Jan 30, 2014 - script r (by fixing r = 2) and the subscript k, thus re- stricting the analysis to the ..... [19] Barnum H. and Linden N., J. Phys. A: Math. Gen., 34.
A field-theoretical approach to entanglement in neutrino mixing and oscillations Massimo Blasone1,2 , Fabio Dell’Anno2,3 , Silvio De Siena2,3,4 and Fabrizio Illuminati2,3,4 1 2
arXiv:1401.7793v1 [quant-ph] 30 Jan 2014
3 4
Dipartimento di Fisica, Universit` a di Salerno, Via Giovanni Paolo II, 132 84084 Fisciano, Italy INFN Sezione di Napoli, Gruppo collegato di Salerno, Italy Dipartimento di Ingegneria Industriale, Universit` a di Salerno, Via Giovanni Paolo II, 132 84084 Fisciano, Italy CNISM Unit` a di Salerno, I-84084 Fisciano (SA), Italy
PACS PACS PACS
03.65.Ud – Entanglement and quantum nonlocality 14.60.Pq – Neutrino mass and mixing 03.70.+k – Theory of quantized fields
Abstract. - The phenomena of particle mixing and flavor oscillations in elementary particle physics can be addressed by the point of view of quantum information theory, and described in terms of multi-mode entanglement of single-particle states. In this paper we show that such a description can be extended to the domain of quantum field theory, where we uncover a fine structure of quantum correlations associated with multi-mode, multi-particle entanglement. By means of an entanglement measure based on the linear entropies associated with all the possible bipartitions, we analyze the entanglement in the states of flavor neutrinos and anti-neutrinos. Remarkably, we show that the entanglement is connected with experimentally measurable quantities, i.e. the variances of the lepton numbers and charges.
Introduction. – In recent years, growing attention has been devoted to the investigation of entanglement in the context of quantum field theory and elementary particle physics [1–3]. Indeed, understanding the nature of nonlocal quantum correlations in infinite-dimensional systems of fields and particles and their physical consequences poses some subtle conceptual and technical questions. The proper characterization and quantification of entanglement are achieved unambiguously only by properly taking into account the algebra of observables besides the tensor product structure of the individual state spaces. In the case of quantum fields, this framework must be extended further by considering correlations among distinguishable physical field modes rather than among indistinguishable particles and excitations [4–11]. For instance, the singleparticle Bell superposition state |0, 1i + |1, 0i between any two modes of the electromagnetic field is a paramount instance of a maximally (bipartite) entangled quantum state [10]. In this case the entanglement is between two different field modes with occupation numbers ranging between 0 and 1. Considering single-particle or multi-particle (e.g., multi-photon) states of many modes leads to straightforward generalizations that allow to consider the bipartite and multipartite multi-mode field entanglement of single-
particle and multi-particle states. Among the most spectacular instances of single-particle, multi-mode entanglement, the one associated to particle mixing and flavor oscillations is remarkably outstanding [2, 3]. Specifically, by considering the physically relevant cases of two- and threeflavor neutrino oscillations, it has been shown that the flavor transition probabilities are simple functions of multimode, single-particle entanglement [3]. This observation has concrete operational consequences in that it allows in principle to design experimental schemes for the transfer of the quantum information encoded in neutrino states to spatially delocalized two-flavor charged lepton states [2,3]. Thus, entangled states of oscillating neutrinos appears to be legitimate physical resources for quantum information tasks. Moreover, the detection of nonlocal quantum correlations in the form of bipartite and multipartite entanglement would provide a decisive test ground for confirming or disproving the quantum nature of fundamental physical phenomena such as particle mixing and flavor oscillations. By exploiting essential tools of entanglement theory, in the present work we aim at quantifying the content of multi-mode flavor entanglement in systems of oscillating neutrinos. In the course, we will broadly generalize and extend the analysis carried out previously in a simplified,
p-1
