arXiv:1410.1707v2 [quant-ph] 15 Dec 2014 nadto ou rdw ursas n rmr strange more or one ( also parity quarks the violate down containin decays or Hyperon baryons quarks. up are to that addition hyperons in t via i.e. decaying interaction, and spin weak fo half-integer the having paper particles This of on problem universe. cus unsolved our in the asymmetry to quan- matter-antimatter itself of property relates counterintuitive mechanics most tum the of one Herewith hoe a erltdt h ilto fthe of violation the to related be B interaction, can weak which theorem the [1], via K-mesons decaying for particles that spinless shown are been has it Recently, try. eato steol neato htbek h aiysym parity charge-conjugation–parity( combined the in the breaks and weak that try The interaction only universe. the our is rules teraction that think we that teractions a,w n htthe part that In find we literature. lar, the clea in and existing unified theory a misunderstandings information into some quantum in China) to FAIR in it at II connects BEPC [2] picture, at experiment [3] PANDA BES-III the ex- and new by Germany the puts (e.g. also data which pected hyperons of process decay weak tnadMdlo lmnayprilspeit lotn c of tiny also tribution predicts particles elementary of Model Standard fteqatmter a ervae nhyperon-antihypero in revealed be can theory we B quantum pro least the via contextuality of the not theories whether realistic but and possible local Last is versus inequalities test freedom. a of whether degrees discuss spin entangl the the witnesses in W that ment system. observable the optimal in entanglement an is introduce there experimen that where suggests case data theoretic two-particle tal the information to quantum proceed we this decay insight with the by Equipped measured imperfectly process. a only is is process particle di decay decaying or the channel that show quantum further noisy can We complementar Bohr’s relation. interferometer of ity asymmetric discussion quantified an a of for arms allowing two the to correspond the ekitrcin r n u ftefu udmna in- fundamental four the of out one are interactions Weak edvlpa nomto hoei ecito fthe of description theoretic information an develop We C P ymtyhsbe pt o xeietlyfound. experimentally now to up been has symmetry C rfrnet aiycnevn rvoaigpoess(e processes violating or corr conserving process parity decay to weak di preference the phase a fixed that and find We visibility fixed spin. with hyperons by ASnmes 36.a 42.n 03.65.Ud 14.20.Jn, 03.67.-a, numbers: PACS revealed. be can cann contextuality literature) circumstances in which i under measured misconception be common can to entanglement contradiction how show Stern-G we imperfect insight an theoretic with pha measurements the of spin part to real corresponds the times visibility the to proportional itiuino h oetmo h agtrprils efi c We are axes quantization particles. two daughter process: the the this carries of of that interpretation momentum channel the quantum of open an distribution as considered be can P eaayeteaheal iiso h unu information quantum the of limits achievable the analyze We iltn rcse,hwvr ovoainof violation no however, processes, violating ekItrcinPoess hc unu nomto sr is Information Quantum Which Processes: Interaction Weak P iltn n o-iltn amplitudes non-violating and violating 1 nvriyo ina aut fPyis otmngse5 Boltzmanngasse Physics, of Faculty Vienna, of University ff rnl ttd h pno the of spin the stated, erently P ff ymty The symmetry. ) rnefrec yeo.Ntr hoe ahrlwvisibili low rather chooses Nature hyperon. each for erence C P C P symmetry. symme- ) .C Hiesmayr C. B. perty ell’s me- icu- on- ell he e- rs g e n - - - - esit Di shift. se oe pnaeul ihprobabilities with spontaneously hosen h nefrnecnrs ftetoitreigamplitude visibility interfering the two predictability the as of considered contrast be interference the can term interference the ewe h rbblt htapril easvateproce the via decays particle a a that probability the between n,ovosy ehave we obviously, and, lmnaypoete,s h etrw nwoeo them, of one know we c better of the pair so a is properties, interf visibility plementary in and duality Predictability of devices. concept complementarity metric related Bohr’s closely of the rephrase or relation quantitative a is which hr ecniee o h ateuto h normalized the equation last the for quantities considered we where u n pndgeso freedoms momen- of between degrees state spin separable and a tum in is hyperon decaying a of system: caying systems. suigtodi two Assuming clyoei