with where f < 1 is a factor parametrizing O’s in- f(λ) < 1 for λ ∈ (0, 1). formation loss relative to the case in which he The binary Shannon entropy H(y) = does not toss a coin and, instead, directly asks −y log y − (1 − y) log(1 − y) in the role of α would S .46 The reason O experiences such a relative 1 fail this requirement. Namely, the measure α thus information loss about the outcome of his inter- factorizes, α(λ (2y − 1)) = f(λ) α(2y − 1), while rogation is, of course, entirely due to the random- i i H(y) would not. Setting λ = λ · λ yields47 ness of the coin flip. But the coin flip is indepen- 1 2 dent of the systems S1,2 and, in particular, of the f(λ1 · λ2) α(2 yi − 1) = α(λ1 · λ2 (2 yi − 1)) states in which these are relative to O; the fac- = f(λ1) α(λ2(2 yi − 1)) tor λ by which the probabilities ~yO→S1 become = f(λ1)f(λ2) α(2 yi − 1) 45We suspect that this result may be derivable from purely group theoretic arguments without an operational and therefore f(λ1) · f(λ2) = f(λ1 · λ2), which setup by employing the mathematical fact that to every implies f(λ) = λp, for some power p ∈ R. But continuous matrix group acting linearly on some space then, α must be a homogeneous function α(2 yi − there corresponds a conserved inner product which is quadratic in the components of the vectors. 47Such a factorization of coin flip probabilities could be 46In fact, one can equivalently interpret the situation as achieved, e.g., if O decided to use one coin, with ‘heads’ follows: O only considers a system S1 which would be in probability λ1, to firstly decide which of two possible con- the state ~yO→S1 if it was present with certainty. However, vex mixtures to prepare where both possible mixtures are

λ in the state λ(2 ~yO→S1 − ~1) represents the probability generated with a second coin with ‘heads’ probability λ2. that S1 ‘is there’ at all. Indeed, in this case, (4) can be If three of the four states within the two mixtures are written as λ(~yO→S1 +~nO→S1 )= λ·~1 such that λ(~yO→S1 − chosen as the state of no information, one would obtain

~nO→S1 )= λ(2 ~yO→S1 − ~1). precisely such an equation.

Accepted in Quantum 2017-11-27, click title to verify 66 p 1) = k (2 yi − 1) with some constant k ∈ R. In derive the quadratic measure more generally by consequence, the information IO→S(2 ~yO→S −~1) is starting from the landscape of information infer- (up to k) the p-norm of the Bloch vector 2 ~yO→S − ence theories and imposing rule 3 of information ~1. preservation and rule 4 of maximality of time evo- We can rule out that p ∈ (−∞, 0] because in lution thereon. In this regard, the present deriva- this case, as one can easily check, it is impossi- tion may similarly be taken as a strong justifica- ble to satisfy all the consistency conditions (i)– tion for the original Brukner-Zeilinger proposal. (vi) of subsection 6.2. Hence, p > 0. At this stage we can make use of (59) and a result by Aaronson [90] which implies that the only vector 6.9 The set of all time evolutions is a subgroup p-norm with p> 0 which is preserved by a contin- of SO(DN) uous matrix group is the 2-norm. Since any given Thus far, we only gathered that a given time evo- ~ time evolution of the Bloch vector 2 ~yO→S − 1 is lution is described by a one-parameter group and, governed by a continuous, one-parameter matrix 2 by theorem 6.6, must have | det T (∆t)| = 1. Now, group, we conclude that α(2 yi −1) = k (2 yi −1) . given (61), we are in the position to say quite a bit Imposing condition (iv) yields k = 1 and there- more. The full matrix group leaving (61) invari- fore ultimately ant is O(DN ). However, a given time evolution

DN is a continuous one-parameter group and must 2 IO→S(~yO→S)= (2 yi − 1) . (61) therefore be connected to the identity. Hence, Xi=1 T (∆t) ∈ SO(DN ), ∀ T (∆t). Consequently, the group corresponding to any fixed time evolution It is straightforward to convince oneself that all is a one-parameter subgroup of SO(D ). We of (i)–(vi) of subsection 6.2 are satisfied by this N also noted before that the set of all possible time quadratic information measure. O’s total amount evolutions T is a group thanks to rule 4. It of information about S is thus the squared length N must therefore likewise satisfy T ⊂ SO(D ). of the generalized Bloch vector, thereby assuming N N This topic is thoroughly discussed in the com- a geometric flavour. It is important to emphasize panion articles, where it is shown that the rules that, had we not imposed continuity of time evo- imply T = PSU(2N) for the D = 3 [1] and lution in rule 4, we would not have been able to N 1 T = PSO(2N) for the D = 2 case [2]. Note arrive at (61); if time evolution was not continu- N 1 that PSU(2N) is a proper subgroup of SO(D = ous, many solutions to α in terms of y would be N i 4N − 1) for N > 1. The generators of T are the possible. N set of possible time evolution generators of S’s The quadratic information measure (61) has states. been proposed earlier by Brukner and Zeilinger in [32,34,91,92] from a different perspective, em- phasizing that this is the most natural measure 6.10 Pure and mixed states taking into account an observer’s uncertainty – due to statistical fluctuations – about the out- The explicit quantification of O’s information come of the next trial of measurements on a now permits us to render the distinction between system in a multiple shot experiment. Further- three informational classes of S’s states – which more, taking the formalism of quantum theory we already loosely referred to as ‘states of max- as given, Brukner and Zeilinger [73] later singled imal knowledge’ or ‘states of non-maximal infor- out the quadratic measure from the set of Tsal- mation’ in previous sections – precise. lis entropies by imposing an ‘information invari- Firstly, we determine the maximally attain- ance principle’, according to which a continuous able (independent and dependent) information transformation among any two complete sets of content within a state ~yO→S of a system of N mutually complementary measurements in quan- gbits. This can be easily counted: once O knows tum theory should leave an observer’s informa- the answers to N mutually independent questions tion about the system invariant. While [73] is (these do not need to be individuals), he will also certainly compatible with the present framework, know the answers to all their bipartite, tripar- here we come from farther away to the same re- tite,... and N-partite correlation questions – all sult: we do not pre-suppose quantum theory and of which are contained in QMN too by theorems

