Bell Length in the Entanglement Geometry

1 23

International Journal of TheoreticalPhysics ISSN 0020-7748 Int J Theor PhysDOI 10.1007/s10773-014-2461-6

Bell Length in the Entanglement Geometry

Davide Fiscaletti & Ignazio Licata

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Int J Theor PhysDOI 10.1007/s10773-014-2461-6

Bell Length in the Entanglement Geometry

Davide Fiscaletti · Ignazio Licata

Received: 4 September 2014 / Accepted: 8 December 2014© Springer Science+Business Media New York 2014

Abstract A geometric approach to entangled qubit pairs is outlined via Bohm Theory. Anentropic quantum correlation distance is here proposed as a mark of the non-local “hand-shaking” between two systems under the action of Quantum Potential. The Bell-CHSHinequalities and Berry Phase are analyzed in terms of this new correlation measure we calledBell Length, in honour of J.S. Bell (1928-1990).

Keywords Quantum potential · Quantum entropy · Entangled qubits · CHSH-Bellinequalities

1 Introduction

In recent years we are assisting at a growing interest in Bohm’s interpretation of quantummechanics. Thanks to the introduction of the quantum potential, Bohm’s approach to quan-tum mechanics can explain the quantum behaviour of matter – which appears mysteriousand obscure in the Copenhagen interpretation – without radically modifying the quantumformalism and altering the predictions. The quantum potential throws new fundamental lightinto the analysis of quantum non-locality. While in the Copenhagen interpretation of quan-tum mechanics non-locality emerges as an ‘unexpected host’ which lies behind the purelyprobabilistic interpretation of the wave function and the mechanism of “casuality” associ-ated with it, Bohm’s approach is intrinsically non-local in virtue of the space-like, active,contextual information of the quantum potential. The quantum potential expresses a non-local, contextual information woven in the background of quantum processes. [1–6]; (Fromthe historical point of view see [7, 8]).

D. FiscalettiSpaceLife Institute, San Lorenzo in Campo, PU, Italye-mail: [email protected]

I. Licata (�)ISEM, Institute for Scientific Methodology, Palermo, Italye-mail: [email protected]

Author's personal copy

mailto:[email protected]

mailto:[email protected]

Int J Theor Phys

In the references [9–12] the authors of this paper showed that the Bohm theory canreceive a new interesting and suggestive reading which is based on the idea that some pecu-liar features of the quantum potential derive from a physical quantity which can be defined“quantum entropy”. This new way of reading the Bohm theory has been appropriately calledas the “entropic version” of Bohm theory or, more briefly, “entropic Bohm theory”.

The entropic approach to Bohm theory starts by defining the logarithmic function

SQ = −1

2ln ρ (1)

where ρ = |ψ (�x, t)|2 is the probability density associated with the wave function ψ (�x, t)

of the physical system. In the entropic version of bohmian mechanics, the space-temporaldistribution of the ensemble of particles describing the physical system under considerationis assumed to generate a modification of the geometry of the configuration space corre-sponding with a quantum entropy given by (1). The quantum entropy (1) can be interpretedas the physical entity that, in the quantum domain, describes the degree of order and chaosof the vacuum supporting the density ρ describing the space-temporal distribution of theensemble of particles associated with the wave function of the physical system under con-sideration [13]. Starting from the quantum entropy (1), a new picture of non-relativisticBohm theory emerges in which the quantum potential can be expressed as

Q = − �2

2m

(∇SQ

)2 + �2

2m

(∇2SQ

)(2)

namely is an information channel given by the sum of two quantum correctors linkedwith the quantum entropy. In fact, the two bohmian equations of motion deriving from thedecomposition of the Schrodinger equation, namely the quantum Hamilton-Jacobi equationdescribing the motion of the corpuscle associated with the wave function and the continuityequation, respectively become

|∇S|22m

− �2

2m

(∇SQ

)2 + V + �2

2m

(∇2SQ

)= −∂S

∂t(3)

and∂SQ

∂t= − (�v · ∇SQ

) + 1

2∇ · �v. (4)

Equation (3) provides an energy conservation law in which the term − �2

2m

(∇SQ

)2 can be

interpreted as the quantum corrector of the kinetic energy |∇S|22m

of the particle while the term�

2

2m

(∇2SQ

)can be interpreted as the quantum corrector of the potential energy V . On the

basis of (2), (3) and (4), the quantum entropy emerges just as the informational line of thequantum potential, describing the deformation of the geometry in the presence of quantumeffects. The geometrical properties of the configuration space associated with the quantumentropy can be also characterized by introducing a quantum-entropic length – evaluatingthe modification of the geometry with respect to the Euclidean geometry characteristic ofclassical physics – given by relation

Lquantum = 1√(∇SQ

)2 − ∇2SQ

. (5)

Once the quantum-entropic length (5) becomes non-negligible the system goes into aquantum regime.


Int J Theor Phys

On the basis of recent research [14–16], the Bohmian quantum potential can be identifiedwith the curvature scalar of the Weyl integrable space, namely is

Q = − �2

2m

∇2�

�(6)

where the scalar function � is linked with the curvature through relation R = 8 ∇2��

. Afundamental equation of motion in the Weyl integrable space can be considered which hasthe following form

∂S

∂t+ 1

2m∇S · ∇S + V − �

2

2m

∇2�

�= 0 (7)

Equation (7) is analogous to the quantum Hamilton-Jacobi equation of Bohm interpretationof quantum mechanics, if one identifies the scalar function � with the amplitude of thewave function. In this picture, the inverse square root of the curvature scalar leads to definea typical length (Weyl length) that can be used to evaluate the strength of quantum effects,in other words the quantity

LW = 1√

∇2��

(8)

can be defined as the quantum length. Here, taking into account the equivalence between thefundamental (7), based on Weyl integrable space, and the fundamental quantum Hamilton-Jacobi (3), one obtains:

− �2

2m

[(∇SQ

)2 − ∇2SQ

]= − �

2

2m

∇2�

�(9)

Therefore, the quantum length (8) may be written as

Lquantum = 1√(∇SQ

)2 − ∇2SQ

(10)

namely depends on the quantum entropy (1). According to (5), the quantum entropy maybe considered as the ultimate visiting card of the quantum length (8). By expressing thequantum length (8) in the form (5), namely in function of the entropy, this relation tellsus much more about the dynamics of the quantum system. In fact, the presence of the twoquantum correctors of the energy seems to suggest that (5) is an indicator of non-localcorrelation (and thus that provides a direct measure of the degree of departure of a systemfrom the Euclidean geometry characteristic of classical physics). The maximum value of (5)is obtained for Lmax

quantum = 1 (see [12]), which corresponds to the maximum de-localizationof a quantum system: for this reason the quantity (5) can be called the Bell length, in honourof John S. Bell (1928-1990).

