Noncontextuality in multipartite entanglement

We discuss several multiport interferometric preparation and measurement configurations and show that they are noncontextual. Generalizations to the n particle case are discussed. PACS numbers: 03.67.Mn,42.50.St Submitted to: J. Phys. A: Math. Gen. ‡ email: [email protected], homepage: http://tph.tuwien.ac.at/ svozil Noncontextuality in multipartite entanglement 2 1. Contextuality in universal quantum networks In addition to recent techniques to prepare engineered entangled states in any arbitrarydimensional Hilbert space [1, 2, 3, 4], multiport interferometers could provide feasible quantum channels for physical questions requiring the utilization of higher than twodimensional states. In what follows, multiport interferometry will be mainly proposed for experimental tests of issues related to proof-of-principle demonstrations of quantum (non)contextuality; in particular to study properties of systems of observables corresponding to interlinked arrangements of tripods in three-dimensional Hilbert space, or interlinked orthogonal bases in higher dimensions. Contextuality [5, 6, 7] has been introduced by Bohr [8] and Bell (Ref. [5], Sec. 5) as the presumption § that the “. . . result of an observation may reasonably depend not only on the state of the system . . . but also on the complete disposition of the apparatus.” That is, the outcome of the measurement of an observable A might depend on which other observables from systems of maximal observables (Ref. [9], p. 173 and Ref. [10], Sec. 84) are measured alongside with A. The simplest such configuration corresponds to an arrangement of five observables A,B,C,D,K with two comeasurable, mutually commuting, systems of operators {A,B,C} and {A,D,K} called contexts, which are interconnected by A. A will be called a link observable. This propositional structure can be represented in three-dimensional Hilbert space by two tripods with a single common leg. The multiport interferometers for the preparation of quantum states and detection schemata corresponding to this configuration are enumerated explicitly in Section 3. Recently, Spekkens has proposed an operational definition of contextuality which generalizes the standard notion based on the quantum contextuality of sharp measurements [11]. Proofs of the Kochen-Specker theorem [12, 13, 14, 15, 5, 16, 17, 18, 19, 20, 21, 22, 23] utilize properly chosen finite systems of interlinked contexts; every single context corresponding to a system of maximal comeasurable observables. The systems of contexts are chosen for the purpose of showing that there does not exist any consistent possibility to ascribe global truth values by considering all conceivable truth values assignable to the individual contexts—the whole cannot be composed of its parts by adhering to the classical rules, such as the independence of truth values of identical propositions occurring in different parts. One way to consistently maintain interlinked contexts is to give up noncontextuality; i.e., to drop the assertion that the outcome of measurements of (link) observables are independent on the context and are not affected by which other observables are measured concurrently ‖. In that way, contextuality is introduced as a way to maintain value definiteness for each one of the § compare Bohr’s remarks in Ref. [8] about “the impossibility of any sharp separation between the behaviour of atomic objects and the interaction with the measuring instruments which serve to define the conditions under which the phenomena appear.” ‖ Other schemata to avoid the Kochen-Specker theorem such as Meyer’s [24] restrict the observables such that the construction of inconsistent schemata of interlinked contexts is no more possible. Still other schemata [25] deny the existence of even this restricted set of contexts by maintaining that an n-ary quantum system is only capable of storing exactly one nit of quantum information. Thereby only a single context appears relevant; e.g., the context associated with the particular basis of n-dimensional Hilbert space in which this nit is encoded. Noncontextuality in multipartite entanglement 3 individual contexts alone. Indeed, if contextuality is a physically meaningful principle for the finite systems of observables employed in proofs of the Kochen-Specker theorem, then it is interesting to understand why contextuality should not already be detectable in the simplest system of observables {A,B,C} and {A,D,K} representable by two interlinked tripods as discussed above. Furthermore, in extension of the two-context configuration, also systems of three interlinked contexts such as {A,B,C}, {A,D,K} and {K,L,M} interconnected at A and K ¶ will be discussed in Section 4. In what follows, the schema of the proposed experiment will be briefly outlined; for more details, the reader is referred to Refs. [26, 27]. Any unitary operator in finite dimensional Hilbert space can be composed from a succession of two-parameter unitary transformations in two-dimensional subspaces and a multiplication of a single diagonal matrix with elements of modulus 1 in an algorithmic, constructive and tractable manner. The method is similar to Gaussian elimination and facilitates the parameterization of elements of the unitary group in arbitrary dimensions (e.g., Ref. [28], Chapter 2). Reck, Zeilinger, Bernstein and Bertani have suggested to implement these group theoretic results by realizing interferometric analogues of any discrete unitary and hermitean operators in a unified and experimentally feasible way [26, 29]. Early on, one of the goals was to achieve experimentally realizable multiport analogues of multipartite correlation experiments; in particular for particle states in dimensions higher than two. The multiport analogues of many such experiments with higher than two-particle two-dimensional entangled states have been discussed by Zukowski, Zeilinger and Horne [27]. The multiport analogues of multipartite configurations are serial compositions of a preparation and an analyzing multiport interferometer operating with single particles at a time. In the preparation phase, a particle enters a multiport interferometer; its wave function undergoing a unitary transformation which generates the state required for a successive measurement. In a second phase, this state is the input of another multiport interferometer which corresponds to the self-adjoint transformation corresponding to the observables. If those observables correspond to multipartite joint measurements, then the output ports represent analogues of joint particle properties. The observables of multiport interferometers are physical properties related to single particles passing through the output ports. Particle detectors behind such output ports, one detector per output port, register the event of a particle passing through the detector. The observations indicating that the particle has passed through a particular output port are clicks in the detector associated with that port. In such a framework, the spatial locatedness and apartness of the analogous multipartite configuration is not preserved, as single particle events correspond to multipartite measurements. Rather, the emphasis lies on issues such as value definiteness of conceivable physical properties and on contextuality, as discussed above. There are many forms of suitable two-parameter unitary transformations corresponding to generalized two-dimensional “beam splitters” capable of being the factors of higher ¶ Too tightly interconnected systems such as {A,B,C}, {A,D,K} and {K,L,C} have no representation as operators in Hilbert space. Noncontextuality in multipartite entanglement 4 than two-dimensional unitary transformations (operating in the respective two-dimensional subspaces). The following considerations are based on the two-dimensional matrix T(ω,φ) = ( sinω cosω e−iφ cosω −e−iφ sinω )