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and the distance becomes DQG Wy, L(y =4 z[1-(ol)? ~(o3)? = (ol)*]. Since Q(|w)) contains a sum over all qubits [see Eq. (C3)], we finally obtain where * denotes complex conjugate. After some simple cal- culations, we obtain the following relations:

Figure 23 and the distance becomes DQG Wy, L(y =4 z[1-(ol)? ~(o3)? = (ol)*]. Since Q(|w)) contains a sum over all qubits [see Eq. (C3)], we finally obtain where * denotes complex conjugate. After some simple cal- culations, we obtain the following relations:

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where (A_,) denotes the expectation value of the observable A., in the pure state |). “An important property following is that the -purity is invariant under group transformations: if a new basis for b is introduced by letting A,=D'A,D, with D =expGid}_tgA,), D'D =1, and tg real numbers, then one finds
where (A_,) denotes the expectation value of the observable A., in the pure state |). “An important property following is that the -purity is invariant under group transformations: if a new basis for b is introduced by letting A,=D'A,D, with D =expGid}_tgA,), D'D =1, and tg real numbers, then one finds
For simplicity, we begin by studying the GE of a two- qubit system (two-spin- 5), where the most general, pure 5) +b] 5 a =) +c|-3,5)+d|-5,-4), with the complex numbers a,b,c, and d satisfying al? +b/?+[0)2+(4)2=1. The traditional measures of pure-state entanglement in this case are well understood, indicating that the Bell states |GHZ? 1/2) [36] (and its local spin rotations) are maximally entangled with respect to the local Hilbert space decomposition 7;® Hy. On the other hand, calculating the purity relative to the (irreducible) Lie algebra of all local observables h=su(2); @su(2), ={o),;1:1,2;a=x,y,z} classifies the pure two- spin- 5 states in the same way as the traditional measures do (see Fig. 1). Here, the operators ol} =0,,@1 and o2=1@0, are the Pauli operators acting on spin 1 “and 2, respectively, and where Bell’s states are maximally entangled (P,=0) and product states of the form |~)=|¢1);®|d2)2 (GCSs of the local algebra h above) are generalized unentangled, with maximum purity. Therefore, the normalization factor K =} may be obtained by setting P,, =1 in such a product state. As explained in Sec. II, P,, is invariant under group operations, i.e., in this case, local rotations. Since all GCSs of h belong to the same orbit generated by the application of group op- erations to a particular product state (a reference state like ;, S)= 1,7)), they all consistently have maximum h-purity (Py =1 ) .
For simplicity, we begin by studying the GE of a two- qubit system (two-spin- 5), where the most general, pure 5) +b] 5 a =) +c|-3,5)+d|-5,-4), with the complex numbers a,b,c, and d satisfying al? +b/?+[0)2+(4)2=1. The traditional measures of pure-state entanglement in this case are well understood, indicating that the Bell states |GHZ? 1/2) [36] (and its local spin rotations) are maximally entangled with respect to the local Hilbert space decomposition 7;® Hy. On the other hand, calculating the purity relative to the (irreducible) Lie algebra of all local observables h=su(2); @su(2), ={o),;1:1,2;a=x,y,z} classifies the pure two- spin- 5 states in the same way as the traditional measures do (see Fig. 1). Here, the operators ol} =0,,@1 and o2=1@0, are the Pauli operators acting on spin 1 “and 2, respectively, and where Bell’s states are maximally entangled (P,=0) and product states of the form |~)=|¢1);®|d2)2 (GCSs of the local algebra h above) are generalized unentangled, with maximum purity. Therefore, the normalization factor K =} may be obtained by setting P,, =1 in such a product state. As explained in Sec. II, P,, is invariant under group operations, i.e., in this case, local rotations. Since all GCSs of h belong to the same orbit generated by the application of group op- erations to a particular product state (a reference state like ;, S)= 1,7)), they all consistently have maximum h-purity (Py =1 ) .