In this paper, our aim is to analyse spin-spin correlations in entangled qubit pairs in thepicture of the entropic version of Bohm theory whose fundamental entity is the Bell lengthand to show the perspectives in the reading of Bell’s inequalities. The paper is structured inthe following manner. In chapter 2 we will develop the entropic bohmian formalism describ-ing spin-spin correlations of entangled qubit pairs. In chapter 3, we will see how the Belllength for systems of entangled qubit pairs acts as an entropic quantum correlation distancemeasuring the entanglement properties and we will analyse its applications as regards thereading of Bell’s inequalities. In chapter 4 we will propose a geometric entropic re-readingof the Berry phase. Finally, in chapter 5 we will conclude by making some considerationsabout the perspectives introduced by the quantum entropy and the Bell length intended asentropic quantum correlation distance in quantum computing.


Int J Theor Phys

2 The Quantum-Entropic Length for Spin-Spin Correlations in Entangled QubitPairs

As shown in [17], quantum entanglement of a qubir pair of spin ½ particles can bedescribed in a bohmian framework by starting from the mapping between a quantum spher-ically symmetric rigid rotor and a classical spinning top in the presence of a quantumpotential.

By expressing the wave function as ψ = ReiS , where R (ξ) and S (ξ) are real functionsof Euler angles ξ = (α, β, γ ) which specify the orientation of a rigid body, in a Bohmianpicture the angular momentum is given by a real three-dimensional vector

�M = i �MS. (11)

The dynamics of the spherically symmetric rigid rotor is determined from the Hamilton-Jacobi-type equations for the classical Hamiltonian with an additional quantum potential Q,namely

H = �M2

2I+ Q, (12)

where

Q = �M2R

2IR(13)

is the quantum potential, I is the moment of inertia. The quantum potential (13) generatesa quantum torque

�T = −i �MQ (14)

which rotates the angular momentum vector via the equation of motion

d �Mdt

= �T (15)

along the trajectory ξ (t).In the geometric entropic approach, the quantum potential (9) may be directly expressed

as

Q = 1

2I

[( �MSQ

)2−

( �M2SQ

)](16)

namely can be seen as an information channel into the behaviour of the rigid rotor whichis determined by the quantum entropy. According to (16), one can say that the quantumeffects of the rigid rotor correspond to a quantum entropy describing the degree of order andchaos of the vacuum – a storage of virtual trajectories supplying optimal ones for particlemovement – supporting the space-temporal distribution of the density of particles in theelement d3ξ along the trajectory ξ (t). As regards the configuration space of the rigid rotor,a deformation of the geometry corresponds to the quantum entropy in the sense that thequantum entropy of the rigid rotor generates a quantum torque

�T = − i

2I

[( �MSQ

)2−

( �M2SQ

)] �M (17)

which rotates the angular momentum vector via the equation of motion

d �Mdt

= − i

2I

[( �MSQ

)2−

( �M2SQ

)] �M (18)


Int J Theor Phys

along the trajectory ξ (t). Moreover, the modification of the geometry associated with thequantum entropy as regards a spherically symmetric rigid rotor can be also characterized byintroducing the quantum-entropic length (or Bell length) given by relation

Lquantum = 1√

12I

(( �M2SQ

)−

( �MSQ

)2) . (19)

Once the quantum-entropic length (19) becomes non-negligible the rigid rotor goes into aquantum regime.

Now, let us analyse the dynamics and the quantum effects of a general two qubits state,with vanishing total angular momentum projections, given by the following equation

|ψ〉 = cosϑ

2|↑↓〉 + eiφ sin

ϑ

2|↑↓〉 (20)

where |↑↓〉 corresponds to the state of the system when the qubit is in the “up” state, i.e.,in the direction of the z-axis, and the second qubit is in the “down” state, while |↑↓〉 corre-sponds to the state of the system when the first qubit is in the “down” state and the secondqubit is in the “up” state.

In a bohmian framework the guiding wave function

ψ (ξ) = cosϑ

2u↑ (ξ1) u↓ (ξ2) + eiφ sin

ϑ

2u↓ (ξ1) u↑ (ξ2) (21)

works in a six-dimensional space spanned by ξ = {ξ1, ξ2}, where ξ1, ξ2 are the coordinatesof the first and the second rotor, respectively. The corresponding Hamiltonian is given byrelation

H =�M2

1 + �M22

2I+ Q (22)

where

Q =(M2

1 + M22

)R

2IR(23)

is the quantum potential, �M1 = i �M1S and �M2 = i �M2S being the angular momenta of thetwo qubits 1 and 2 respectively, S being the phase of the wave function (21) of the system.In the geometric entropic approach, the quantum potential (23) may be expressed as

Q = 1

2I

[( �M1SQ

)2+

( �M2SQ

)2−

( �M21 SQ

)−

( �M22 SQ

)](24)

namely is generated by the quantum entropy of the system. Moreover, the deformation ofthe geometry associated with the entangled qubit pair can be described by the followingquantum-entropic length (or Bell length)

Lquantum = 1√

12I

(( �M21 SQ

)−

( �M1SQ

)2+

( �M22 SQ

)−

( �M2SQ

)2) (25)

and can be seen as a consequence of the fact that the quantum entropy generates the quantumtorques

�T1 = − i

2I

[( �M1SQ

)2+

( �M2SQ

)2−

( �M21 SQ

)−

( �M22 SQ

)] �M1 (26)


Int J Theor Phys

and

�T2 = − i

2I

[( �M1SQ

)2+

( �M2SQ

)2−

( �M21 SQ

)−

( �M22 SQ

)] �M2 (27)

According to the geometric entropic approach here proposed, the quantum torques (26) and(27) can be in fact associated with the quantum-entropic length (25) on the basis of equations

�T1 = i

L2quantum

�M1 (28)

�T2 = i

L2quantum

�M2 (29)

respectively.The total angular momentum projection M1z + M2z of the entangled qubit pair is zero

while the angular momenta owed to the action of the non-local quantum potential (24) andthe corresponding quantum torques (28) and (29) exhibit a complex precessional motion.