neetdi h nua itiuino h d the of distribution ( angular products the in interested is one ically eo atce ..apoeto noteaoestate above the onto projection a i.e. particle, peron where n nua oetmsae hc ednt by di denote respective we the which of states momentum superposition angular a ent be can di projector have the can state final the cp o h decay the for xcept and tb eeldi yeo eas esuyalso study We decays. hyperon in revealed be ot nomto hoei otn fa nefrn n de- and interfering an of content theoretic Information rahaprts qipdwt hsinformation this with Equipped apparatus. erlach hs ytm n h elsnnoaiy(in nonlocality Bell’s why and systems these n nomto ftehprnsi oteangular the to spin hyperon the of information dasml emtia nomto theoretic information geometrical simple a nd 1 b P V I I T ( ( rcsigo h ekitrcinrevealed interaction weak the of processing = sod oa nefrmti eiewt a with device interferometric an to esponds ,φ θ, ,φ θ, 2 ff =  suulydbe ea rtasto arx Often matrix. transition or decay dubbed usually is 1 rnl ttdtewa neato process interaction weak the stated erently + ,φ θ, T (

ˆ ) ) + k k P a T T k / = eedn nteplrzto tt ftehy- the of state polarization the on depending ) 2 T b = = 2 P a a k 00Ven,Austria Vienna, 1090 , k k a Tr 2 2 k − k T = = e ssatb suigta h nta state initial the that assuming by start us Let a edrvdb optn h di the computing by derived be can k ff Tr Tr + T a + spin Σ rn nsw obtain we ones erent k a 1 T + spin spin 2 k k k · k T a T T , + −→ / ( b b b b T Tr k k k k k T T 2 a 2 T T ) b p mom

+ b a ρ ff k π · , k b spin and rn nua oet,therefore momenta, angular erent 0 2 T k .Tedcyprocess decay The ). − Re b | Ψ 1 ) T h coe The . { ρ evealed? ih V Tr † spin ρ Ψ isexpressing ties 1 hyperon = spin ± 2 | α ρ ( 2 T where mom k T a ˆ T k a + = T a ⊗ ffi ρ k a T 2 spin ρ k · k α ρ in nfotof front in cient b mom + ) spin is † k T T ˆ T ⊗ b † b b } k ρ a k  2 , spin , b ff , , erence Typ- . c . . . , ecay .A s. ff om- ero- (2) (3) (1) V er- ss , . 2

trace operation

I(θ,φ) = Trspin (Ta + Tb) ρspin (Ta + Tb)†

Trspin(K+ρspinK+ + K ρspinK ) ≡ − − (a) where the Kraus operators have the conceptually simple form (ω > 0) ± #» #» #» #» #» #» K = √ω ω1 ω2 ω1 ω2 := √ω Π ω ω ± ± | ± ih ± | ± 1± 2 with ω + ω = 1. The two Blochvectors have to be orthogo- #» + #» nal, ω ω =− 0, since the transition is completely positive and 1 2 #» #» are chosen· such that they have maximal length ω ω 2 = | 1 ± 2| s(2s + 1) (s...spin number). A Blochvector expansion#» #» of a 1

density matrix is generally given by ρ = ½ + b Γ where d { d · } (b) d is the dimension of the system [14]. Since we are dealing with spin-degrees of freedom we have d = 2s + 1 and we FIG. 1: (Color online) (a) depicts an interferometer for a spin- 1 #» 2 can choose as a set of orthonormal basis the#» generalized Her- particle with spin oriented along s propagating through two beam- mitian and traceless Gell-Mann matrices Γ (for s = 1 they splitters (BS) and picking up a relative phase χ. In (b) the Bloch’s 2 correspond to the Pauli matrices). Given this structure we can sphere is depicted and shows how a given interferometric setup changes the initial spin. Depending on the initial spin the interfer- reinterpret the weak decay process as an incomplete spin mea- ometer reveals “wave-particle” property, quantified by the visibility, surement of the decaying particle or “which-way” information, quantified by the predictability. Both #» #» #» #» I(θ,φ) = ω+ Tr(Π ω + ω ρspin) + ω Tr(Π ω ω ρspin) quantities add up to one for pure spin states. 1 2 − 1− 2 1 # » # » #» = 1 + (ω1 + (ω+ ω ) ω2) s (4) (2s + 1){ − − · } #» #» the less we can determine the other one. One may relate pre- = where#» s is the Bloch vector representation of ρspin, i.e. s dictability to the “particle-like property”,i.e. a “which way”– Tr( Γ ρ ). With probability ω the spin state of the hyperon spin #» + #» information, and visibility to the “wave-like property”, i.e. the is projected onto direction ω1 + ω2 or with the remaining interference contrast or sharpness. The first step in bringing probability ω the initial spin state is measured along the di- #» −#» the qualitative statement “the observation of an interference rection ω1 ω2. Thus the weak process can be associated to pattern and the acquisition of which-way information are mu- a spin measurement− with an imperfect Stern-Gerlach appara- tually exclusive” into a quantitative one was taken by Green- tus (switching with probability ω the magnetic field) which bergerand Yasin [4] and refined by Englert[5]. The authorsof is geometrically depicted in Fig. 2.