M. Blasone, F. Dell’Anno, S. De Siena and F. Illuminati purely quantum-mechanical setting to the framework of Quantum Field Theory (QFT), and we will develop a fully quantum-field theoretical treatment. Flavor states of mixed particles have been extensively discussed and a proper QFT framework for their description has already been worked out successfully since some years [12–14]. Within such a QFT framework, we show that the phenomena of flavor oscillations exhibit a richer structure of quantum correlations with respect to the corresponding quantum mechanical (QM) case previously studied [3]. Indeed, while in QM, flavor neutrino states are singleparticle states possessing multi-mode flavor entanglement, in QFT neutrino states are multi-mode flavor-entangled multi-particle states, as the presence of anti-neutrino particle species provides further degrees of freedom to the oscillating neutrino system. It is important to remark that entanglement is an observable-induced, relative physical quantity [6, 7], possessing a specific operational meaning according to the reference quantum observables that are singled out in each physical situation, and to the quantum subsystems selected as suitable parties in the partition (bipartite or multipartite) of a physical system. By assuming the particleantiparticle species as further quantum modes, we investigate the entanglement content of the neutrino system in the instance of multi-particle flavor-species entanglement associated with flavor oscillations. We find that the entanglement quantified via entropic measures is directly related to experimentally measurable quantities, the variances of the lepton numbers and charges. The paper is organized as follows: we first review the quantum information tools exploited in the paper and the main results previously achieved in the simplified QM framework. We then investigate the entanglement phenomenology of neutrino mixing and flavor oscillations adopting a fully QFT framework, and we discuss the novel, nontrivial structure of flavor entanglement that emerges in the QFT framework. Finally, we discuss briefly possible outlooks on future research. Entanglement and particle mixing: Quantum Mechanics. – In this section we briefly review the results obtained within the quantum mechanical framework [2, 3]. In the two flavor case, neutrino mixing is described by the 2 × 2 rotation matrix U(θ): cos θ sin θ U(θ) = , (1) − sin θ cos θ where θ is the mixing angle. The two-flavor neutrino states are defined as |ν (f ) i = U(θ) |ν (m) i (2) T
where |ν (f ) i = (|νe i, |νµ i) are the states with definite T flavors e, µ and |ν (m) i = (|ν1 i, |ν2 i) those with definite masses m1 , m2 . Both |να i (α = e, µ) and |νj i (j = 1, 2) are orthonormal. By describing the free propagation of the neutrino mass eigenstates with plane waves of the form
|νj (t)i = e−iωj t |νj i, ωj denoting the frequency associated with the mass eigenstate |νj i, the time evolution of the flavor states is given by |ν (f ) (t)i = U(t)|ν (f ) i ≡ U(θ)U0 (t)U−1 (θ)|ν (f ) i
(3)
where |ν (f ) i are the flavor states at t = 0, and U0 (t) = diag(e−iω1 t , e−iω2 t ). At time t the average neutrino number of the state |να (t)i in the mode a is: hNa (t)ib ≡ hνb (t)|Na |νb (t)i = |Uba (t)|2 , (a, b = e, µ). (4) By assuming the neutrino occupation number associated with a given mass or flavor (mode) as reference quantum number, one can establish the following correspondences with two-qubit states: |ν1 i ≡ |1iν1 |0iν2 ≡ |10i12 , |ν2 i ≡ |0iν1 |1iν2 ≡ |01i12 , (5) |νe i ≡ |1iνe |0iνµ ≡ |10ieµ , |νµ i ≡ |0iνe |1iνµ ≡ |01ieµ ,(6) where |jiνα stands for a j-occupation number state of a neutrino in mode α. Entanglement is thus established among mass or flavor modes, in a single-particle setting. For instance, the free evolution of the electron-neutrino state |νe (t)i can be written in the alternative forms: |νe (t)i
= e−iω1 t cos θ|10i12 + e−iω2 t sin θ|01i12 , (7)
|νe (t)i
= Uee (t)|10ieµ + Ueµ (t)|01ieµ ,
(8)
where |Uee (t)|2 + |Ueµ (t)|2 = 1. Thus, the states |ν (f ) (t)i can be seen as entangled Bell-like superpositions of the mass eigenstates with time-independent coefficients or of the flavor eigenstates with time-dependent coefficients. The entanglement present in the pure two-qubit states, Eq. (7) and Eq. (8), is quantified by the von Neumann entropy of the reduced density matrix, or by any other monotonic function of it, as the linear entropy [15]. Indeed, such a measure can be suitably exploited to construct measures of multipartite entanglement in multipartite states [16–18], and global measures encompassing both bipartite and multipartite contributions [19–23]. For example, the global entanglement is associated to the set of linear entropies associated to all possible bi-partitions of the whole system. We briefly review the definition of the global entanglement, which will be used in this work. Let ρ = |ψihψ| be the density operator corresponding to a pure state |ψi, describing the system S partitioned into N parties. We consider the bipartition of the N -partite system S in two subsystems SAn , constituted by n parties (1 ≤ n < N ), and SBN −n , constituted by the remaining N − n parties. Let ρAn ≡ T rBN −n [ρ] denote the reduced density matrix of subsystem SAn after tracing over subsystem SBN −n . The linear entropy associated to such a bipartition is defined as
p-2
(An ;BN −n )
SL
(ρ) =
d (1 − T rAn [ρ2An ]) , d−1
(9)
QFT entanglement in neutrino oscillations Ha; bL
SL,e
the entanglement is zero, and the global state of the system is factorized. For τ > 0, flavor oscillations occur, and the linear entropy exhibits a typical oscillatory behavior; the entanglement is maximal at largest mixing, that is hNe ie = hNµ ie = 0.5. We find that the linear entropies (a;b) SL coincide, apart from a constant factor, to the variances associated with the average neutrino number, i.e.