As regards a geometric description of entangled qubit pairs in a bohmian picture, inter-esting results have been obtained recently by Ramsak [18, 19]. In the papers Geometricalview of quantum entanglement and Spin-spin correlations of entangled qubit pairs in theBohm interpretation of quantum mechanics, Ramsak analysed the dynamics of the quantumentanglement of a pair of qubits in the framework of the Bohm interpretation of quantummechanics, showing that the angular momenta of two qubits can be viewed geometricallyin the bohmian space of hidden variables and characterized by their relative angles. For per-fectly entangled pairs, the qubits exhibit a unison precession determining a constant anglebetween their angular momenta. Ramsak computed trajectories �M1,2 [ξ (t)] covering the fullconfiguration space with ≈ 106 initial values ξ (0) per |ψ〉, namely for a particular choiceof ϑ and φ, finding that, although they exhibit extremely rich variety, they are characterizedby the following common properties:

(i) The quasi-periodic motion appears chaotic and, except in special cases, the projec-tions of the total momentum �M onto the xy-plane winds around the origin an infinitenumber of times in a spirographic manner, creating a dense annulus limited by fixedouter and inner radii;

(ii) The curve corresponding to the relative momentum �M2 − �M1 is closed and periodic.

These results suggest that the entanglement properties of a qubit pair in the state (20) derivefrom the dynamics of the azhimutal angles ϕ1 [ξ (t)] and ϕ2 [ξ (t)] of the angular momenta,which precess in a well-defined way with some initial probability distribution. In particular,on the basis of Ramsak’s results, the probability distribution

dP (ϕ)

dϕ=

∫δ [ϕ − ϕ (ξ)] R2 (ξ) dξ (30)

of the ensemble average difference of azimuthal angles ϕ [ξ (t)] = ϕ2 − ϕ1 is constant forunentangled qubits and becomes progressively peaked at φ for increasing entanglement,culminating in precession of angular momenta at equal relative angle ϕ [ξ (t)] = φ for all ξ

consistent with perfect entanglement. The shape of the distribution is independent of φ.In the geometric approach here proposed the quantum entropy of the entangled sys-

tem can be considered the fundamental key of reading determining the behaviour of theprobability distribution (30) of Ramsak’s approach according to equation

dP (ϕ)

dϕ=

∫δ [ϕ − ϕ (ξ)] exp

(−2SQ

)dξ. (31)


Int J Theor Phys

On the basis of (31), the entanglement properties of a qubit pair are determined by thedeformation of the geometrical properties of the configuration space. Moreover, one can saythat the peaking of the probability distribution (31) for increasing entanglement between thetwo qubits is associated with the precessional motion of the quantum torques (28) and (29).In particular, the probability distribution (31) is peaked in correspondence to the maximumvalue of the Bell length (25).

In the approach based on (24)–(29), it becomes therefore permissible the following re-reading of Ramsak’s results, and in particular of the behaviour of the probability distribution(30): as regards a system of two entangled qubits the quantum potential (24) emerges whichcorresponds to the deformation of the geometry described by the quantum-entropic length(25), and the quantum information associated with the quantum potential (24) and thus thequantum-entropic length (25) generate the quantum torques (28) and (29) which, by exhibit-ing a precessional motion, imply the precession of angular momenta at equal relative angleϕ [ξ (t)] = φ for all ξ consistent with perfect entanglement regarding the probability dis-tribution (30). The dynamics of the azimuthal angles ϕ1 [ξ (t)] and ϕ2 [ξ (t)] of angularmomenta – and, consequently, the chaotic and spirographic features of the trajectories deter-mined by the projections of the total angular momentum onto the xy-plane and the closedand periodic curve corresponding to the relative momentum �M2 − �M1 – are therefore deter-mined by the quantum torques (28) and (29), and thus by the quantum potential (24), namelyby the deformation of the geometrical properties of the configuration space described bythe quantum entropy.

Moreover, by classifying the ensemble representatives of the system of two qubits onthe basis of the relative direction of angular momenta precession of the two qubits, Ramsakfound that in the low entanglement regime at each moment the xy-plane projection of theangular momenta �M1 and �M2 for a given trajectory ξ (t) precess mainly in opposite direc-tions, namely precess either in the same or in the opposite direction. The concurrency of thetrajectories ξ (t) is evaluated by introducing the quantity

Cξ = 1

τ

∫ τ

0sign

dϕ1 [ξ (t)]

dt

dϕ2 [ξ (t)]

dtdt. (32)

In particular, one has Cξ = ±1 for perfectly concurrent movers (movers that always precessin the same direction) and perfectly anti-concurrent movers (movers that always precess inthe opposite direction), respectively, and Cξ > 0 for trajectories where angular momentamove concurrently more than half of the time for some members of the ensemble. In thelow-entanglement regime anticoncurrent movers dominate whereas the distribution of con-current movers progressively dominates as entanglement increases. The concurrency (31) ischaracterized by the probability distribution

dP(Cξ

)

dCξ

= P+δ(Cξ − 1

) + P−δ(Cξ + 1

) + ρ(Cξ

)(33)

where P± is the probability that the concurrency is exactly ±1, respectively, and ρ(Cξ

)

is a continuous function for which motion is sometimes concurrent and sometimes anti-concurrent as t changes. In the context of the geometric entropic approach based on(24)–(29), the concurrency (32) of the trajectories of the angular momentum vectors andtheir distribution probability (33) are the results of the behaviours of the quantum torques(28) and (29) and thus of the geometry of the configuration space associated with the quan-tum entropy: the quantum information associated with the quantum potential (24) and thusthe quantum-entropic length (25) generates the quantum torques (28) and (29) which, byexhibiting a precessional motion, determine the fact that the angular momentum vectors of


Int J Theor Phys

each representative of the ensemble of a two qubits pair precess in unison or not as well asthe characteristic probabilities of these concurrent or anti-concurrent motions.