Accepted in Quantum 2017-11-27, click title to verify 67 5.27 and 5.31. But these are then are pairwise but not necessarily mutually inde- pendent. N N N N N Using this observation, we shall characterize + + · · · = = 2N − 1 1 ! 2 ! N! i=1 i ! S’s states according to their information content, X i.e. squared length of the Bloch vector. By rules 3 answered questions from QMN , while all remain- and 4, this distinction applies to all states which ing questions in QMN will be maximally comple- are connected via some time evolution to the mentary to at least one of the known ones. O’s to- states above, including those for which the infor- tal information, as quantified by IO→S (47), thus mation may be distributed partially over many contains plenty of dependent bits of information elements of a fixed QMN . Specifically, we shall

– a result of the fact that the questions in QMN refer to a state ~yO→S as a

pure state: if it is a state of maximal information content, i.e. maximal length

DN 2 N IO→S(~yO→S)= (2 yi − 1) = (2 − 1) bits, Xi=1

mixed state: if it is a state of non-extremal information content, i.e. non-extremal length

DN 2 N 0 bit

1~ totally mixed state: if it is the state of no information ~yO→S = 2 1 with zero length

1 IO→S ~yO→S = ~1 = 0 bit.  2 

We note that these characterizations of states in ory – are, indeed, the extremal states, requires terms of their length are indeed true in quantum more work. For N = 1 we shall demonstrate this theory. In particular, N qubit pure states actu- shortly, while the discussion for N > 1 is deferred ally have a Bloch vector squared length equal to to [1]. 2N − 1. Our reconstruction gives this peculiar fact a clear informational interpretation. 7 The N =1 case and the Bloch ball One may wonder whether the above definition Before closing this toolkit for now, we quickly give of a pure state is directly equivalent to being ex- a flavor of the capabilities of the newly developed tremal in Σ and thereby to the usual defini- N concepts and tools by applying them to show that tion of pure states in GPTs or quantum theory. in the simplest case of a single gbit (N = 1) rules This is not obvious from what we have estab- 1–4 indeed only have two solutions within L , lished thus far. However, since clearly we as- gbit namely the qubit and the rebit state space includ- sume I = (2N − 1) bits to be the maxi- O→S ing their respective time evolution groups. This mally attainable information, we can already con- proves the claim of section 4.1 for N = 1. clude that pure states so defined must lie on the To this end, recall theorem 5.6 which asserts boundary of Σ because a non-constant convex N that the dimension of the N = 1 state space Σ function cannot have its maximum in the inte- 1 is either D = 2 or D = 3 which thus far we rior of its convex domain [89]. Showing that the 1 1 suggestively referred to as the ‘rebit’ and ‘qubit pure states, as defined above, of the theory sur- case’, respectively. viving the imposition of all rules – quantum the-

Accepted in Quantum 2017-11-27, click title to verify 68 7.1 A single qubit and the Bloch ball

We begin with the D1 = 3 case. Σ1 will be parametrized by a three-dimensional vector

y1 ~yO→S =  y2  , y  2    where y1,y2,y3 are the ‘yes’-probabilities of a mutually maximally complementary question set Q1,Q2,Q3 constituting an informationally complete Q1 (11). The informational distinction of states introduced in section 6.10 reads in this case

pure states: ~yO→S such that

2 2 2 IO→S = (2 y1 − 1) + (2 y2 − 1) + (2 y3 − 1) = 1 bit,

mixed states: ~yO→S such that

2 2 2 0 bit

1 ~ totally mixed state: ~yO→S = 2 1 such that

2 2 2 IO→S = (2 y1 − 1) + (2 y2 − 1) + (2 y3 − 1) = 0 bit.

Recall from section 6.9 that the set of possible theory. time evolutions T1 must be contained in SO(3) The set of allowed states populates the entire and that the time evolution rule 4 requires the unit ball in the three-dimensional Bloch ball, i.e. 3 set of states into which any state ~yO→S can evolve Σ1 ≃ B . This follows from the fact that apply- to be maximal, while being compatible with the ing the full group T1 = SO(3) to (1, 0, 0) gener- rules. Thus, in particular, the image of the cer- ates the entire Bloch sphere as a closed set of ex- tainly legal pure state (1, 0, 0) (rules 1 and 2 im- tremal states and the fact that Σ1 must be closed ply its existence in Σ1) under T1 must be maxi- convex according to assumption 2. Hence, we mal. Applying all of SO(3) to this state generates recover the well-known three-dimensional Bloch 2 all states with |2~y − ~1| = 1 bit and these cer- ball state space of a single qubit of standard quan- tainly abide by the rules. Consequently, the set tum theory with the set of all pure states defining of all possible time evolutions, compatible with the boundary sphere S2, the totally mixed state rules 1–4, is the rotation group (as the state of no information) constituting the center and the set of mixed states filling the inte- T ≃ SO(3) ≃ PSU(2). 1 rior in between, as illustrated in figure 6a. This This is the component of the isometry group of is precisely the geometry of the set of all normal- C2 the Bloch ball which is connected to the identity ized density matrices on . Notice that the pure 2 C 1 and it is required in full in order to maximize the state space S ≃ P indeed coincides with the C2 number of states into which (1, 0, 0) can evolve. set of all unit vectors in (modulo phase). However, since time evolution is state indepen- Given the complete symmetry of the Bloch ball dent (c.f. assumption 8), this is the time evolu- as the state space for N = 1, there should not ex- tion group for all states. PSU(2) is precisely the ist a distinguished informationally complete ques- adjoint action of SU(2) on density matrices ρ2×2 tion set QM1 , corresponding to a distinguished or- 2 † over C , ρ2×2 7→ U ρ2×2 U , U ∈ SU(2) and thus thonormal Bloch vector basis, by means of which coincides with the set of all possible unitary time O can interrogate S. While it is an additional evolutions of a single qubit in standard quantum assumption, it is thus natural to stipulate that