vs XNa \e
1
0.75
0.5
(a;b)
h(∆Na )2 ie ≡ hNa2 ie − hNa i2e = 4−1 SL,e .
0.25
(13)
Remarkably, the above relation represents a direct experimental connection between entanglement and funda0 Π 2Π mental physical quantities in the physics of elementary Τ particles. This observation remains true and is further (a;b) Fig. 1: (Color online) QM instance: The linear entropies SL,e strengthened when one moves to the QFT setting, as we as functions of the scaled time τ = (ω2 −ω1 )t, with a, b = νe , νµ do in the following Section. It is worth noticing that the (full line), and a, b = ν1 , ν2 (dotted line) The mixing angle θ above results are in complete agreement with the definiis fixed at the experimental value sin2 θ = 0.314. The av- tions of entanglement in terms of quantum uncertainties erage neutrino flavor-numbers hNe ie (double-dot-dashed line) of physical observables as proposed in Refs. [24]. 0
3Π 2
Π 2
and hNµ ie (dot-dashed line) are also reported.
Entanglement and particle mixing: Quantum Field Theory. – In order to present a generalization where the d is the Hilbert-space dimension given by d = of the above analysis to the QFT framework, we recall min{dim SAn , dim SBN −n } = min{2n , 2N −n }. The corre- the essential features of a specific QFT model of partisponding average linear entropy writes: cle mixing describing the phenomena of neutrino oscilla −1 X tions [12, 13]. The neutrino Dirac fields νe (x) and νµ (x) N (n:N −n) (A ;B ) hSL (ρ)i = SL n N −n (ρ) , (10) are defined through the mixing relations: n An νe (x) = cos θ ν1 (x) + sin θ ν2 (x) , (14) where the sum is intended over all the possible biνµ (x) = − sin θ ν1 (x) + cos θ ν2 (x) , (15) partitions of the system in two subsystems, respectively with n and N − n elements (1 ≤ n < N ) [22]. The linear where, in standard notation, x stands for the four-vector entropies Eq. (9) can be easily computed for the two-qubit x ≡ (t, x), and the free fields ν1 (x) and ν2 (x) with defBell state |νe (t)i, i.e. Eq. (8), with density matrix ρe = inite masses m1 and m2 . The generator of the mixing (ν ;ν ) (ν ;ν ) |νe (t)ihνe (t)|. The linear entropies SL,e1 2 ≡ SL 1 2 (ρe ) transformations is given by: Z associated to the reduced state after tracing over one mode † † 3 (mass) writes: Gθ (t) = exp θ d x ν1 (x)ν2 (x) − ν2 (x)ν1 (x) , (16) (ν ;ν2 )
SL,e1
(ν ;ν1 )
= SL,e2
= sin2 2θ ,
(11)
α νσα (x) = G−1 θ (t) νi (x) Gθ (t),
(17)
