We propose three criteria for the generation of random digital strings from quantum beam splitters: (i) three or more mutually exclusive outcomes corresponding to the invocation of three-and higher dimensional Hilbert spaces; (ii) the mandatory use of pure states in conjugated bases for preparation and detection; and (iii) the use of entangled singlet (unique) states for elimination of bias. From the numerous random number generators based on physical processes (cf. Refs. [7] [8] [9] [10] [11] [12] to name but a few), the use of single photons (or other quanta such as neutrons) subjected to beam splitters appears particularly promising [13] [14] [15] [16] for the following reasons: (i) due to (ideally) single-photon events, the physical systems are "elementary;" (ii) they can be controlled to a great degree; and (iii) they can be certified to be random relative to the postulates of quantum theory [17] . Three features of quantum theory directly relate to random sequences generated from beam splitter experiments: (i) the randomness of individual events (cf. Ref. [18, p. 866] and Ref. [19, p. 804]); (ii) complementarity [20, p. 7]; and (iii) value indefiniteness; i.e., the absence of twovalued states interpretable as "global" (i.e., valid on all observables) truth functions [21] . In order to fully implement these quantum features, we propose three improvements to existing protocols [13] [14] [15] [16] [22] [23] [24] . The first criterion ensures that the quantum random number generators can be certified to be subjected to quantum value indefiniteness. A necessary condition for this to apply is the possibility of three or more mutually exclusive outcomes in measurements of single quanta. Formally, this is due to the fact that violations of Bell-type inequalities, as well as proofs of Gleason's and KochenSpecker-type theorems are only realizable [25] in three-and higher dimensional Hilbert spaces. Only from three-dimensional vector space onwards it is possible to nontrivially interconnect bases 2 through one (or up to n − 2 for n-dimensional Hilbert space) common base element(s). This can be explicitly demonstrated by certain, even dense [26] [27] [28] , "dilutions" of bases, which break up the possibility to interconnect, thus allowing value definiteness. In more operational terms, if some "exotic" scenarios (e.g., Ref. [29, 30] ) are excluded, the violation of Bell-type inequalities for two two-state particles (corresponding to two outcomes on each side) is a sufficient criterion for quantum value indefiniteness. Of course, one could argue that protocols based on two outcomes are still protected by quantum complementarity, and the full range of quantum indeterminism, in particular quantum value indefiniteness, is not needed. There is also the possibility that the Born rule might be derived through some other argument (possibly from another set of axioms) than Gleason's theorem [31] [32] [33] [34] . However, there exist sufficiently many two-valued states on propositional structures with two outcomes to allow for a homeomorphic embedding of this structure into a classical Boolean algebra. In any case, it appears prudent to use all the "mind-boggling" features of quantum mechanics against cryptananalytic attacks on some quantum-generated sequence. The resulting trivalent or multi-valued sequence can be easily "downgraded" or "translated" to binary sequences through elimination or identification without loss of randomness: systematically eliminating n − 2 symbol(s) will transform a random sequence on an alphabet with n ≥ 3 symbols into a random sequence on an alphabet with two symbols [1]. The second criterion proposes the mandatory use of pure states from maximally conjugated bases for preparation and detection. This requirement deals with the single particle source of quantum random number generators. Indeed, many two-particle experiments have been using this criterion already, as full tomography is performed to characterize the state as completely as possible. These experiments use a (Bell) state which is as pure and maximally entangled as operationally feasible; quite often they produce the singlet Bell state (which, due to technical issues related to other degrees of freedom, can never be ideally pure). Tomography is used to characterize the state and hence certify the randomness of outcomes. Hence in this sense and in these experiments, the criterion is already implicitly implemented. Although it is generally believed that mixed (nonpure) quantum states can be "produced" and operationalized "for all practical purposes," one might cautiously argue that this may actually be a subjective statement on behalf of the observer: whereas the experimenter might "pretend" that the exact state leaving the particle source is unknown, it might still be possible to conceive of the state to be in some, albeit unknown but not principally unknowable, unique pure state. Even if one is unwilling to accept these principal concerns, it remains prudent not to expose the protocols for generating quantum randomness to the possibility of hidden regularities of the source. After all, beam splitters are just one-to-one bijective devices representable by reversible unitary operators [36] [37] [38] ; a fact which can be seen by recombining the two paths by a second beam splitter in a Mach-Zender interferometer, thereby recovering the original signal. Thus, in order to assure quantum randomness, the beam splitter should not be considered as an isolated element, but has to be examined in combination with the source. In accordance with this principle, a mismatch between state preparation and measurement guarantees that quantum complementarity ensures the indeterministic outcome. This can, for instance, be implemented by preparing the single particle in a pure state which corresponds to an element of a certain basis, and then measuring it in a different basis, in which the original state is in a coherent superposition of more than one states (cf. Ref. [13] and the first protocol using beam splitting polarizers in Ref. [15] ). Third and finally, in order to eliminate any possible bias (for some "classical" methods to eliminate bias, we refer to Refs. [39] [40] [41] [42] ), we propose to utilize Einstein-Podolsky-Rosen type measurements of two quanta in a unique entangled state. Any state satisfying the uniqueness property [43] in at least two directions, such as the singlet states 1 √ 2
In REF , Svozil proposed three criteria for building quantum random number generators based on beam splitters: (A) Have three or more mutually exclusive outcomes correspond to the invocation of Hilbert spaces with dimension at least 3; (B) Use pure states in conjugated bases for preparation and detection; (C) Use entangled singlet (unique) states to eliminate bias.
Three criteria for quantum random number generators based on beam splitters
Phys. Rev. A 79, 054306 (2009)
Physics