Another relevant quantity describing the correlations in a two qubits system is the proba-bility distribution of the average cosine dP 〈cos(ϕ−φ)〉

d〈cos(ϕ−φ)〉 which provides an excellent measure ofentanglement, ranging from zero for an unentangled state to unity for a maximally entangledstate. In this regard, one can prove the following expressions

φ = 〈ϕ〉B , (34)

CB = 〈cos (ϕ − φ)〉B =√

〈cos ϕ〉2B + 〈sin φ〉2

B (35)

where CB is dependent only of ϑ . A higher degree of entanglement can be thus visualizedas a highly correlated distribution of angular momenta making azimuthal angles differenceclose to φ, with suppressed fluctuations for progressively increasing entanglement. In theentropic approach here proposed, the quantity CB given by (35) can be expressed as

CB = 2∣∣∣〈ψ | �T1 · �T2 |〉

∣∣∣ = |sin ϑ | (36)

namely

CB = 2 〈ψ | − 1

L4quantum

�M1 · �M2 |〉 = |sin ϑ | (37)

According to (36) and (37) the quantity CB associated with the probability distribution ofthe average cosine is determined by the quantum torques (28) and (29), in other words bythe Bell length, that is to say by the quantum entropy. For CB → 1 the system tends to reachthe maximum correlation between the two qubits whilst for CB → 0 the entanglement tendsto be suppressed.

Moreover, Ramsak showed also that the angular momenta �M1 and �M2 of the two qubitsform an angle � in the Bohmian space whose average cosine is given by the followingexpression:

〈cos �〉 =⟨ �M1 · �M2

|M1M2|

⟩

. (38)

As regards the correlator (38), it is interesting to underline that it has a similar behaviour tothe one characterizing the quantity CB and thus, in the geometric approach suggested in thisarticle, also the behaviour of this correlator can be seen as a consequence of the quantumtorques (28) and (29), in other words of the Bell length (25).

Finally, one has to emphasize that the Bohmian version of Bell’s inequalities leads tothe same results of standard quantum mechanics as regards the entanglement propertiesof a qubit pair. In the picture proposed by Ramsak, non-locality in entangled qubit pairsis associated with an instantaneous coupling between their angular momenta via quantumpotential. The angular momenta are characterized by a precession which forms a constantrelative azimuthal angle φ and it is this precession which determines correlators yieldingthe violation of the Bell inequalities. In Ramsak’s approach the Bohmian correlator has thefollowing form

B(�a, �b

)= 3

⟨(�a · �M1

) (�b · �M2

)

|M1| |M2|

⟩

= (axbx + ayby

)Bx cos φ

+ (aybx + axby

)Bx sin φ − azbzBz (39)

(where Bx,z are functions of ϑ). In the geometric entropic approach to quantum entangle-ment based on (24)–(29), the behaviour of the functions Bx,z appearing in the Bohmian


Int J Theor Phys

correlator (39) is determined by the quantum torques (28) and (29) which determine thecoupling of the angular momenta of entangled qubit pairs, and thus by the quantum-entropiclength (25). In synthesis, one can say that, on the basis of the geometric entropic approachto entangled qubit pairs based on (24)–(29), the ultimate source of quantum information isrepresented by the quantum torques (28) and (29), and thus by the quantum-entropic lengthdescribing the deformation of the geometry of the background described by the quantumentropy.

3 The Entropic Quantum Correlation Distance in Bell’s Inequalities

In 1964 John Stewart Bell moved the concept of entanglement from the philosophical debateto laboratory, introducing inequalities – known as Bell inequalities – which investigatecorrelations between two particles who share a quantum state [20]. The Bell inequalitiesconcern measurements made by observers on pairs of particles that have interacted and thenare separated. According to the quantum mechanical formalism they are entangled, whilelocal realism would limit the correlation of subsequent measurements of the particles. Onthe basis of the Bell inequalities, entanglement became thus a property of states which couldnot be described by local realistic theories. Many formulations of the Bell inequalities exist.For example, the well known Clauser, Horne, Shimony, and Holt (CHSH) inequality andClauser and Horne (CH) inequality [21, 22] are used for the verification of nonlocal corre-lations in a two-dimensional Hilbert space. The Bell inequality of a qubit pair in the formderived by Clauser, Horne, Shimony and Holt can be formulated as

T r |(ρBCHSH )| ≤ 2 (40)

where the Bell-CHSH operator is

BCHSH = �a · �σ ⊗(�b + �b′) · �σ + �a′ · �σ ⊗

(�b − �b′) · �σ , (41)

and its expected value is maximized over real-valued three-dimensional unit vectors �a, �a′, �band �b′ (here �σ are the usual Pauli matrices). The states for which T r |(ρBCHSH )| > 2 arethe ones which cannot be described by a realistic local theory and are thus associated withnon-classical correlation and non-locality.

Another important evolution of the concept of entanglement came in 1989, with thework of R. F. Werner [23]. Werner defined entangled states as the ones which could notbe created using local operations on the systems with classical communication between theparts. In other words, Werner proposed a definition of separability for mixed states froman operational point of view: separable states are the ones which can be produced by localoperations and classical communication. They can be written as

ρAB =∑

k

pkρ(k)A ⊗ ρ

(k)B (42)

with 0 ≤ pk ≤ 1 and∑

k

pk = 1. Separable mixed states can be correlated, but this corre-

lation is a classical one that comes from the probabilities pk . Moreover, Werner showed thatentanglement does not provide a sufficient condition to guarantee the nonlocal features ofa mixed state, by constructing entangled states which do not violate the Bell inequality. Asregards the link between entanglement and nonlocality, it was only in 1991 that, by virtueof Gisin’s results about pure bipartite states, entanglement emerged as a necessary and suf-ficient condition to determine nonlocality [24]. In the light of these research, the interesting


Int J Theor Phys

perspective was opened to provide a generalization of the Bell inequality which can yield aunifying picture between non-locality and entanglement. In this regards, one can mention,for example, Popescu’s 1995 work (which shows that, by starting from an entangled statewhich does not violate the Bell inequality, one may obtain a new state which violates theBell inequality, by means of local operations and classical communication), the works ofCollins and Gisin, of Yeong-Cherng Liang which explore a new inequality that is violatedby states which do not violate the Bell-CHSH inequality, as well as the work of Methot andScarani which introduces other definitions of non-locality [25–28]

In the recent paper Two-qubit mixed states more entangled than pure states: comparisonof the relative entropy of entanglement for a given nonlocality by Horst, Bartkiewicz andMiranowicz [29], a degree of violation of Bell-CHSH inequality and thus of quantum non-locality was evaluated by using the function