which, by fixing θ at the experimental value sin2 θ = 0.314, where (σ, i) = (e, 1), (µ, 2), and the superscript α = (ν ;ν ) the spinorial component. At finite voltakes the specific value SL,e1 2 ≃ 0.861. The linear en- 1, . . . , 4 denotes † ume V , G−1 (t) is a unitary operator, i.e. G−1 (νe ;νµ ) (νe ;νµ ) θ θ (t) = Gθ (t), tropy SL,e ≡ SL (ρe ) associated to the reduced preserves the canonical anti-commutation relations, and state after tracing over one mode (flavor) writes: maps the Hilbert space for free fields H1,2 to the Hilbert space for mixed fields He,µ , i.e. G−1 (t) : H1,2 7→ He,µ . (νe ;νµ ) (νµ ;νe ) SL,e = SL,e = 4|Uee (t)|2 |Ueµ (t)|2 (12) In particular, the flavor vacuum isθ given by |0(t)i e,µ = G−1 (t) |0i at finite V . We denote by |0i the flavor 1,2 e,µ θ (Here and in the following we often omit time dependence vacuum at t = 0. The free fields νi (x) (i = 1, 2) are given for simplicity.) Thus the linear entropy is strictly related by the following expansions to the average neutrino flavor numbers Eq. (4). In Fig. 1 i (νe ;νµ ) 1 Xh r r we show the behavior of SL,e as functions of the scaled, r† r uk,i αk,i (t) + v−k,i β−k,i (t) eik·x , (18) νi (x) = √ dimensionless time τ = (ω2 − ω1 )t; we also report the V k,r neutrino flavor-numbers hNe ie and hNµ ie . As expected, (ν ;ν ) the entanglement among mass modes, i.e. SL,e1 2 , re- where k is the momentum vector, r = 1, 2 denotes the r† r† mains constant during the free propagation. On the con- helicity, αrk,i (t) = αrk,i e−iωk,i t , βk,i (t) = βk,i eiωk,i t , and p r trary, as regards the entanglement among flavor modes, ωk,i = k2 + m2i . The operators αrk,i and βk,i are the (νe ;νµ ) i.e. SL,e , at time τ = 0, the two flavors are not mixed, annihilation operators for the vacuum state |0im ≡ |0i1 ⊗ p-3
M. Blasone, F. Dell’Anno, S. De Siena and F. Illuminati r |0i2 , i.e. αrk,i |0im = βk,i |0im = 0. For further details see Refs. [12, 13]. By use of Gθ (t), the flavor fields can be expanded as: i 1 Xh r r r† r νσ (x) = √ uk,i αk,σ (t) + v−k,i β−k,σ (t) eik.x . (19) V k,r
The flavor annihilation operators are defined as αrk,σ (t) ≡ r† r† −1 r G−1 θ (t)αk,i Gθ (t) and βk,σ (t) ≡ Gθ (t)βk,i Gθ (t). For simplicity, let us choose the reference frame such that k = (0, 0, |k|), we have αrk,e (t) = cos θ αrk,1 (t) r† + sin θ |Uk | αrk,2 (t) + ǫr |Vk | β−k,2 (t) , (20) where ǫr = (−1)r and |Uk | ≡ |Vk | ≡ =
r† r r ur† k,i uk,j = v−k,i v−k,j r r r† r ǫr ur† k,1 v−k,2 = −ǫ uk,2 v−k,1
2
(ωk,1 + m1 ) − (ωk,2 + m2 ) p |k|, ωk,1 ωk,2 (ωk,1 + m1 )(ωk,2 + m2 ) 2
two masses eigenvalues; p expresses the ratio between the momentum and the masses geometrical mean and corresponds to the relativistic limit for p ≫ 1. Evidently, the |νe (t)i is a multi-particle entangled state both in the mass eigenstates Hilbert space, i.e. Eq. (23), and in the flavor eigenstates Hilbert space, i.e. Eq. (24). Analogously with the Pontecorvo states (2) and (8), we assume the neutrino