± The imperfection has two Ref. [6] investigated physical situations for which the comple- causes: Firstly, the difference (ω+ ω ) equals an asymme- mentary expressions depend only linearly on some variable y. try (denoted in the following by α)− which− corresponds to the This included interference patterns of various types of double characteristics of the interferometer. Secondly, the two direc- #» #» slit experiments (y is linked to position), but also oscillations tions ω1 ω2 are characteristical for the weak decay. Explicit due to particle mixing (y is linked to time), e.g. by the neu- examples± are given below. tral K-meson system, and also Mott scattering experiments of 1 Let us remark here that for a spin- 2 particle the decay dy- identical particles or nuclei (y is linked to a scattering angle). namics simplify considerably since the real and imaginary For the K-meson system the effect of the P symmetry vi- part of the transition amplitude T + T are equal, leading to C #» a b olation was investigated [7] showing that it shifts obtainable ω = 0. Thus we can say that Nature chooses between two 1 #» information about our reality to different aspects, without vi- opposite directions ω, albeit between two different handed olating the complementarity principle, i.e. from predictability coordinate systems or± chirality. Of course, we could also rein- to visibility and vice versa. All these two-state systems terpret equation (4) if and only if the initial spin state is pure P V belonging to distinct fields of physics can then be treated via stating that the spin direction is projected onto the mixed mo- the generalized complementarity relation in a unified way. In #» #» #» #» mentum state ω+Π ω 1+ ω 2 + ω Π ω 1 ω 2 . This interpretation Ref. [8] the authors investigated how the un-stability due toa does not hold if there is an initial− correlation− between spin decay within the interferometer reduces the visibility. and momentum degrees of freedom, e.g. for decay cascades A simple open quantum formalism for decaying hyper- (Example III). 1 ons: Quite generally, we can rewrite the dynamics in the spin Example I: A spin- 2 particle propagating through an 1 space by switching to the formalism of open quantum systems interferometer: Let us consider a spin- 2 particle that passes that allows a fast computation and transparent interpretation. a beam-splitter (BS). The interaction can be reasonably well i π σ In particular, we can always find two hermitian Kraus oper- described by the unitary operator UBS = e− 4 y . Then the i χ σ ators K being the sum of the real and imaginary part of the particle picks up usually a phase shift χ (Uphase(χ) = e− 2 x ) decay matrix± such that the following equation holds under the and passes again a beam-splitter. A pure initial spin state 3

With Eq. (2) we obtain a quantified quantum information the- oretic interpretation of Bohr’s complementarity relation of the interferometer in terms of 2+ 2 = 1, i.e. howmuch interfer- ence contrast correspondsP to aV certain interferometric choice. Note the conceptual difference to the previous scenario where referred to the chosen initial spin. We are therefore not V 1 limited to spin- 2 decays since we are interested in the two (a) (b) amplitude process of the weak interaction. Let us apply now this view to hyperon decays. FIG. 2: (Color online) (a) shows the three dimensional Bloch’s Example II: Spin- 1 hyperon nonleptonic decays: The sphere for spin- 1 particles and (b) the eight dimensional Bloch’s 2 2 conservation of the total spin implies that the final states can sphere for spin- 3 particles. Weak interaction depicts two quantiza- #»2 #» have two different angular momentum eigenstates. We denote tion directions ω ω along which the initial spin of the hyperon 1 2 #» with S/P the amplitude of the parity violating/conservating is projected. For spin-± 1 particles ω = 0 and with probability ω the 2 #» 1 #» hyperon spin is projected onto n where n corresponds to the± mo- process (corresponding to the S, P wave with angular mo- ± 3 mentum direction of the daughter particle. In case of spin- particles mentum l = 0, 1), then the decay matrices are Ta = S ½ 2 #» #» the real and imaginary parts of the transition amplitude differ result- and Tb = P n σ computed via the corresponding spherical ing in a contribution to the quantization directions that is independent harmonics and· Clebsch-Gordan coefficients [9]. The angular of difference of the two probabilities ω . Hence the hyperon spin is #» ± #» #» distribution of the normalized momentum n of the daugh- projected onto two slightly tilted quantization directions ω1 ω2. #» #» #» ± ter particle is given by Eq.