Accepted in Quantum 2017-11-27, click title to verify 69 totally mixed state state space of a single qubit can also be derived from various operational axioms within GPTs mixed states [15, 16, 18, 21, 23, 59, 93] and constitutes a cru- ~r = 2 ~y − ~1 O→S cial step in most GPT based reconstructions of PSfrag replacements ~r quantum theory [14–16, 20, 21]. The principle of continuous reversibility, according to which every pure state of the convex set can be mapped into any other by means of a continuous and reversible transformation, usually assumes a crucial role in pure states such derivations. Here we offer a novel perspec- tive on the origin of the Bloch ball by deriving it from elementary rules for the informational rela- (a) 3D qubit Bloch ball tion between O and S; in particular, we recover continuous reversibility as a by-product.

totally mixed state 7.2 A single rebit and the Bloch disc

mixed states The analogous result holds for the D1 = 2 case: ~r = 2 ~yO→S − ~1 Σ will be parametrized by a two-dimensional ~r 1 PSfrag replacements vector

y1 ~yO→S = , y2 !

where y1,y2 are the ‘yes’-probabilities of two pure states maximally complementary questions Q1,Q2 con- stituting an informationally complete Q1 (11). We then have (b) 2D rebit Bloch disc pure states: ~yO→S such that Figure 6: The three-dimensional Bloch ball (a) and the two-dimensional Bloch disc (b) are the correct state 2 2 IO→S = (2 y1 − 1) + (2 y2 − 1) = 1 bit, spaces Σ1 of a single qubit in standard quantum theory and a single rebit in real quantum theory, respectively. The vector ~r parametrizing the states is the Bloch vector mixed states: ~yO→S such that 2 ~yO→S − ~1. 2 2 0 bit

and the to every Q ∈ Q1 there exists a unique pure state 1 ~ totally mixed state: ~yO→S = 2 1 such that in Σ1 which represents the truth value Q =‘yes’ and, conversely, that every pure state of this sys- 2 2 IO→S = (2 y1 − 1) + (2 y2 − 1) = 0 bit. tem corresponds to the definite answer to one question in Q . But then Q ≃ S2 which also 1 1 In analogy to the qubit case, the time evolution coincides with the set of all possible projective rule 4 implies that measurements onto the +1 (or, equivalently, the −1) eigenspaces of the Pauli operators ~n · ~σ over (a) the set of all possible time evolutions is the 2 3 2 C which is parametrized by ~n ∈ R , |~n| = 1, (projective) rotation group and where ~σ = (σx,σy,σz) are the usual Pauli matrices. This set of permissible questions QN # T1 ≃ SO(2) ≃ PSO(2) ≃ SO(2)/Z2 for all N will be discussed more thoroughly in [1] together with a derivation of the Born rule for because SO(2) is the full (connected com- projective measurements. ponent of the orientation preserving) isom- The ‘ballness’ and three-dimensionality of the etry group of the Bloch disc and all time