occupation number as reference quantum number. However, with respect to Eqs. (7) and (8), we have still two masses and flavors, but we have a further degree of freedom that is the neutrino species, i.e. particles and anti-particles. Therefore, we obtain multipartite entanglement in a four-qubit state. In the instance of the Hilbert space H12 , Eq. (23) can be written as: |νe (t)i = e−iω1 t cos θ|1000i12¯1¯2 + e−i(ω2 −ω1 )t U |0100i12¯1¯2 +e−i(ω2 +ω1 )t V |1110i12¯1¯2 ,
(26)
(21) where |ijkhi12¯1¯2 denotes the four-qubit vector |iiν1 |jiν2 |kiν¯1 |hiν¯2 with i, j, k, h = 0, 1. On the other hand, in the instance He,µ , Eq. (24) can be written as:
(22)
|νe (t)i = Uee (t)|1000ieµ¯eµ¯ + Ueµ (t)|0100ieµ¯eµ¯ + e µ¯ µ Ue¯ eµ ¯ + Uee (t)|1101ieµ¯ eµ ¯, eµ (t)|1110ieµ¯
2
with i, j = 1, 2, i 6= j, |Uk | + |Vk | = 1. The state of an r electron neutrino is defined as |νk,e i ≡ αr† k,e |0ie,µ . In the √ 2 relativistic limit |k| ≫ m1 m2 , one gets |Uk | −→ 1 and 2 |Vk | −→ 0, and the corresponding (quantum-mechanical) Pontecorvo state is recovered. We now omit the superscript r (by fixing r = 2) and the subscript k, thus restricting the analysis to the flavor neutrino state |νe i of fixed momentum and helicity. In the Hilbert space H1,2 , the neutrino state at time t can be written in the form: |νe (t)i = e−iω1 t cos θ α†1 + e−i(ω2 −ω1 )t U α†2 + +e−i(ω2 +ω1 )t V sin θ α†1 α†2 β1† |0i1,2 . (23)
where |ijkhieµ¯eµ¯ denotes the four-qubit vector |iiνe |jiνµ |kiν¯e |hiν¯µ with i, j, k, h = 0, 1. Let us analyze the multipartite entanglement possessed by |νe i, i.e. Eq. (27), by using the global entanglement defined by Eqs. (9) and (10): we compute the linear entropies (a;b,c,d) SL associated with the bipartition of a singleparticle subsystem and a three-particle subsystem, and (1:3) the corresponding average linear entropy hSL i. Next (a,b;c,d) we compute the linear entropies SL associated with the balanced bipartition of two-particle subsystems, and (2:2) the corresponding average linear entropy hSL i. The (a;b,c,d) linear entropies SL,e associated with Eq. (26) write:
In the Hilbert space He,µ , Eq. (23) is rewritten as: † † † e |νe (t)i = Uee (t) α†e + Ueµ (t) ᆵ + Ue¯ eµ (t) αe αµ βe µ¯ µ (24) +Uee (t) α†e ᆵ ⵆ |0ie,µ , where the time-dependent coefficients are given by: Uee (t) = e−iω1 t cos2 θ + sin2 θ e−i(ω2 −ω1 )t |U |2 +e−i(ω2 +ω1 )t |V |2 , Ueµ (t) = e−iω1 t U cos θ sin θ e−i(ω2 −ω1 )t − 1 , e −iω1 t Ue¯ V cos θ sin θ 1 − e−i(ω2 +ω1 )t , eµ (t) = e
(27)
(ν ;ν2 ,¯ ν1 ,¯ ν2 )
SL,e1
(ν ;ν ,¯ ν ,¯ ν ) SL,e2 1 1 2 (¯ ν ;ν ,ν ,¯ ν ) SL,e1 1 2 2 (¯ ν ;ν1 ,ν2 ,¯ ν1 )
SL,e2
= 4U 2 sin2 θ(1 − U 2 sin2 θ) ,
= sin2 2θ , 2
2
(28) (29)
2
2
= 4V sin θ(1 − V sin θ) ,
(30)
= 0.