(4) with ω1 = 0, ω2 = n and ω+ ω = α. Depending on the production process sym- − − #» #» 1 #» #» metries on s , n may be superimposed. In the standard phe- ρspin = (½ + s (θ,φ) σ) is then changed by a total unitary 2 · nomenology of hyperon decays the following decay parame- operation UIF = UBS Uphase(χ) UBS into a final state · · ters are introduced [16] ρ = U ρ U† f IF spin IF 2Re S P 2Im S P ∗ ∗ √ 2 s (θ,φ)cos χ + sy(θ,φ) sin χ α = { } , β = { } = 1 α sin φ, 1 − x #» S 2 + P 2 S 2 + P 2 − = ½ +  sx(θ,φ) sin χ + sy(θ,φ)cos χ  σ . | |2 | |2 | | | | 2{   } S P  sz(θ,φ)  γ = | | −| | = √1 α2 cos φ.  −  S 2 + P 2 − A measurement in x- or y-direction reveals the interference | | | | and the measured values of the parameters α, φ are given 1 σ I(θ,φ) = Tr( ± x ρ ) = 1/2(1 sin θ cos(φ + χ)) , in Ref. [15]. We can connect these parameters to our in- 2 f ∓ · formation theoretic quantities = α2 + β2, = γ and (5) V P | | α = cos χ . Time reversal symmetryp requires, in the ab- V SP 2 2 sence of final-state interactions, that the S - and P-wave am- and therefore the visibility is in this case = sx + sy = V q plitudes should be relatively real, hence φ 0, which means sin θ. Whereas a measurement in z-direction reveals the prob- ≈ ability to propagate via the upper or lower path of the interfer- that the asymmetry parameter α approximately equals the vis- ibility of the S and P wave. ometer V 1 σ I(θ,φ) = Tr( ± z ρ ) = 1/2(1 cos θ) , (6) 2 f ∓ and therefore the “which way” information is quantified by In the table I we list all hyperon decays for which α and the phase φ have been measured. Let us remark here that if φ the predictability = sz = cos θ, i.e. the initial spin contribution in z-direction.P | − Thus| , can be chosen via the is not measured we cannot say anything about the visibility preparation of the initial spin (givenV theP interferometric setup). since it depends strongly on φ except for α close to 1. We In the following we go one step further and assume that one observe that the phase shifts χSP, which reduce the total visi- has an interferometric device including a measurement along bility of the decay, are rather small. For some decays, where x- or y-direction, albeit revealing the “wave particle” infor- the phase shifts are known with limited accuracy, a larger val- ues of the phase shifts (up to 0.5π) are still possible. The mation of the spin, however, we now change the probability ≈ of the beam-splitter to a T 2 : T 2 = S 2 : P 2 one and predictability of all weak decay processes is rather large ex- a b + find (see Eq. (1)) k k k k | | | | cept for the Σ particle. This particle has –in contrary to the other hyperons–two decay channels with nearly equal branch- 2 2 I(θ,φ) = ( S + P ) ing ratios. Due to the rule for the hyper charge Y (∆Y = 1 ) | | | | · 2 2 S P #» #» and the spurious-kaon rule [16] identical α’s and φ’s are ex- 1 | || | n (0, χSP) s (θ,φ) (7) 0 ∓ S 2 + P 2 · ! pected for the following two pairs of decays: Λ π− p; π n | | | | and Ξ ,0 Λπ ,0. That is significantly different−→ to the case by choosing the following T matrices − − of the Σ −→decays where the visibility is close to zero. This #» #» ±

T = T ½ and T = T U† n (0, χ ) σ U . implies that one of these decays must be mainly a S -wave and a k ak b k bk IF SP · IF 4

hyperon (quarks) decay channel (frequency) phase shift χ in [π] visibility predictability S P V P Λ(uds) pπ− (63.9%) (0.043 0.023) 0.648 0.014 0.762 0.012 0 − ± ± ± nπ (35.8%) (0.042 0.023) 0.656 0.040∗ 0.755 0.034∗ + − ± ± ± Λ¯ (¯ud¯s¯) p¯π (63.9%) 0.036 0.021 0.714 0.079∗ 0.700 0.080∗ ± ± ± Σ−(dds) nπ− (99.8%) (0.38 0.16) 0.19 0.24 0.98 0.05 − ± ± ± Σ+(uus) pπ0 (51.6%) (0.038 0.035) 0.976 0.016 0.161 0.097 − ± ± ± nπ+ (48.3%) 0.41 0.13 0.24 0.33 0.972 0.078 ± ± ± Ξ0(uss) Λπ0 (99.5%) 0.214 0.085 0.53 0.11 0.85 0.07 ± ± ± Ξ−(dss) Λπ− (99.8%) 0.0226 0.0086 0.459 0.012 0.8884 0.0062 ± ± ± TABLE I: The information theoretic content of the weak decay processes violating the strangeness number. The values and errors are taken from the particle data book [15]. The asterisk ∗ denotes that the phase φ was not independently measured but deduced via conservation laws ¯ 0 3 from the other decay modes. Λ also decays into nπ , however, it has not yet been measured. The only non-resonant hyperons with spin- 2 are the Ω± particles, the measured asymmetry are close to zero (αΩ = 0.0180 0.0024) and no φ information is available. ±