B (ρ) = √max [0, M (ρ) − 1] (43)

whereM (ρ) = max

j<k

{hj + hk

} ≤ 2 (44)

where hj are the eigenvalues of the real symmetric matrix U = T T T , Tij =T r

[ρ(σi ⊗ σj

)]being the correlation matrix, whilst the degree of entanglement was

measured by using the relative entropy of entanglement defined as

ER (ρ) = minσ∈D

S (ρ//σ) (45)

which is the relative entropy

S (ρ//σ) = T r(ρ log2 ρ − ρ log2 σ

)(46)

minimized over the set D of separable states σ . However, the relative entropy of entan-glement used by Horst, Bartkiewicz and Miranowicz is limited to distinguishing a densitymatrix ρ from the closest separable state σ only and does not function to provide a measureof the relative entropy of entanglement of a general two-qubit mixed state. In the case oftwo qubits A and B described by the density operator ρAB and reduced density operatorsρA and ρB , an interesting measuring system of the degree of entanglement is provided byHall’s quantum correlation distance [30]

C (ρAB) > 2√(

1 − T r[ρ2

A

]) (1 − T r

[ρ2

B

])(47)

which can also be expressed as

C (ρAB) = 1

4T r

∣∣∣∣∣∣

∑

j,k

Tjkσj ⊗ σk

∣∣∣∣∣∣

(48)

where T denotes the 3 × 3 spin covariance matrix with coefficients

Tjk = ⟨σj ⊗ σk

⟩ − ⟨σj ⊗ I

⟩ 〈I ⊗ σk〉 . (49)

Hall’s quantum correlation distance (47) leads to the following expression for the lowerbound of the quantum mutual information shared by the two qubits A and B:

I (ρAB)≥⎧⎨

⎩

log 2 − H(

1+Lquantum

2 ,1−Lquantum

2

), C(ρAB)≤0,72654

log 4−H(

14 + Lquantum

2 , 14 − Lquantum

6 , 14− Lquantum

6 , 14 − Lquantum

6

), C(ρAB)>0,72654

(50)


Int J Theor Phys

where H is the Shannon entropy of the probability distribution of the system of the twoqubits into consideration.

The deformation of the geometry – with respect to the classical case – determined by anentangled qubit pair is associated with the quantum entropic Bell length (25) and thus thestrong condition for the entanglement between the two qubits is represented by the conditionthat the quantum entropic length (25) satisfies relation (47). By using (47) and taking intoaccount that, in terms of the quantum entropy, ρA = exp

(−2SQA

)and ρB = exp

(−2SQB

),

one obtains

1√

12I

(( �M21 SQ

)−

( �M1SQ

)2+

( �M22 SQ

)−

( �M2SQ

)2)

> 2

√(1 − T r

[(exp

(−2SQA

))2]) (

1 − T r[(

exp(−2SQB

))2])

(51)

which may be expressed also as

1( �M2

1 SQ

)−

( �M1SQ

)2+

( �M22 SQ

)−

( �M2SQ

)2

>2

I

(1 − T r

[(exp

(−2SQA

))2]) (

1 − T r[(

exp(−2SQB

))2])

. (52)

In analogy with Hall’s approach (where the quantum correlation distance (47) leads to thequantum mutual information (49)), a tight lower bound for the quantum mutual informationshared by two qubits corresponds to the quantum entropic Bell length (25), which is givenby relations:

I(Lquantum

)≥⎧⎨

⎩

log 2 − SQ

(1+Lquantum

2 ,1−Lquantum

2

), Lquantum ≤ 0, 72654

log 4−SQ

(14 + Lquantum

2 , 14 − Lquantum

6 , 14 − Lquantum

6 , 14 − Lquantum

6

),Lquantum > 0, 72654

(53)where I

(Lquantum

) = SQA+ SQB

− SQAB. For Lquantum ≤ 0, 72654 this lower bound

can only be achieved by entangled states, and cannot be achieved by any classical distri-bution PAB having the same correlation distance. It is also interesting to remark that, forLquantum > 0, 72654, the bound is tight if only one of the reduced states is maximallymixed. We suggest that the bound in relation (53) holds for all two-qubit states.

The crucial feature of quantum correlations is that they cannot be explained by anyunderlying statistical model that satisfies three physically very plausible properties: (i) noinformation transfer which takes place faster than the speed of light, (ii) free choice ofmeasurement settings, and (iii) independence of local outcomes. Various interpretations ofquantum mechanics differ in regard to which of these properties should be given up. It isof interest to determine how many information-theoretic resources are required to simulatea given quantum correlation. To be more explicit, one could ask himself how many bits ofcommunication, or bits of correlation between the source and the measurement settings, orbits of correlation between the outcomes, are required.

In the approach here proposed, the correlation between the two qubits is the Bell length(25), which indeed plays the role of Hall’s quantum correlation distance (47). In otherwords, according to the approach here proposed, the quantum correlation distance (47) isassimilated to the Bell length (25), which can be considered as the ultimate visiting card


Int J Theor Phys

determining the quantum correlations. In order to clarify this point, before all, let us con-sider the joint probability PAB (a, b) of outcomes a and b, for measurements of variables Aand B on respective spacelike-separated qubits and λ any underlying variables relevant forthe correlations. On the basis of the Bayes theorem the probability distribution PAB (a, b)

has the following form

PAB (a, b) =∑

λ

pAB (λ) PAB (a, b |λ) (54)

where summation is replaced by integration over any continuous values of λ. The prop-erty that there is no information transfer faster than light speed requires that the underlyingmarginal distribution of A, pA (a |λ), is independent of whether B or B′ was measured onthe second system (and vice versa), while the free-choice property requires that λ is inde-pendent of the choice of the measured variables A and B, i.e., that pAB (λ) = pA′B ′ (λ) forany A, A′, B, B′. Finally, the outcome independence property requires that any observedcorrelation between A and B arises from ignorance of the underlying variable, namelythat PAB (a, b |λ) = PA (a |λ) PB (b |λ) for all A, B and λ. Thus the quantum entropiccorrelation distance of PAB (a, b |λ) vanishes identically:

Lquantum

(PAB|λ

) = 0. (55)

As is well known, the assumption of all three properties implies that two-valued randomvariables with values ±1 must satisfy the Bell-CHSH inequality

CHSH ≡ 〈AB〉 + ⟨AB ′⟩ + ⟨

A′B⟩ − ⟨

A′B ′⟩ ≤ 2 (56)

whereas quantum correlations can violate this inequality by as much as a factor of√