(31)
Of course, in the quantum mechanical limit, Eqs. (28) and (29) reduces to Eq. (11), while Eqs. (30) and (31) go to zero. By fixing the mixing angle θ at the experimental value sin2 θ = 0.314, and the parameters x and p at x = 10 and p = 5, we get: (ν ;ν ,¯ ν ,¯ ν )
µ −iω1 t Uµ¯ U V sin2 θ e−i(ω2 +ω1 )t − e−i(ω2 −ω1 )t , ee (t) = e
e 2 µ¯ µ 2 |Uee (t)|2 + |Ueµ (t)|2 + |Ue¯ eµ (t)| + |Uee (t)| = 1. (25)
SL,e1 2 1 2 ≃ 0.831, (¯ ν ;ν ,ν ,¯ ν ) SL,e1 1 2 2 ≃ 0.075, (1:3) hSL,e i ≃ 0.442. (a;b,c,d)
(ν ;ν ,¯ ν ,¯ ν )
SL,e2 1 1 2 ≃ 0.861, (¯ ν ;ν ,ν ,¯ ν ) SL,e2 1 2 1 = 0,
(32)
The linear entropies SL,e associated with Eq. (27) are In the following, for convenience, we introduce the real given by simple generalizations of Eq. (12): √ parameters x = m2 /m1 , p = |k|/ m1 m2 , and τ = (νe ;νµ ,¯ νe ,¯ νµ ) (ω2 − ω1 )t. Therefore, x represents the ratio between the SL = 4|Ueµ |2 (1 − |Ueµ |2 ) , (33) p-4
QFT entanglement in neutrino oscillations Ha,b; c,dL
Ha; b,c,dL
SLe
SL,e
0.75
1
0.75 0.5
0.5
0.25
0.25
0 Π 2
0
3Π 2
Π
0
2Π
Π 2
0
Τ
Fig. 2: (Color online) QFT instance: The linear entropies (ν ;ν ,¯ ν ,¯ ν ) (ν ;ν ,¯ ν ,¯ ν ) SL,ee µ e µ (double-dot-dashed line), SL,eµ e e µ (dot(¯ ν ;νe ,νµ ,¯ νµ )
dashed line), SL,ee
(¯ ν ;νe ,νµ ,¯ νe )
(dotted line), SL,eµ
(dashed
(1:3)
line), and the average linear entropy hSL,e i (full line) as functions of the scaled time τ = (ω2 − ω1 )t. The values of parameters are those of Eq.(32).
3Π 2
2Π
Fig. 3: (Color online) QFT instance: The linear en(ν ,ν ;¯ ν ,¯ ν ) (ν ,¯ ν ;ν ,¯ ν ) tropies SL,ee µ e µ (double-dot-dashed line), SL,ee e µ µ (ν ,¯ νµ ;νµ ,¯ νe )
(dot-dashed line), SL,ee
2
2
= 4|Uee | (1 − |Uee | ) ,
(34)
=
(35)
e 2 4|Ue¯ eµ | (1
−
e 2 |Ue¯ eµ | ) ,
(dotted line), and the average
(2:2)
linear entropy hSL,e i (full line) as functions of the scaled time τ = (ω2 − ω1 )t. The values of parameters are those of Eq.(32). (ν ,¯ ν1 ;ν2 ,¯ ν2 )
(ν ;ν ,¯ ν ,¯ ν ) SL µ e e µ (¯ ν ;ν ,ν ,¯ ν ) SL e e µ µ (¯ ν ;ν ,ν ,¯ ν ) SL µ e µ e
Π
Τ
SL,e1
(ν ,¯ ν2 ;ν2 ,¯ ν1 )
SL,e1
2 sin2 2θ , (39) 3 4 2 2 = U sin θ(2 − U 2 + U 2 cos 2θ) . (40) 3 =
µ 2 µ¯ µ 2 = 4|Uµ¯ ee | (1 − |Uee | ) .
(36) By fixing the mixing angle θ at the experimental value sin2 θ = 0.314, and the parameters x and p at x = 10 and Of course, in the quantum mechanical limit, Eqs. (33) and p = 5, the above quantities take the values: (34) reduce to the Pontecorvo analogs, while Eqs. (35) and (ν ,¯ ν ;ν ,¯ ν ) (ν ,ν ;¯ ν ,¯ ν ) (36) go to zero. In Fig. 2 we plot the quantities (33)-(36) SL,e1 1 2 2 ≃ 0.574, SL,e1 2 1 2 ≃ 0.050, (41) as functions of the scaled time τ for x = 10 and p = 5; it is (ν ,¯ ν ;ν ,¯ ν ) (2:2) SL,e1 2 2 1 ≃ 0.554, hSL,e i ≃ 0.393. worth noticing that, such a choice of the parameters corresponds to the following assumptions: mass m2 greater (a,b;c,d) associated with Eq. (27) write: than mass m1 of one order of magnitude, and momentum The entropies SL,e of the same order of magnitude as the masses geometri4h (ν ,ν ;¯ ν ,¯ ν ) (ν ;ν ,¯ ν ,¯ ν ) SL,ee µ e µ = 1 − (|Uee |2 + |Ueµ |2 )2 cal mean. All the linear entropies, except SL,ee µ e µ , 3 i exhibit rapidly oscillating components, due to the phee¯ e 2 µ¯ µ 2 2 −(|U | + |U | ) , (42) eµ ee nomenon of oscillations involving antineutrinos. Comparing the QM and QFT instances, i.e. Figs. 1 and 2 respec4h (ν ,¯ ν ;ν ,¯ ν ) µ 2 2 (ν ;ν ) (ν ;ν ,¯ ν ,¯ ν ) SL,ee e µ µ = 1 − (|Uee |2 + |Uµ¯ tively, the linear entropies SL,ee µ and SL,ee µ e µ show ee | ) 3 i a very similar shape and behavior. By straightforward 2 e¯ e 2 2 −(|U | + |U | ) , (43) eµ eµ analytical calculation we find that the linear entropies (a;b,c,d) , both in the instance of H12 and Hab , coincide, SL 4h (ν ,¯ ν ;ν ,¯ ν ) e 2 2 SL,ee µ µ e = 1 − (|Uee |2 + |Ue¯ eµ | ) apart from a constant factor, with the variances associated 3 i with the particle number: µ 2 2 (44) −(|Ueµ |2 + |Uµ¯ ee | ) . (a;b,c,d)