#» #» the other one mainly a P-wave. This means also that the im- The angle between the momenta of the two hyperons n µ n ν perfection of the spin measurement is low in strong contrast is fixed by the request to orientate the two reference systems· to Ξ− decays where the probability of both directions is close with respect to the initial hyperon spin. We obtain a concep- to 1 . tually simple interpretation of the decay cascade as in Exam- 2 #» In summary, the weak decay process of hyperons with spin ple II: the two quantization directions are τ and the mo- ± can be viewed as an interferometer device with a fixed phase mentum direction#» of the baryon is measured with probabil- 1 τ #» and a beam splitter with a fixed splitting. This corresponds ity (1 | | ) along them. Notable that τ has a contribu- 2 ± τ0 to fixed visibilities and predictabilities independently of the tion (1 µ) along the direction of the momentum of the − P #» initial spin of the decaying hyperon. daughter hyperon n , whereas the direction of the momen- #»µ Example III: Cascade of hyperon decays: Considering tum of the baryon n ν is multiplied by the predictability µ. P subsequent decays of hyperons into hyperons and finally into For example the predictabilities of the two decay cascades ,0 ,0 ,0 ,0 0 non-strange particles we can apply the transition matrices Ξ− Λπ− pπ− π−, nπ− π are large µ 0.8 and therefore−→ the quantization−→ axis becomes to a goodP approxima≈ - Tµ, Tν in a straightforward way by #» #» tion αµ n µ + αν n ν. µ ≈ I(θν,φν; θµ,φµ) = Tr Tν Tµ ρ T † T † Entanglement: Entangled ΛΛ¯ pairs can be produced for · spin µ · ν  µ  µ example via proton-antiproton annihilation and the full spin Tr K ρ K† + Tr K ρ K† . + spin + spin structure can be experimentally determined [17]. In this ≡    − − The last equation holds since we have again a complete posi- section we analyse the entanglement properties and discuss tive evolution. Let us remark here that these Kraus operators whether Bell’s nonlocality or contextuality can be revealed in K are not a product of the Kraus operators of each decay. such experiments. Let us assume that (i) there is no initial The± reason is that after the first decay we have correlations correlation between the momentum degrees of freedom and between the spin and momentum degrees of freedom, i.e. the the spin degrees of freedom and (ii) there is no entanglement between the momentum degrees of freedom. Our formalism crucial initial condition ρmom ρspin does not hold. Contrary to the dynamics of closed systems⊗ open quantum systems have extends straightforwardly to the two-particle case by taking no continuity of time and therefore it is not always possible to the tensor product of the Kraus operators. Experiments [17– formulate the general dynamics by means of differential equa- 19] suggest that the initial spin state is a maximally entangled tions generating contractive families. On the contrary cas- Bell state (except for backward scattering angles). Therefore cades allow to reveal also more information on the S -wave without loss of generality we can choose the antisymmetric Bell state ψ = 1 and the angular momentum and P-wave coefficients, i.e. after a straightforward computa- | −i √2 {| ⇑⇓i−| ⇓⇑i} 1 distribution becomes tion we find for the decay of a hyperon µ (spin- 2 ) into a hy- 1 1 peron ν (spin- ) and finally into a baryon (spin- ) the follow- 2 2 2 2 #» 2 ( S + P ) #» #» 1 τ #» #» #» I(θ ,φ ; θ ,φ ) = | | | | 1 αΛα ¯ n n . ing Kraus coefficients ω = (1 | | ) and ω1 = 0, ω2 = τ 1 1 2 2 Λ 1 2 ± 2 ± τ0 4 − · with  (8) 2 2 2 2 #» #» #» #» τ0 = ( S µ + Pµ )( S ν + Pν )(1 + αµαν( n µ n ν)) Since n 1 n 2 is multiplied by a constant αΛαΛ¯ T¨ornquist [22] #» | |2 | |2 | |2 | |2 · · τ = ( S µ + Pµ )( S ν + Pν ) concluded that Λ decays “as if it had a polarization αΛ tagged | | | | | | #» | #»| #» in the direction of the π+ (coming from the Λ¯ ) and vice versa”. (αµ + αν(1 µ) n µ n ν) n µ −P · The knowledgeof how one of the Λ s decayed –or shall decay  #» #» #» ′ +αν µ n ν + ανβµ n µ n ν . (since time ordering is not relevant)– reveals the polarization P ×  5 of the second Λ. He concludes that this is the well-known Einstein-Podolsky-Rosen scenario. In general entanglement is detected by a certain observable that can witness the entanglement content, i.e. a Hermitian operator for which holds Tr( ρ) < 0 for at least one state W W ρ and Tr( ρsep) 0 for all separable states ρsep. For the antisymmetricW Bell≥ state such an optical entanglement witness