2.Now, the Bell-CHSH inequality (56) can be easily expressed in terms of the Bell length(25). In fact, assuming that no-signalling and measurement independence hold, and definingLmax

quantum to be the maximum value of the Bell length Lquantum

(PAB|λ

)over all A, B and

λ, the standard Bell-CHSH inequality given by (56) may be generalized as

〈AB〉 + ⟨AB ′⟩ + ⟨

A′B⟩ − ⟨

A′B ′⟩ ≤ 4

2 − Lmaxquantum

. (57)

In the approach based on the quantum entropic correlation distance (51), it follows thereforethat a Bell inequality violation can be simulated by the following relation

〈AB〉 + ⟨AB ′⟩ + ⟨

A′B⟩ − ⟨

A′B ′⟩ = 2 + V (58)

for some V> 0, if the observers share random variables having a quantum entropiccorrelation distance of at least

Lmaxquantum ≥ 2V

2 + V. (59)

As a consequence, the observers must share a minimum mutual information of

Imin = log 2−SQ

(1 + Lmax

quantum

2,

1 − Lmaxquantum

2

)≥ log 2−SQ

(2 + 3V

4 + 2V,

2 − V

4 + 2V

). (60)

The mutual information (60) reduces to zero in the limit of no violation of Bell inequality,i.e. when V= 0, and reaches a maximum of 1 bit of information in the limit of the maximumpossible violation, V= 2, namely for Lmax

quantum = 1, which is the limit value of the Belllength, beyond which the de-correlation between the two qubits begins. Therefore, on the


Int J Theor Phys

basis of the quantum entropic correlation distance (51), an entropic length Bell inequalitycan be introduced which corresponds to the Bell inequality (56)

CHSHentropic ≤ 0 (61)

where

CHSHentropic = 〈AB〉 + ⟨AB ′⟩ + ⟨

A′B⟩ − ⟨

A′B ′⟩ − 4

2 − Lmaxquantum

. (62)

The violation of the entropic length Bell inequality (61) essentially coincides withthe one of the standard CHSH inequality. As regards the entangled state of a twoqubit pair (20), the maximal violation of the entropic length Bell inequality (61),which corresponds to V=2, namely to Lmax

quantum = 1, is obtained for ϑ = π2 ,

on which one gets CHSHentropic ≈ +0, 237. For other values of ϑ , the max-imal violation of (61) when optimized over the measurements, follows the exactsame profile as for the standard Bell-CHSH inequality (56). However, the measure-ments that maximize the violation of CHSH are not the ones which give the max-imal violation of CHSHentropic. In general, for the standard Bell-CHSH scenario,the violation of the standard inequality (56) is a necessary but not sufficient condition forthe violation of (61). The advantage of the entropic length Bell inequality (61) lies in thefact that the results depend directly on the Bell length which emerges directly as correla-tion degree in quantum systems and which provides just an entropic quantum correlationdistance.

The maximum correlation in the two qubits system described by the general state (21), isdetermined by the limit value 1 of the Bell length. On the other hand, the maximum degreeof entanglement corresponds also to the limit value 1 of the probability distribution of theaverage cosine (35) (or (36), that is the same). This implies that there is a fundamentallink between the Bell length and the probability distribution of the average cosine (35). Inparticular, on the basis of (36) and (37), one can say that the peaking of the entanglementcorresponding to the maximum value of the Bell length Lmax

quantum = 1 is associated with theprecessional motion of the quantum torques (28) and (29). In other words, one can say thatthe geometrical properties associated with the quantum potential (24), expressed by the Belllength (25), generate the quantum torques (28) and (29) whose precessional motions lead toa peaking of the probability distribution (31) in correspondence of the maximum value ofthe Bell length Lmax

quantum = 1. Finally, through the Bell length it is possible to introduce ageneralization of Bell inequalities, which takes explicitly account of the physical complexityof the interplay between information and entropy in a quantum system.

It becomes therefore permissible the following re-reading of the spin-spin correlations ina two qubits system characterized by the wave function (20): the quantum entropy leads tothe quantum potential (24) which corresponds to a deformation of the geometry describedby the Bell length (25); the ultimate visiting card describing the geometrical properties asso-ciated with the quantum potential (24), namely the Bell length (25), generate the quantumtorques (28) and (29) whose precessional motions lead to a peaking of the probability dis-tribution (31) in correspondence of the maximum value of the Bell length Lmax

quantum = 1;this maximum value of the Bell length leads to the maximum violation of an entropic Bellinequality of the form (61) which may be therefore considered as a generalization of thestandard Bell inequality.

Moreover, as shown recently by Chaves and Fritz [31, 32], violations of entropic inequal-ities introduce a very particular kind of contextuality : if a probabilistic model violates anentropic inequality, then so does every other probabilistic model obtained by permutingthe outcome probabilities of a joint measurement, if the permuted joint distribution has the


Int J Theor Phys

same marginal scenarios defined by specifying a set of observables for which certain sub-sets are known to be compatible and can be jointly measured. For example this leads to thephenomenon that the Popescu-Rohrlich box

P PR (a, b |x, y ) = 1

4

[1 + (−1)a⊕b⊕xy

](63)

defined to be the unique marginal model which maximally violates the standard Bell-CHSHinequality (56), is entropically indistinguishable from the marginal model

P C (a, b |x, y ) = 1

4

[1 + (−1)a⊕b

](64)

describing classical correlations. On the basis of Chaves’ and Fritz’s results, one can saythat, in the entropic approach of Bell’s inequalities here suggested, entropic quantities asso-ciated with the quantum entropic length (or Bell length) cannot distinguish between theperfect anti-correlation of A′ and B′ as it appears in (63), and the perfect correlation of A′and B′ as in (58). In the light of Chaves’ and Fritz’s results, for any marginal model withtwo-outcome measurements one has

CHSHentropic ≤ 1 (65)

namely the maximal violation of (61) is +1.This boundary for the violation of Bell-CHSH scenario can be obtained by the no-

signalling box

P max = 1

2P PR + 1

2P C. (66)

which is an equal mixture of the Popescu-Rohrlich box with classical correlations. P max canbe seen as the probabilistic model in which each of the three pairs (A,B), (A,B′) and (A′,B)displays perfect correlation, while the fourth pair (A′,B′) is uncorrelated. P max achievesa value of 3 on the standard CHSH scenario, and therefore does not have a quantum-mechanical realization since it is beyond Tsirelson’s bound of 2

√2. This example shows

that a convex combination of two non-violating marginal models may violate an entropicinequality like (61). This puts in evidence the non-linear character of the entropic lengthBell inequality (61) which is associated with the non-Euclidean geometry described by theBell length (25).