Accepted in Quantum 2017-11-27, click title to verify 70 evolutions are permitted which preserve the system) has access to by interaction with other Bloch vector length, representing O’s total systems, we have developed a novel framework information about S, and the orientation of for characterizing and (re-)constructing the quan- the Bloch vector basis. Orientation preser- tum theory of qubit systems. We have laid down, vation means that O’s convention about the without ontological statements, the mathemati- ‘yes’-‘no’-labelling of the question outcomes cal and conceptual foundations for a landscape is preserved: The isomorphism denoted with of theories describing how an observer acquires a # requires some explaining because SO(2) information about physical systems through in- is actually a double cover of PSO(2) as indi- terrogation with yes-no-questions. cated on the right. Indeed, the real density Within this landscape four elementary rules 1 1 matrix of a single rebit, ρ2×2 = 2 ( +rxσx + have been given which constrain the observer’s 2 rzσz) on R where ri = (2yi − 1), evolves acquisition of information for qubit (and rebit) under the adjoint action of SO(2), ρ2×2 7→ systems. The rule of limited information and the T iσy t O ρ2×2 O for O = e ∈ SO(2). This is complementarity rule imply an independence and equivalent to an action of PSO(2) because compatibility structure of the binary questions the non-trivial center element −1 ∈ SO(2) which, in fact, reproduces that of projective mea- acts trivially on ρ2×2 and thus factors out. surements onto the +1 eigenspaces of Pauli op- However, as a subcase of the single qubit erators in quantum theory. In particular, these case this adjoint action of SO(2) on ρ2×2 is rules entail in a constructive and simple way also equivalent to O′ ~r for O′ ∈ SO(2) in the 1. a novel argument for the dimensionality of Bloch vector representation. the Bloch ball, (b) the state space Σ1 coincides with the two- dimensional Bloch disc, as depicted in figure 2. a new method for determining the 6b, with the totally mixed state in the center, correct number of independent ques- the pure states on the boundary circle S1 and tions/measurements necessary to describe a the mixed states in the interior in between. system of N qubits (or rebits), This is precisely the geometry of the set of 3. a natural explanation for entanglement, unit trace, positive-semidefinite, symmetric monogamy of entanglement and quantum R2 matrices over – the space of density ma- non-locality, trices of a single rebit. Similarly, the pure state space S1 ≃ RP 1 coincides with the set 4. the explicit correlation structure of two of all unit vectors in the Hilbert space R2 qubits and rebits, and modulo Z . 2 5. more generally the correlation structure for Again, given the symmetry of the Bloch disc, arbitrarily many qubits and rebits. there is no reason for a distinguished informa- Furthermore, the rules of information preserva- tionally complete question basis QM1 on Q1, in- carnated as a distinguished Bloch vector basis, tion and maximality of time evolution are shown to exist. Accordingly, we assume that any pure to result 1 state on S corresponds to the definite answer 6. in a reversible time evolution, and of some question in Q1 which O may ask the 1 rebit. In this case, Q1 ≃ S , which coincides 7. under elementary consistency conditions, in with the set of projective measurements onto the a quadratic information measure, quantify- +1 eigenspaces of the Pauli operators ~n · ~σ over ing the observer’s prior information about R2 which are parametrized by ~n ∈ R2, |~n|2 = 1 the answers to the various questions he may and where ~σ = (σx,σz) are the two real and sym- ask the system. metric Pauli matrices. This measure has been earlier proposed by Brukner and Zeilinger from a different perspec- 8 Discussion and outlook tive [32,34,73,91,92], complementing our present derivation. Based on the premise to only speak about in- Combining these results, we then show, as the formation which an observer (or more generally simplest example, how the rules entail that

Accepted in Quantum 2017-11-27, click title to verify 71 8. the Bloch ball and Bloch disc are recovered formation and additional inputs are necessary in as state spaces for a single qubit and rebit, order to do proper physics. respectively, together with the correct time Nevertheless, despite its current limitations, evolution groups and question sets. this informational approach teaches us something non-trivial about the structure and physical con- The full reconstruction of qubit quantum the- tent of quantum theory. ory, following from the present four rules (and two additional ones), is completed in the companion While this work is more generally motivated by paper [1] (the rebit case which violates one of the the effort to understand physics from an infor- additional rules is considered separately [2]). In mational and operational perspective and high- conjunction, this derivation highlights the partial lights a partial interpretation of quantum the- interpretation of quantum theory as a law book, ory, it clearly does not single out ‘the right one’ describing and governing an observer’s acquisi- (see also the discussion in [94]). For example, by tion of information about physical systems. In speaking exclusively about the information acces- particular, it highlights the quantum state as the sible to an observer, we are by default silent on observer’s ‘catalogue of knowledge’. the fate of hidden variables and on whether they Certainly, there are some limitations to the could give rise to the assumptions and rules which present approach: First of all, we employ a clear we impose. The status of hidden variables is sim- distinction between observer and system which ply not relevant here (other than that local hidden cannot be considered as fundamental. Secondly, variables are ruled out). the construction is specifically engineered for the Let us, nonetheless, make a few possible quantum theory of qubit systems. While arbi- (but not inescapable) interpretational state- trary finite dimensional quantum systems could, ments. Given the absence of ontological commit- in principle, be immediately encompassed by im- ments, this informational approach is generally posing the so-called subspace axiom of GPTs compatible with (but does not rely on) the re- [14, 16], the latter does not naturally fit into our lational [28, 29], Brukner-Zeilinger informational set of principles, mostly because rules 1 and 2 [31–35], or QBist [36–39] interpretations of quan- quantify the information content of a system in tum mechanics. They depart from the traditional terms of N ∈ N bits which is only suitable for idea that systems necessarily have, absolute, i.e. composite gbit systems. More generally, the lim- observer independent properties (or, more gen- itation is that the current approach only encom- erally, properties independent of their relations passes finite dimensional quantum theory, but not with other systems). Instead, many physical quantum mechanics. As it stands, the mechani- properties are interpreted as relational; the inter- cal phase space language does not naturally fit action between systems establishes a relation be- into the present framework and more sophisti- tween them, permitting an information exchange cated tools are required in order to cover mechan- which reveals certain physical properties relative ical systems, let alone anything beyond that. to one another and in the absence of hidden vari- While this informational construction recovers ables this would be all there is. Certainly, in or- the state spaces, the set of possible time evolu- der not to render such a view hopelessly solip- tions and projective measurements of qubit quan- sistic, systems ought to have certain intrinsic at- tum theory, in other words, the architecture of tributes, e.g. a corresponding state space or set the theory, it clearly does not tell us much about of permissible interactions/measurements, such the concrete physics (other than demanding it to that different observers have a basis for agree- fit into this general construction). This purely ing or disagreeing on the description of physical informational framework is rather universal and objects. However, a state of a system or measure- information carrier independent. But qubits as ment/question outcome is taken relative to who- information carriers can be physically incarnated ever performs the measurement or asks the ques- in many different ways: as electron or muon spins, tion. Different observers may agree on states or photon polarization, quantum dots, etc. The measurement outcomes by communication (i.e., framework cannot distinguish among the differ- physical interaction) but if one rejects the idea of ent physical incarnations, underlining the obser- an absolute and omniscient observer it is natural vation that not everything can be reduced to in- to also abandon the idea of an absolute and ex-