1 ½ is given by = (½ + σ σ ) (any other witness can W 3 ⊗ i i ⊗ i (a) (b) be obtained by local unitaryP transformations). Since the weak interaction only allows for an imperfect spin measurement we FIG. 3: (Color online) The big (green) tetrahedron corresponds to the geometrical illustration of all local maximally mixed bipartite qubit have to multiply the spin part by αΛαΛ¯ . Thus Tr( αρΛΛ¯ ) results in W states which can be represented by three real numbers. The four cor- ners represent the four maximally entangled Bell states which are 1 the only pure states in the picture. Separable states are those inside αΛα ¯ 0 ρ , (9) 3 − Λ ≥ ∀ sep the (blue) double pyramid where the surfaces correspond to optimal entanglement witnesses. For the ΛΛ¯ system we can interpret that which is clearly violated since αΛα ¯ = 0.46 0.06 [15]. (a) the Hilbert space shrinks with the factor αΛαΛ¯ visualized by the Λ ± Therefore, the measurement of the correlation functions σi small (red) tetrahedron. Since the corners are still outside of the opti- h ⊗ mal entanglement witnesses (double pyramid), entanglement can be σi in x, x and y,y and z, z directions of the Λ and Λ¯ reveals en- detected via the weak decaying process. Or equivalently, as illus- tanglement.i Generally, we can say that the asymmetries lead trated in (b) we could also say that weak interactions correspond to to imperfect spin measurements which shrink the observable imperfect spin measurements blowing up the optimal entanglement space. Equivalently, we can say that the given interferometric witnesses, huge (red) double pyramid. device leads to a shrinking of the Hilbert space of the acces- sible spin states. In Fig. 3 we have visualized the geometry of the Hilbert space for all locally mixed states of bipartite In the following we analyze these inequalities for the hyperon- qubits forming a magic simplex [11–13]. Locally mixed states antihyperon system. are those for which any partial trace reduces to the maximally We considered Bell inequalities for qubits with two, three mixed state. This set of states can be described by three real or four different choices of observables for both particles, i.e. numbers. Positivity requires that these three numbers form the following expressions a tetrahedron with the four maximally entangled Bell states #» #» #» #» #» #» I2 = Prob( a 1, b 1) + Prob( a 1, b 2) + Prob( a 2, b 1) in the corners. The separability condition corresponds to a #» #» #» #» double pyramid. The surfaces of the pyramid correspond to Prob( a 2, b 2) Prob( a 1) Prob( b 1) , − #» #» − #» #»− #» #» the optimal entanglement witnesses. As shown in Fig. 3 the I3 = Prob( a 1, b 1) + Prob( a 1, b 2) + Prob( a 1, b 3) #» #» #» #» #» #» factor αΛαΛ¯ (a) shrinks the total state space (smaller red tetra- +Prob( a 2, b 1) + Prob( a 2, b 2) Prob( a 2, b 3) hedron) or (b) blows up the optimal entanglement witnesses. #» #» #» #» − #» 1 +Prob( a 3, b 1) Prob( a 3, b 2) Prob( a 1) Since αΛαΛ¯ > 3 one can distinguish between entangled and #» − #» − separable states directly, i.e. without the additional informa- 2Prob( b 1) Prob( b 2) , − − tion coming from the two interferometer devices. #» #» #» #» #» #» Testing for local realism or contextuality: The next ques- I = Prob( a , b ) + Prob( a , b ) + Prob( a , b ) 4 1 #»1 1 #»2 1 #»3 tion is whether the entanglement observed in this system man- #» #» #» +Prob( a 1, b 4) + Prob( a 2, b 1) + Prob( a 2, b 2) ifests itself in the most counterintuitive properties of quantum #» #» #» #» #» #» +Prob( a 2, b 3) Prob( a 2, b 4) + Prob( a 3, b 1) theory, i.e. Bell’s nonlocality or contextuality. The notion #» #» − #» #» #» #» of contextuality, introduced by John Bell [24] and by Kochen +Prob( a 3, b 2) Prob( a 3, b 3) + Prob( a 4, b 1) #» #» − #» #» and Specker [25], can be explained as follows. Suppose that Prob( a , b ) Prob( a ) 3Prob( b ) − #»4 2 − #» 1 − 1 a measurement A can be jointly performed with either mea- 2Prob( b ) Prob( b ) , − 2 − 3 surement B or C, i.e. without disturbing the measurement A. #» #» where Prob( a , b ) is the probability that the first particle is Measurements B and C are said to provide a context for the i #» j measurement A. The measurement A is contextual if its out- measured along a i giving e.g. the plus result#» and the second come depends on whether it was performed together with B or particle is measured along the direction b j and gives e.g. the with C. Therefore, the essence of contextuality is the