4 The Berry Phase in the Geometric Entropic Approach

In the paper Gauge Theory and Two Level Systems, Bruno, Capolupo, Kak, Raimondo andVitiello showed that a gauge structure and a Berry-like geometric phase characterizes theinitialization process of a qubit in quantum computing [33]. Their approach suggests a shiftof paradigm in the study of quantum computing: the Berry-like phase description of thesystem and its gauge invariant behaviour are considered as properties which derive from thecollective behaviour of fluctuating quantum phases of qubit states.

The Berry-like phase involved in the qubit time evolution as regards the entangled state(20) for the n-cycle case is

β(n)φ = −2πω1

ϕ+

∫ nT

0

⟨φ (t)

∣∣∣∣i

∂

∂t

∣∣∣∣φ (t)

⟩dt = 2πn sin2 ϑ

2(67)


Int J Theor Phys

where T is the period of oscillation,

|φ (t)〉 = eiω1t

(cos

ϑ

2|↑↓〉 + eiω sin

ϑ

2|↑↓〉

), (68)

ϕ = ω1 − ω2, (69)

ω1 and ω2 being the eigenvalues (having the dimensions of frequencies) of the generator ofphase transformation

H = ω1 |↑↓〉 + ω2 |↑↓〉 . (70)

The Berry-like phase turns out to be physically equivalent to the Anandan-Aharonovinvariant

s = 2πn sin ϑ (71)

representing the distance between qubit evolution states, as measured by the Fubini-Studymetric, in the projective Hilbert space (which is the set of rays of the Hilbert space intoconsideration). Equation (67) can also be rewritten in the following convenient form

β(n)φ =

∫ nT

0

⟨φ (t)

∣∣∣∣i

∂

∂t

∣∣∣∣ φ (t)


2(72)

where ∣∣∣φ (t)⟩= e−if(t)

(cos

ϑ

2|↑↓〉 + eiω sin

ϑ

2|↑↓〉

), (73)

f (t) = f (0) − ω1t , where f(0) is a constant.Now, in the geometric entropic approach to entangled qubit pairs illustrated in chapter 2,

the ultimate source of quantum non-local information are the quantum torques (28) and (29)which derive from the deformation of the geometry associated with the quantum entropyand expressed by the quantum-entropic length (25). We propose therefore that the geometricinvariant representing the distance between qubit evolution states, and thus the Berry-likephase, emerges from the Bell length, which is the ultimate visiting card describing the cor-relations between the two qubits. Taking account of (36), the Anandan-Aharonov invariant(71) may be conveniently written as

s = 4πn 〈ψ | �T1 · �T2 |ψ〉 (74)

which shows in what sense it emerges from the quantum torques (28) and (29). Takingaccount (28) and (29), equation (74) may also be explicitly expressed as

s = 4πn 〈ψ | − 1

L4quantum

�M1 · �M2 |ψ〉 . (75)

In virtue of the equivalence between the Anandan-Aharonov invariant (71) and the Berry-like phase (67), according to the authors it becomes thus legitimate to assume that thegeometric invariant representing the distance between qubit evolution states, and thus theBerry-like phase, corresponds to the quantity (74). Therefore one has

β(n)φ =

∫ nT

0

⟨φ (t)

∣∣∣ i

∂

∂t

∣∣∣φ (t)


24πn 〈ψ | − 1

L4quantum

�M1 · �M2 |ψ〉 (76)

which can be considered the definition of the Berry-like phase in the entropic approachproposed by the authors. According to (76), one can say that the ultimate source determiningthe Berry-like phase involved in the qubit time evolution as regards the entangled state(20) is the Bell length and thus the quantum entropy. In other words, one can say that theBerry-like phase characterization of the system into consideration can be seen as a propertywhich emerges from a more fundamental deformation of the geometrical properties of the


Int J Theor Phys

configuration space associated with the Bell length and thus with the quantum entropy ofthe system. We can define the quantity (76) as the geometric entropic Berry-like phaseregarding the two qubit system in the state (20).

Moreover, one can consider |φ (t)〉 → e−if(t) |φ (t)〉 =∣∣∣φ (t)

⟩as a local gauge transfor-

mation of the state |φ (t)〉. On the basis of (70), one can observe that time evolution only

affects the |↑↓〉 component of the state∣∣∣φ (t)

⟩. The gauge transformation acts as a “fil-

ter” freezing out (or compensating) time evolution of the |↑↓〉 state component, so that oneobtains

i�∂

∂t

∣∣∣φ (t)⟩=−ϕe−if(0)eiϕ sin

ϑ

2|↑↓〉 = (H − ω1) e−if(0)

(cos

ϑ

e|↑↓〉+ eiϕ sin

ϑ

2|↑↓〉

)

= (H − ω1)

∣∣∣φ (t)

⟩(77)

Now, considering the state |φ(t)〉 given by (68) and the state

|ψ(t)〉 = e−iω1t

(− sin

ϑ

2|↑↓〉 + eiω cos

ϑ

2|↑↓〉

)(78)

one has

H |φ(t)〉 = ωφφ |φ(t)〉 + ωφψ |ψ(t)〉 , (79)

H |ψ(t)〉 = ωψψ |ψ(t)〉 + ωφψ |φ(t)〉 (80)

where

ωφφ = ω1 cos2 ϑ

2+ ω2 sin2 ϑ

2= 〈φ (t)|H |φ(t)〉 (81)

ωψψ = ω1 sin2 ϑ

2+ ω2 cos2 ϑ

2= 〈ψ (t)| H |ψ(t)〉 (82)

ωφψ = 1

2(ω2 − ω1) sin ϑ = 〈ψ (t)|H |φ(t)〉 . (83)

Equations (79) and (80) can be written in the following compact form(H − ωφψ (|φ(t)〉〈ψ(t)| + |ψ(t)〉〈φ(t)|)) |ζ(t)〉 = ωd |ζ(t)〉 (84)

where ωd = diag(ωφφ, ωψψ

). In (77) the operator

F = (H − ωφψ (|φ(t)〉〈ψ(t)| + |ψ(t)〉〈φ(t)|)) (85)

may be interpreted as the free energy operator and may also be expressed as

F = λSQ (86)

where λ is the temperature of the entangled qubit pair and SQ is the quantum entropy givenby (1).