Accepted in Quantum 2017-11-27, click title to verify 72 ternal standard by means of which properties of absolute observer systems could be defined. PSfrag replacements From such a perspective, one could say that S5 the relational character of the qubits’ proper- S6 S4 S7 ties is a consequence of a universal limit on the amount of information accessible to the observer communication and the mere existence of complementary infor- mation such that the observer can not know the S1 S3 answers to all his questions simultaneously. It is the observer who determines which questions he S will ask and thereby what kind of system prop- 2 erty the interrogation will reveal and thus, ulti- mately, which kind of information he will acquire Figure 7: The universe as an information exchange net- (although clearly he does not determine what the work of subsytems without absolute observer. question outcome is). But relative to the observer the system of qubits does not have properties other than those accessible to him. admitting any register in the network to act as ‘observer’, the self-reference problem [98, 99] im- 8.1 An operational alternative to the wave pedes a given register to infer the global state of function of the universe the entire network – including itself – from its in- Pushing an informational interpretation of quan- teractions with the rest. Accordingly, relative to tum theory to the extreme, one may speculate any subsystem, one could assign a state to the whether the quantum state could also represent rest of the network, but a global state and thus a state of information in a gravitational or cosmo- a global Hilbert space would not arise. The ab- logical context. For instance, is such an interpre- sence of a global state in quantum gravity has tation adequate for the ‘wave function of the uni- been proposed before [100–103] – albeit from a verse’ which is ubiquitous in standard approaches different, less informational perspective. to quantum cosmology (e.g., see [95–97])? Such This offers an operational alternative to the an interpretation would require the existence of problematic concept of the ‘wave function of the an absolute and omniscient observer, an idea universe’ which is ubiquitous in quantum cosmol- which we just abandoned. ogy. However, while a ‘wave function of the uni- Alternatively, one could adopt one of the cen- verse’ might not make sense as a fundamental tral ideas of relational quantum mechanics [28, concept, it could still be given meaning as an ef- 29], according to which all physical systems can fective notion, namely as the state of the large assume the role of an ‘observer’, recording infor- scale structure of our universe on whose descrip- mation about other systems, thereby relieving the tion ‘late time’ observers better agree. clear distinction used in this manuscript. Ex- Clearly, if this was to offer a coherent picture tending this idea to a space-time context, one of physics, there would need to exist non-trivial could interpret the universe as an abstract net- consistency relations among the different regis- work of subsystems/subregions, viewed as infor- ters’ descriptions such as, e.g., the requirement mation registers, which can communicate and ex- that ‘late time observers’ agree on the state of change information through interaction (see fig- the large scale structure. This is not a practically ure 7). In this background independent context, unrealistic expectation as a concrete playground any information acquisition by any register is in- for this idea has recently been constructed from ternal, i.e. occurs within the network; a global a different motivation [104]: a scalar field on the observer outside the network becomes meaning- background geometry of elliptic de Sitter space less. This may appear as a purely philosophical can only be quantized in an observer dependent observation, but it implies concrete consequences manner. A global Hilbert space for the quantum for the description of the network: there should field does not exist, but consistency conditions be no global state (aka ‘wave function of the uni- between different observers’ descriptions can be verse’) for the entire network at once. Indeed, derived. (Observer consistency has also been dis-

Accepted in Quantum 2017-11-27, click title to verify 73 cussed in general terms within Relational Quan- [6] T. Jacobson, “Gravitation and vacuum tum Mechanics [28, 29] and QBism [64, 65].) Al- entanglement entropy,” though not being a quantum gravity model, it Int.J.Mod.Phys. D21 (2012) 1242006, may serve as a platform for concretizing this pro- arXiv:1204.6349. posal further. [7] E. Bianchi and R. C. Myers, “On the Architecture of Spacetime Geometry,” Acknowledgements Class.Quant.Grav. 31 no. 21, (2014) 214002, arXiv:1212.5183. This work has benefited from interactions with [8] E. T. Jaynes, “Information theory and many colleagues. First and foremost, it is a statistical mechanics," pleasure to thank Markus Müller for uncount- Phys. Rev. 106 (1957) 620 ably many insightful discussions, patient expla- nations of his own work and suggestions. I am [9] C. H. Bennett, “The thermodynamics of grateful to Chris Wever for collaboration on [1,2] computation—a review,” and important discussions, to Sylvain Carrozza Intern. J. Theor. Phys. 21, (1982) 905–940. for drawing my attention to Relational Quantum [10] K. Maruyama, F. Nori, and V. Vedral, Mechanics in the first place and many related “Colloquium: The physics of maxwell’s conversations, to Tobias Fritz and Lucien Hardy demon and information,” for asking numerous clarifying questions, and to Reviews of Modern Physics 81, (2009) 1. Rob Spekkens for many careful comments and [11] S. Popescu, A. J. Short, and A. Winter, suggestions. I would also like to thank Caslav “Entanglement and the foundations of Brukner, Borivoje Dakic, Martin Kliesch, Matt statistical mechanics,” Leifer, Miguel Navascués, Matteo Smerlak and Nature Physics 2, (2006) 754–758. Rafael Sorkin for useful discussions. Research at Perimeter Institute is supported by the Govern- [12] M. Horodecki and J. Oppenheim, ment of Canada through Industry Canada and by “Fundamental limitations for quantum and the Province of Ontario through the Ministry of nanoscale thermodynamics,” Research and Innovation. Nature communications 4 (2013). [13] M. A. Nielsen and I. L. Chuang, Quantum References computation and quantum information. Cambridge university press, 2010. [1] P. A. Höhn and C. S. P. Wever, “Quantum [14] L. Hardy, “Quantum theory from five theory from reasonable axioms,” questions,”Phys.Rev. A95 (2017) 012102, arXiv:quant-ph/0101012 [quant-ph]. Preprint: arXiv:1511.01130. [15] B. Dakic and C. Brukner, “Quantum theory [2] P. A. Höhn and C. S. P. Wever, “A and beyond: Is entanglement special?,” reconstruction of real quantum theory from Deep Beauty: Understanding the Quantum rules on information acquisition," to appear World through Mathematical Innovation, [3] R. D. Sorkin, “On the Entropy of the Ed. H. Halvorson (Cambridge University Vacuum outside a Horizon,” 10th Int. Conf. Press, 2011) 365-392 (11, 2009), Gen. Rel. Grav. Contributed Papers II arXiv:0911.0695. (1983) 734, arXiv:1402.3589. [16] L. Masanes and M. P. Müller, “A derivation [4] L. Bombelli, R. K. Koul, J. Lee, and R. D. of quantum theory from physical Sorkin, “A Quantum Source of Entropy for requirements,” Black Holes,” New Journal of Physics 13, (2011) 063001. Phys.Rev. D34 (1986) 373–383. [17] M. P. Müller and L. Masanes, [5] T. Jacobson, “Thermodynamics of “Information-theoretic postulates for space-time: The Einstein equation of quantum theory,” in Quantum Theory: state,” Phys.Rev.Lett. 75 (1995) 1260–1263, Informational Foundations and Foils. arXiv:gr-qc/9504004. Chiribella G., Spekkens R. (eds)