lack of plus result. For any local realistic theories Ii 0 has to hold. ≤ possibility to assign an outcome to A prior to its measurement We find that the strongest constraint is found for the famous and independently of the context in which it was performed. CHSH-Bell inequality [20] (I2), i.e. The seemingly different Bell theorem is in fact a special in- 1 stance of the Kochen-Specker theorem where contexts natu- αΛαΛ¯ (0.46 0.06) (10) ≈ ± ≤ √ rally arise from the spatial separation of measurements. The 2 usual toolboxes for revealing nonlocality and contextuality are that has to hold for all local realistic theories. Clearly, this state in-dependent or dependent inequalities of probabilities. reveals no violation for the ΛΛ¯ system. However, we have 6 to remark that for testing theories based on local hidden pa- for each hyperon an interferometric device with a fixed visi- rameters versus the predictions of quantum mechanics there bility, i.e. interference contrast, and a fixed phase shift. The are two important requirements: Firstly, one is not allowed visibility results from a non-symmetric beam-splitter. Based to refer to results that are deduced from quantum mechani- on Bohr’s complementarity relation we can derive the related cal considerations. In our case this means that we cannot re- predictability, quantifying the “particle-like” property, which normalize to the asymmetry parameters αΛ,Λ¯ since they are turns out to be in general high (except for one decay mode of obtained from quantum mechanical properties. Secondly, any the Σ+ hyperon). Thus weak interaction distinguishes strongly conclusive Bell’s test requires that an active control of the ex- between processes conserving and not conserving the parity perimenter about measurement setting is given, i.e. a choice symmetry. of freedom which property of the state will be measured. Oth- Applying the open quantum formalism we find a simple erwise it is straightforward to construct a local realistic hid- and transparent method to describe weakly decaying hyper- den variable theory to explain the data. This active control ons. We find that the decay via the weak interaction corre- is clearly not available for weakly decaying hyperons since sponds to an imperfect spin measurement, where right and the quantization axes are spontaneously chosen. Some au- left-handed coordinate systems mix. thors [23] argued that one can circumvent the requirement Equipped with this quantum information theoretic knowl- since an observer can choose a coordinate system at will at edge we proceeded to two-particle system, e.g. the ΛΛ¯ sys- each side. We disagree since one needs always one common tem. Entanglement can be proven experimentally although reference system to describe the two-particle decay. Note that the imperfection in the spin measurements. We show further if there would be entanglement in the momentum state then that it is impossible to find a conclusive contradiction between this entanglement can (in some cases) be transferred to the quantum theory and local realistic theories via Bell inequali- spin states for a relativistically boosted observer [21]. How- ties since weak decays offer no active control over the quan- ever, in this case also the operators corresponding to the mea- tization directions. Last but not least we investigate contextu- surements are boosted in exactly such a way that the expecta- ality in the ΛΛ¯ system. We find that it cannot be revealed tion value is independent of the considered reference frame in by the Mermin-Peres square if one assumes that the viola- agreement with special relativity. Hence a violation of a Bell tion of the parity symmetry shrinks the accessible observable inequality found in one reference system is also violated in space. The second possible interpretation —violation of the another one. In summary, in contrary to the weakly decaying parity symmetry corresponds to the shrinking of accessible spinless K-meson system [1] the active measurement proce- state space— would reveal the contextual nature of quantum dure for hyperons decays are not available and therefore no theory, however, this interpretation does not hold if there are contradiction to the premisses of local realism based on Bell correlations between the spin and momentum degrees of free- inequalities can be derived. dom. In opposition two the weakly decaying K-meson system Contextuality tests usually do not suffer from these require- the observation of the violation of the symmetry would ments. Due to the imperfect spin