Another possible manner to generate a Berry phase has been shown theoretically inqubits flux by applying an external field . As a consequence, varying the interactionstrengths and applying an external field may provoke both the entanglement and the phasedynamics of the subsystems to change [34, 35]. In Relation between quantum entanglementand the Berry phase in systems of two interacting qubits [36], Chen, Cho and Su inves-tigated quantum entanglement and the Berry phase in systems of two interacting qubits,where the interaction has the form of an XXZ-type exchange interaction, which makes thequbits entangled.


Int J Theor Phys

In the context of a XXZ Hamiltonian of the form

H =Hx+Hz+HB =

⎛

⎜⎜⎝

2B0 cos θ + Jz B0 sin θe−iφ B0 sin θe−iφ 0B0 sin θeiϕ −Jz 2Jx B0 sin θe−iφ

B0 sin θeiφ 2Jx −Jz B0 sin θe−iφ

0 B0 sin θeiφ B0 sin θeiφ −2B0 cos θ + Jz

⎞

⎟⎟⎠ (87)

where Hx = Jx

(S1xS2x + S1yS2y

), Hz = JzS1zS2z and HB = μ

(�S1 + �S2

)· �B (t),

μ being the gyromagnetic ratio, Jx and Jz the exchange interaction strengths betweenthe qubits, �B (t) = Bn (t) a slowly rotating external field with the unit vector n (t) =(sin θ cos ϕ (t), sin θ sin ϕ (t), cos θ), the geometry of the system determines a Berry phaseof the form

βn =∫ 2π

0dφ 〈ψn| i∂φ |ψn〉 = 16πB3

0εn sin θ sin 2θ

ε4n − 2 (1 + 3 cos 2θ) B2

0ε2n + 16B4

0 cos2 θ(88)

where

εn = 2√

Q cos

(P + 2nπ

3

)+ 2

3J, (89)

Q = 4(J 2 + 3B2

0

)/9, (90)

P = cos−1

(R

√Q3

)

, (91)

R = 4J(

2J 2 + 9B20

(1 − 3 cos2 θ

))/27. (92)

In the geometric entropic approach suggested in this article, the Berry phase (88) charac-terizing systems of interacting qubits where the interaction Hamiltonian has the form (87)may be written as:

βn = 16πB30εn sin θ sin 2θ

ε4n − 2 (1 + 3 cos 2θ) B2

0ε2n + 16B4

0 cos2 θ= 4πn 〈ψ | − 1

L4quantum

�M1 · �M2 |ψ〉 .

(93)According to (93), one can say that the interaction between the two qubits determines adeformation of the geometrical properties of the configuration space which correspondsto the Bell length and thus to the quantum entropy of the system. Equation (93) allowsus to provide a geometrical re-reading to Chen’s, Cho’s and Su’s model of entangledqubit pairs and Berry phase in which the external field and the exchange interactionstrengths describing the interaction between the two qubits are associated with a defor-mation of the configuration space expressed by the Bell length and thus by the quantumentropy.

5 Conclusions: Perspectives in Geometric Approach to Quantum Information

In the light of the treatment of this paper, the geometric entropic Berry-like phase (76),the geometric entropic Berry phase regarding two interacting qubits with external field(93) and the Bell length acting as a quantum correlation distance (51) (and its corre-sponding quantum mutual information (53) can be considered as the starting-points tothrow new light on geometric quantum computation. According to (76) and (93) onecan say that the deformation of the geometrical properties of the configuration spaceexpressed by the Bell length plays a crucial role in the analysis and characterization of


Int J Theor Phys

quantum computing. In particular, they imply that the so-called initialization problemin qubit preparation turns out to be strictly related with the Bell length. According to(51) the Bell length associated with the quantum entropy and acting as a quantum cor-relation distance emerges as entity providing the strong condition for the entanglementbetween two qubits. The quantum correlation distance (51) and its corresponding quan-tum mutual information (53) emerge as the ultimate visiting cards which determine a Bellinequality violation of a two-qubit pair. In particular, they define an entropic length Bellinequality which can be regarded as a generalization of the standard Bell-CHSH inequal-ity and whose violation essentially coincides with the one of the standard Bell-CHSHinequality.

On the other hand, it was recently suggested that the complexity of quantum gatetesting is exponentially related to the number of observational variables [18]. Here, theperspective is opened that the complexity of the quantum gate testing is directly relatedwith the quantum entropy of the system and, above all, with the Bell length. The Belllength determines the action of the Berry phase (76) and the corresponding Berry phasewith external field (93) and this makes the information space associated with the quan-tum entropy the fundamental arena which is related with the complexity of quantumgate testing. In particular, if the Berry phases (76) or (93) satisfy the condition (51),namely

2πn sin2 ϑ

2= 4πn 〈ψ | − 1

L4quantum

�M1 · �M2 |ψ〉

> 2

√(1 − T r

[(exp

(−2SQA

))2]) (

1 − T r[(

exp(−2SQB

))2])

(94)

and

16πB30εn sin θ sin 2θ

ε4n − 2 (1 + 3 cos 2θ) B2

0ε2n + 16B4

0 cos2 θ= 4πn 〈ψ | − 1

L4quantum

�M1 · �M2 |ψ〉

> 2

√(1 − T r

[(exp

(−2SQA

))2]) (

1 − T r[(

exp(−2SQB

))2])

(95)

a suggestive picture directly emerges in which the complexity of the gate testing is associ-ated with the entropic quantum correlation distance (Bell length) defining the non-localityand correlation of the qubits under consideration.

Focusing the attention on the global geometry suggests an alternative view of com-putation, not as a “counting” of qbits, but – in line with the etymological significanceof “information” – as a modification of the configuration space in specific experimentalconfigurations. The so discussed hypercomputational potentialities – far from being limitsituations only occurring in exotic theoretical scenarios – become in this way actual possi-bilities [19]. In this picture, the term “programming” acquires a completely new meaningjust in relation to the particular geometry of the configuration space associated with theentropic quantum correlation distance defining the non-locality between the qubits. We areproposing here that the difference between classical (Shannon-Turing) and quantum com-puting is similar to the difference between Euclidean and non-Euclidean geometry in theconfiguration space associated with the quantum entropy, in the condition of the quantumcorrelation distance (51).


Int J Theor Phys

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Bell Length in the Entanglement Geometry

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