Accepted in Quantum 2017-11-27, click title to verify 74 Fund. Theories Phys., vol 181, Springer [29] M. Smerlak and C. Rovelli, “Relational arXiv:1203.4516. EPR,” Found.Phys. 37 (2007) 427–445, arXiv:quant-ph/0604064. [18] L. Masanes, M. P. Müller, R. Augusiak, and D. Perez-Garcia, “Existence of an [30] A. Peres, Quantum theory: concepts and information unit as a postulate of quantum methods, vol. 57. Springer, 1995. theory,” PNAS 110, 16373 (2013), [31] A. Zeilinger, “A foundational principle for arXiv:1208.0493. quantum mechanics,” Foundations of Physics 29, (1999) 631–643. [19] G. Chiribella, G. M. D’Ariano, and P. Perinotti, “Informational derivation of [32] C. Brukner and A. Zeilinger, “Operationally quantum theory,” invariant information in quantum Physical Review A 84, (2011) 012311. measurements,” Phys. Rev. Lett. 83 (1999) 3354–3357, [20] G. de la Torre, L. Masanes, A. J. Short, arXiv:quant-ph/0005084. and M. P. Müller, “Deriving quantum the- ory from its local structure and reversibility,” [33] C. Brukner, M. Zukowski, and A. Zeilinger, Physical Review Letters 109, (2012) 090403. “The essence of entanglement,” arXiv:quant-ph/0106119. [21] M. P. Müller and L. Masanes, [34] C. Brukner and A. Zeilinger, “Information “Three-dimensionality of space and the and fundamental elements of the structure quantum bit: how to derive both from of quantum theory,” information-theoretic postulates,” in “Time, Quantum, Information”, eds. by New J. Phys. 15, 053040 (2013), L. Castell and O. Ischebeck (Springer, arXiv:1206.0630. 2003), arXiv:quant-ph/0212084. [22] L. Hardy, “Reconstructing quantum [35] C. Brukner and A. Zeilinger, “Young’s theory,” arXiv:1303.1538. experiment and the finiteness of [23] H. Barnum, M. P. Müller, and C. Ududec, information,” “Higher-order interference and single-system Phil. Trans. R. Soc. Lond. A 360 (2002) 1061, postulates characterizing quantum theory,” arXiv:quant-ph/0201026. New J. Phys. 16 123029 (2014) [36] C. A. Fuchs, “Quantum mechanics as arXiv:1403.4147. quantum information (and only a little more),” arXiv:quant-ph/0205039. [24] S. Kochen, “A reconstruction of quantum mechanics,” in Bertlmann R., Zeilinger A. [37] C. M. Caves and C. A. Fuchs, “Quantum (eds) Quantum [Un]Speakables II. (2017) information: How much information in a The Frontiers Collection. Springer, Cham, state vector?,” arXiv:quant-ph/9601025. arXiv:1306.3951. [38] C. M. Caves, C. A. Fuchs, and R. Schack, [25] R. Oeckl, “A local and operational “Quantum probabilities as bayesian framework for the foundations of physics ,” probabilities,” arXiv:1610.09052. Phys. Rev. A 65 (2002) 022305, arXiv:quant-ph/0106133. [26] J. B. Hartle, “Quantum mechanics of [39] C. M. Caves, C. A. Fuchs, and R. Schack, individual systems,” “Unknown quantum states: the quantum de Am. J. of Phys. 36, (1968) 704–712. finetti representation,” [27] Q. Zheng and T. Kobayashi, “Quantum J. Math. Phys. 43, (2002) 4537–4559. optics as a relativistic theory of light,” [40] R. W. Spekkens, “Evidence for the Physics Essays 9 (1996) 447–459. epistemic view of quantum states: A toy [28] C. Rovelli, “Relational quantum theory,” mechanics,” Physical Review A 75, (2007) 032110. Int.J.Theor.Phys. 35 (1996) 1637–1678, [41] R. W. Spekkens, “Quasi-quantization: arXiv:quant-ph/9609002. classical statistical theories with an