measurements we expect for not considerably alter the results. CP the hyperonsa decrease of probabilities and thereforea depen- We believe that the presented information theoretic ana- dence of the violation on the asymmetry term. The tests need lyzes of hyperon decays and entanglement helps to bring the the property of joint measurements that is available for the data from the upcoming experiments in a unified picture and hyperon system since we have the tensor product between the contributes to the understanding of weak interactions. observables. If one follows the interpretation that the accessi- ble state space is shrunken (that holds true only if there is no Acknowledgements: The author is grateful to Andzej ¯ initial correlation between the momentum degrees of freedom Kupsc (Uppsala University) for introducing her to the ΛΛ sys- and the spin degrees of freedom), then all state independent tem and many fruitful discussions and comments to the pa- proofs such as the Mermin-Peres square [26–28] hold also for per. She also thanks Reinhold A. Bertlmann (University of the ΛΛ¯ system and non-contextuality is revealed. Applying Vienna) for discussions and gratefully acknowledges the Aus- the other interpretation that holds also in the general case (see trian Science Fund (FWF-P26783). example III) the observables have to be multiplied by αΛ or αΛ¯ , respectively, and the Mermin-Peres inequality leads to 2 2 2 3 3 Icontextuality = (αΛ + α ¯ ) + 2 αΛα ¯ 4 . (11) Λ Λ ≤ [1] B.C. Hiesmayr, A. Di Domenico, C. Curceanu, A. Gabriel, M. which is not violated. Assuming conservation contextual- Huber, J.-Å. Larsson, and P. Moskal, Eur. Phys. J. C 72, 1856 ity would be revealed if α > 0.88CP and hence greater than the (2012). bound from the Bell inequality α> 0.84. [2] M.F.M. Lutz et al. [PANDA Collaboration], arXiv:0903.3905 Conclusion and Summary: In this contribution we intro- [hep-ex]. [3] D.M. Asner, T. Barnes, J.M. Bian, I.I. Bigi, N. Brambilla, duce an information theoretic approach to hyperons decay- I.R. Boyko, V. Bytev and K.T. Chao et al., Int. J. Mod. Phys. ing via the weak interaction. The parity violating and non- A 24, S1 (2009). violating processes can be considered as two different paths [4] D. Greenberger and A. Yasin, Phys. Lett. A 128, 391 (1988). in an interferometer. We find that weak interaction chooses [5] B.-G. Englert, Rev. Lett. 77, 2154 (1996). 7

[6] A. Bramon, G. Garbarino, B.C. Hiesmayr, Phys. Rev. A 69, 212302 (2002). 022112 (2004). [18] K. D. Paschke et al. (PS185 Collaboration), Phys. Rev. C 74, [7] B.C. Hiesmayr and M. Huber, Phys. Lett. A 372, 3608 (2008). 015206 (2006). [8] D.E. Krause, E. Fischbach and Z. J. Rohrbach, Physics Letters [19] T. Johansson, “Antihyperon-hyperon production in antiproton- A 378, 2490 (2014). protn collisions”, in AIP Conf. Proc. Eigth Int. Conf. on Low [9] Erik Thome, Multi-Strange and Charmed Antihyperon- Energy Antiproton Physics, 2003. Hyperon Physics for PANDA, PhD thesis, Acta Univeritatis Up- [20] J.F. Clauser, M.A. Horne, A. Shimony, R.A. Holt, Phys. Rev. saliensis. Uppsala Dissertations from the Facutly of Science and Lett. 23, 880 (1969). Tehcnology, 2012. [21] N. Friis, R. A. Bertlmann, M. Huber and B. C. Hiesmayr, Phys. [10] C.S. Wu et al., Phys. Rev. 105, 1413 (1957). Rev. A 81, 042114 (2010). [11] B. Baumgartner, B.C. Hiesmayr and H. Narnhofer, Phys. Rev. [22] N. A. T¨ornquitst, Foundations of Physics 11, 171 (1981). A 74, 032327 (2006). [23] S. Chen, Y. Nakaguchi and S. Komamiya, High Energy Physics [12] B. Baumgartner, B.C. Hiesmayr and H. Narnhofer, J. Phys. A: - Phenomenology Testing Bell's Inequality using Charmo- Math. Theor. 40, 7919 (2007). nium Decays, PTEP (Progress of Theoretical and Experimental [13] B. Baumgartner, B.C. Hiesmayr, H. Narnhofer Physics Letters Physics); e-print:arXiv:1302.6438 A, Vol. 372, Issue 13, 2190(2008). [24] J.S. Bell, Rev. Mod. Phys. 38, 447 (1966). [14] R.A. Bertlmann and Ph. Krammer, J. Phys. A: Math.Theor. 41, [25] S. Kochen and E.P. Specker, J. Math. Mech. 17, 59 (1967). 235303 (2008). [26] A. Peres, J. Phys. A: Math. Gen. 24, 175 (1991). [15] K.A. Olive et al. (Particle Data Group), Chin. Phys. C 38, [27] N.D. Mermin, Physics Today 43, 911 (1990). 090001 (2014). [28] J. Thompson, P. Kurzynski, S.-Y. Lee, A. Soeda and D. Kas- [16] H. Pilkuhn, “The Interactions of Hadrons”, North-Holland Pub- zlikowski, “Recent advances in contextuality tests”, e-print: lishing Company-Amsterdam (1967). arXiv:1304.1292. [17] B. Bassalleck et al. (PS185 Collaboration), Phys. Rev. Lett. 89,