h(∆Na )2 i = hNa2 i − hNa i2 = 4−1 SL
. (37)
Therefore, also in the QFT framework, Eq. (37) provides a clear operational meaning for the above entanglement measures throughout a direct connection with measurable physical quantities. Let us now analyze the linear entropies associated with balanced bipartitions. The linear (a,b;c,d) associated with Eq. (26) write: entropies SL,e (ν ,ν2 ;¯ ν1 ,¯ ν2 )
SL,e1
=
4 2 2 V sin θ(2 − V 2 + V 2 cos 2θ) , 3
(38)
The above quantities are plotted in Fig. 3. We observe that, during the time evolution, most of entanglement lies in the bipartitions (νe , ν¯e ; νµ , ν¯µ ) and (νe , ν¯µ ; νµ , ν¯e ), and, thus, between bipartitions each containing a particle. This (a;b,c,d) , property also characterizes the linear entropies SL,e see Fig. 2; in fact, the unbalanced bipartitions constituted by a single antiparticle-mode versus the remaining modes, i.e. (¯ νe ; νe , νµ , ν¯µ ) and (¯ νµ ; νe , νµ , ν¯e ), possess sensibly less entanglement than the other bipartitions. Therefore, most entanglement is shared between the two particle-modes.
p-5
M. Blasone, F. Dell’Anno, S. De Siena and F. Illuminati Finally, it can be analytically demonstrated the existence REFERENCES of a connection between the balanced bipartite linear entropies and the variances of the neutrino numbers and [1] Bertlmann R.A. and Grimus W., Phys. Lett. B, 392 (1997) 426; Bertlmann R.A., Grimus W. and Hiesmayr charges. Indeed, the following relations hold: 3 (νe ,νµ ;¯νe ,¯νµ ) h[∆(Ne + Nµ )] i = S , (45) 4 L,e 3 (ν ,¯ν ;ν ,¯ν ) h[∆(Ne − Ne¯)]2 i ≡ h[∆Qe ]2 i = SL,ee e µ µ , (46) 4 3 (νe ,¯νµ ;νµ ,¯νe ) 2 h[∆(Ne − Nµ¯ )] i = S . (47) 4 L,e 2
In particular, in Eq. (46) we have denoted by Qe ≡ Ne −Ne¯ the charge operator associated with the electronic flavor. Such an equation clearly refers to the entanglement between flavors independently from the species. Discussion and Outlook. – In this paper we studied the entanglement associated to neutrino mixing and oscillations in a field-theoretical framework: the nontrivial nature of QFT flavor states [12] gives rise to a complex structure of quantum correlations, due to the presence of particle-antiparticle pairs. This implies that one deals with multi-mode, multi-particle entangled states. Characterizing the entanglement by linear entropies, we obtained exact expressions relating entanglement to experimentally measurable quantities, as the variances of flavor charges. The main observable quantities in particle mixing and flavor oscillations are thus directly related to entanglement and nonclassicality. The present analysis has been carried out for the case of two generations. We plan to extend the study to the case of three flavors, including CP violation, for which the structure of QFT flavor states becomes considerably more involved [14]. It is also worth to remark that the (exact) results obtained in our paper represent a canonical example of evaluation of entanglement for a relativistic system. In this respect, they could be useful