Accepted in Quantum 2017-11-27, click title to verify 75 epistemic restriction,” in Quantum Theory: [53] M. Pawłowski, T. Paterek, D. Kaszlikowski, Informational Foundations and Foils. V. Scarani, A. Winter, and M. Żukowski, Chiribella G., Spekkens R. (eds) “Information causality as a physical Fund. Theories Phys., vol 181, Springer principle,” Nature 461, (2009) 1101–1104. arXiv:1409.5041 . [54] T. Paterek, B. Dakic, and C. Brukner, [42] P. A. Höhn and M. P. Müller, “An “Theories of systems with limited operational approach to spacetime information content,” symmetries: Lorentz transformations from New J. Phys. 12 (2010) 053037, quantum communication,” arXiv:0804.1423. New J. Phys. 18 (2016), 063026, [55] J. Barrett, “Information processing in arXiv:1412.8462. generalized probabilistic theories,” [43] C. Rovelli, Quantum Gravity. Cambridge Physical Review A 75, (2007) 032304. University Press, 2004. [56] L. Hardy, “Foliable operational structures [44] C. Rovelli, “Time in quantum gravity: for general probabilistic theories,” Deep Physics beyond the Schrödinger regime,” Beauty: Understanding the Quantum World Phys.Rev. D43 (1991) 442–456. through Mathematical Innovation; Halvorson, H., Ed (2011) 409, [45] C. Rovelli, “What is observable in classical arXiv:0912.4740. and quantum gravity?,” Class.Quant.Grav. 8 (1991) 297–316. [57] L. Hardy, “A formalism-local framework for general probabilistic theories, including [46] C. Rovelli, “Quantum reference systems,” quantum theory,” Class.Quant.Grav. 8 (1991) 317–332. Math. Struct. Comp. Science 23, (2013) 399, [47] B. Dittrich, “Partial and complete arXiv:1005.5164. observables for canonical General [58] G. Chiribella, G. M. D’Ariano, and Relativity,” P. Perinotti, “Probabilistic theories with Class.Quant.Grav. 23 (2006) 6155–6184, purification,” arXiv:gr-qc/0507106. Physical Review A 81, (2010) 062348. [48] J. Tambornino, “Relational observables in [59] L. Masanes, M. P. Müller, D. Perez-Garcia, gravity: A review,” SIGMA 8 (2012) 017, and R. Augusiak, “Entanglement and the arXiv:1109.0740. three-dimensionality of the bloch ball,” [49] M. Bojowald, P. A. Höhn, and J. Math. Phys. 55, 122203 (2014), A. Tsobanjan, “Effective approach to the arXiv:1111.4060. problem of time: general features and [60] C. Pfister and S. Wehner, “An examples,” Phys.Rev. D83 (2011) 125023, information-theoretic principle implies that arXiv:1011.3040. any discrete physical theory is classical,” [50] M. Bojowald, P. A. Höhn, and Nature communications 4 (2013) 1851. A. Tsobanjan, “An Effective approach to [61] H. Barnum, J. Barrett, L. O. Clark, the problem of time,” M. Leifer, R. Spekkens, N. Stepanik, Class.Quant.Grav. 28 (2011) 035006, A. Wilce, and R. Wilke, “Entropy and arXiv:1009.5953. information causality in general [51] P. A. Höhn, E. Kubalova, and probabilistic theories,” A. Tsobanjan, “Effective relational New J. Phys. 12 (2010) 033024, dynamics of a nonintegrable cosmological arXiv:0909.5075. model,” Phys.Rev. D86 (2012) 065014, [62] A. S. Holevo, Probabilistic and Statistical arXiv:1111.5193. Aspects of Quantum Theory, [52] S. Popescu and D. Röhrlich, “Quantum North-Holland, Amsterdam, 1982 nonlocality as an axiom,” [63] R. W. Spekkens, “Contextuality for Foundations of Physics 24, (1994) 379–385. preparations, transformations, and unsharp

Accepted in Quantum 2017-11-27, click title to verify 76 measurements,” [77] C. H. Bennett, D. Leung, G. Smith, and Physical Review A 71, (2005) 052108. J. A. Smolin, “Can closed timelike curves or [64] C. Caves, C. Fuchs, and R. Schack, nonlinear quantum mechanics improve “Conditions for compatibility of quantum state discrimination or help solve quantum-state assignments,” hard problems?,” Phys. Rev. A 66 (Dec, 2002) 062111. Phys.Rev.Lett. 103 (2009) 170502, arXiv:0908.3023. [65] N. D. Mermin, “Compatibility of state assignments,” [78] C. F. von Weizsäcker, The Structure of J. Math. Phys. 43, (2002) 4560. Physics. Springer-Verlag, Dordrecht, 2006. [66] C. A. Fuchs and A. Peres, “Quantum state [79] T. Görnitz and O. Ischebeck, An disturbance versus information gain: Introduction to Carl Friedrich von Uncertainty relations for quantum Weizsäcker’s Program for a Reconstruction information,” Phys.Rev. A53 (1996) 2038, of Quantum Theory. arXiv:quant-ph/9512023. in “Time, Quantum, Information”, eds. by L. Castell and O. Ischebeck (Springer, 2003) [67] C. A. Fuchs, “Information gain versus state disturbance in quantum theory,” [80] H. Lyre, “Quantum theory of ur-objects as arXiv:quant-ph/9611010. a theory of information,” [68] E. Specker, “Die logik nicht gleichzeitig Int. J. Theor. Physics 34, (1995) 1541. entscheidbarer aussagen,” [81] B. Dakic and C. Brukner, “The classical Dialectica 14 (1960) 239. limit of a physical theory and the [69] Y.-C. Liang, R. W. Spekkens, and H. M. dimensionality of space,” in Quantum Wiseman, “Specker’s parable of the Theory: Informational Foundations and overprotective seer: A road to contextuality, Foils. Chiribella G., Spekkens R. (eds) nonlocality and complementarity,” Fund. Theories Phys., vol 181, Springer Physics Reports 506, (2011) 1–39. arXiv:1307.3984. [70] A. Cabello, “Simple explanation of the [82] N. Bohr, Atomic Theory and the quantum violation of a fundamental Description of Nature. Cambridge inequality,” University Press, 1961 (Reprint). Physical review letters 110, (2013) 060402. [83] C. M. Caves, C. A. Fuchs, and P. Rungta, [71] A. Cabello, “Specker’s fundamental “Entanglement of principle of quantum mechanics,” formation of an arbitrary state of two rebits,” arXiv:1212.1756. Foundations of Physics Letters 14, (2001) 199. [72] G. Chiribella and X. Yuan, “Measurement [84] V. Coffman, J. Kundu, and W. K. sharpness cuts nonlocality and Wootters, “Distributed entanglement,” contextuality in every physical theory,” Phys.Rev.A 61 (2000) 052306, arXiv:1404.3348. arXiv:quant-ph/9907047. [73] Č. Brukner and A. Zeilinger, “Information [85] B. Regula, S. D. Martino, S. Lee, and invariance and quantum probabilities,” G. Adesso, “Strong monogamy conjecture Foundations of Physics 39, (2009) 677–689. for multiqubit entanglement: The [74] S. Weinberg, “Precision Tests of Quantum four-qubit case,” Mechanics,” Phys.Rev.Lett. 62 (1989) 485. Phys. Rev. Lett. 113 (2014) 110501, arXiv:1405.3989. [75] N. Gisin, “Weinberg’s non-linear quantum mechanics and supraluminal [86] W. K. Wootters, “Entanglement sharing in communications,” real-vector-space quantum theory,” Physics Letters A 143, (1990) 1–2. Foundations of Physics 42, (2012) 19. [76] J. Polchinski, “Weinberg’s nonlinear [87] W. K. Wootters, “The rebit three-tangle quantum mechanics and the EPR paradox,” and its relation to two-qubit entanglement,” Phys.Rev.Lett. 66 (1991) 397–400. J. Phys. A: Math. Theor. 47 (2014) 424037.

Accepted in Quantum 2017-11-27, click title to verify 77 [88] A. Peres, “Separability criterion for density J.Math.Phys. 36 (1995) 6180, matrices,” arXiv:gr-qc/9504038. Phys.Rev.Lett. 77 (1996) 1413–1415, [101] F. Markopoulou, “Quantum causal arXiv:quant-ph/9604005. histories,” [89] R. Webster, Convexity. Oxford University Class.Quant.Grav. 17 (2000) 2059, Press, Oxford, 1994. arXiv:hep-th/9904009. [90] S. Aaronson, “Is quantum mechanics an [102] F. Markopoulou, “Planck scale models of island in theoryspace?,” the universe,” arXiv:gr-qc/0210086. arXiv:quant-ph/0401062. [103] F. Markopoulou, “New directions in [91] C. Brukner and A. Zeilinger, “Quantum background independent quantum gravity,” measurement and shannon information, a in Approaches to Quantum Gravity: reply to m. j. w. hall,” Toward a New Understanding of Space, arXiv:quant-ph/0008091. Time and Matter, ed. D. Oriti (2009), Cambridge University Press, 129 [92] C. Brukner and A. Zeilinger, “Conceptual arXiv:gr-qc/0703097. inadequacy of the shannon information in quantum measurements,” [104] L. F. Hackl and Y. Neiman, “Horizon Phys.Rev.A 63 (2001) 022113, complementarity in elliptic de Sitter space,” arXiv:quant-ph/0006087. Phys. Rev. D 91 (2015), 044016, arXiv:1409.6753. [93] A. J. P. Garner, M. P. Müller, and O. C. O. Dahlsten, “The quantum bit from relativity of simultaneity on an interferometer,” Proc. R. Soc. A473 (2017) 20170596, arXiv:1412.7112. [94] P. A. Höhn, “Reflections on the information paradigm in quantum and gravitational physics,” J. Phys. Conf. Ser. 880, 012014 (2017), arXiv:1706.06882. [95] J. Hartle and S. Hawking, “Wave function of the universe,” Phys.Rev. D28 (1983) 2960–2975. [96] M. Bojowald, “Quantum cosmology,” Lect.Notes Phys. 835 (2011) 1–308. [97] A. Ashtekar and P. Singh, “Loop Quantum Cosmology: A status report,” Class.Quant.Grav. 28 (2011) 213001, arXiv:1108.0893. [98] M. L. Dalla Chiara, “Logical self reference, set theoretical paradoxes and the measurement problem in quantum mechanics,” J. Philosophical Logic 6, (1977) 331. [99] T. Breuer, “The impossibility of accurate state self-measurements,” Phil. Science (1995) 197. [100] L. Crane, “Clock and category: Is quantum gravity algebraic?,”

Accepted in Quantum 2017-11-27, click title to verify 78