for further studies on relativistic properties of entanglement, e.g. its behavior in connection with Lorentz transformations [25]. The results of the present analysis, which extend those of ref. [3] in the framework of QM to a fully QFT setting, may be exploited to investigate quantum information protocols based on oscillating neutrinos, to be realized in analogy to the use of mode entanglement of the electromagnetic field, in particular those protocols based on single-photon entanglement. Concerning experimental tests and implementations, entanglement can be in principle experimentally accessed via a transfer scheme from single-neutrino states to two-flavor charged lepton states [3]. Finally, one could devise extensions to singleneutrino systems of the experimental schemes based on single-neutron interferometry that were able to detect Bell nonlocality in single-nucleon systems [26]. ∗∗∗
B.C., Phys. Lett. A, 289 (2001) 21; Kayser B., Kopp J., Hamish Robertson R.G. and Vogel P., Phys. Rev. D, 82 (2010) 093003; Akhmedov E.K. and Smirnov A.Y., Found. Phys., 41 (2011) 1279; Boyanovsky D., Phys. Rev. D, 84 (2011) 065001. [2] Blasone M., Dell’Anno F., De Siena S., Di Mauro M. and Illuminati F., Phys. Rev. D, 77 (2008) 096002. [3] Blasone M., Dell’Anno F., De Siena S. and Illuminati F., Europhys. Lett., 85 (2009) 50002; [4] Ghirardi G.C., Marinatto L. and Weber T., J. Stat Phys., 108 (2002) 49; [5] Benatti F., Floreanini R. and Marzolino U., Ann. Phys., 327 (2012) 1304; Ann. Phys., 325 (2010) 924. [6] Zanardi P., Phys. Rev. A, 65 (2002) 042101; Zanardi P., Lidar D.A. and Lloyd S., Phys. Rev. Lett., 92 (2004) 060402. [7] Wiseman H.M. and Vaccaro J.A., Phys. Rev. Lett., 91 (2003) 097902; [8] Barnum H. et al., Phys. Rev. Lett., 92 (2004) 107902; [9] Banuls M.-C., Cirac J.I. and Wolf M.M., Phys. Rev. A, 76 (2006) 022311. [10] van Enk S.J., Phys. Rev. A, 72 (2005) 064306. [11] Terra Cunha M.O., Dunningham J.A. and Vedral V., Proc. R. Soc. A, 463 (2007) 2277. [12] Blasone M. and Vitiello G., Ann. Phys. (N.Y.), 244 (1995) 283. [13] Blasone M., Henning P.A. and Vitiello G., Phys. Lett. B, 451 (1999) 140. Blasone M., Jizba P. and Vitiello G., Phys. Lett. B, 517 (2001) 471. [14] Blasone M., Capolupo A. and Vitiello G., Phys. Rev. D, 66 (2002) 025033. [15] Horodecki R., Horodecki P., Horodecki M. and Horodecki K., Rev. Mod. Phys., 81 (2009) 865. [16] Coffman V., Kundu J. and Wootters W.K., Phys. Rev. A, 61 (2000) 052306. [17] Osborne T.J. and Verstraete F., Phys. Rev. Lett., 96 (2006) 220503. [18] Adesso G., Serafini A. and Illuminati F., Phys. Rev. A, 73 (2006) 032345. [19] Barnum H. and Linden N., J. Phys. A: Math. Gen., 34 (2001) 6787. [20] Meyer D.A. and Wallach N.R., J. Math. Phys., 43 (2002) 4273. [21] Brennen G.K., Quantum Inf. Comp., 3 (2003) 619. [22] de Oliveira T.R., Rigolin G. and de Oliveira M.C., Phys. Rev. A, 73 (2006) 010305(R). [23] Wei T.C. and Goldbart P.M., Phys. Rev. A, 68 (2003) 042307. Blasone M., Dell’Anno F., De Siena S. and Illuminati F., Phys. Rev. A, 77 (2008) 062304. ¨ [24] Klyachko A., Oztop B. and Shumovsky A.S., Phys. Rev. A, 75 (2007) 032315; Klyachko A. and Shumovsky A.S., J. Opt. B, 6 (2004) S29. [25] Friis N., Bertlmann R.A., Huber M. and Hiesmayr B.C., Phys. Rev. A, 81 (2010) 042114. Huber M. et al, Europhys. Lett., 95 (2011) 20002. [26] Hasegawa Y. et al, Nature, 425 (2003) 45.
The authors acknowledge support from the EU STREP Project iQIT, Grant Agreement No. 270843. p-6