Another important insight may be gained by calculating the purity relative to the algebra of all observables for the system, h=su(4)={o1,, 01, o:i=1,2; a, B=x,y,z} in this case. One finds that any two-spin-5 pure state |) (including Bell’s states) is then generalized unentangled (P, =1, see also Fig. 1). This property is a manifestation of the relative nature of GE, as considering the set of all observables as being physically accessible is equivalent to not making any pre- ferred subsystem decomposition. Accordingly, in this case any pure quantum state becomes a GCS of su(4). In Fig. 1, we also show the GE for systems of two parties of spin S relative to different algebras. We observe that the purity reduces again to the traditional concept of entangle- ment for higher spin if it is calculated relative to the (irre- ducible) Lie algebra of all local observables h =su(2S +1), ®su(2S+1). For example, if we are interested in distin- guishing product states from entangled states in a two-spin-1 system, we need to calculate the purity relative to the (irre- ducible) algebra h =su(3); @su(3).={AlL@1?, 1} @r2(1<a =<8)}, where the 33 Hermitian and traceless matrices ),; are the well-known Gell-Mann matrices [27],
Another important insight may be gained by calculating the purity relative to the algebra of all observables for the system, h=su(4)={o1,, 01, o:i=1,2; a, B=x,y,z} in this case. One finds that any two-spin-5 pure state |) (including Bell’s states) is then generalized unentangled (P, =1, see also Fig. 1). This property is a manifestation of the relative nature of GE, as considering the set of all observables as being physically accessible is equivalent to not making any pre- ferred subsystem decomposition. Accordingly, in this case any pure quantum state becomes a GCS of su(4). In Fig. 1, we also show the GE for systems of two parties of spin S relative to different algebras. We observe that the purity reduces again to the traditional concept of entangle- ment for higher spin if it is calculated relative to the (irre- ducible) Lie algebra of all local observables h =su(2S +1), ®su(2S+1). For example, if we are interested in distin- guishing product states from entangled states in a two-spin-1 system, we need to calculate the purity relative to the (irre- ducible) algebra h =su(3); @su(3).={AlL@1?, 1} @r2(1<a =<8)}, where the 33 Hermitian and traceless matrices ),; are the well-known Gell-Mann matrices [27],
which satisfy Tr[\ ,Ag]=6z,g. In this basis, the computational spin-1 states are represented by the three-dimensional vec- tors Then, the relative purity for a generic pure state |) becomes where (\',) denotes the expectation value of \', in the state |W). In this way, product states like |~)=|¢,);@|d@o)2 are generalized unentangled (P,,=1) and states like |G HZ‘) (and states connected through local spin unitary operations) are maximally entangled in this algebra (P, =0). FIG. 1. Purity relative to different possible al- gebras for a two-spin-S system. The quantum states |GHZ2) and |F 2) are defined in Eqs. (9).
which satisfy Tr[\ ,Ag]=6z,g. In this basis, the computational spin-1 states are represented by the three-dimensional vec- tors Then, the relative purity for a generic pure state |) becomes where (\',) denotes the expectation value of \', in the state |W). In this way, product states like |~)=|¢,);@|d@o)2 are generalized unentangled (P,,=1) and states like |G HZ‘) (and states connected through local spin unitary operations) are maximally entangled in this algebra (P, =0). FIG. 1. Purity relative to different possible al- gebras for a two-spin-S system. The quantum states |GHZ2) and |F 2) are defined in Eqs. (9).
and the 2 x 2 matrices o, and 1 are given in Eq. (10). Then, the local purity becomes The traditional concept of pure multipartite entanglement in an N-spin-S quantum system refers to quantum states that cannot be written as a product of states of each party. The h- purity distinguishes pure product states from entangled ones if it is calculated relative to the (irreducible) algebra of ocal observables h=@N,su(2S+1), (see Appendix I). By Eq. (6), the measure P,, is invariant under local unitary op- erations as desired. In "articular, the usual concept of en- tanglement in an N- qubit quantum state (N-spin- 5) can be recovered if the purity is calculated relative to the local al- gebra h= =e su(2)i= ={o!, oy ,ol, - oN, oy N oN, where the Pauli operators o! iWA@=X,Y, 2) are now
and the 2 x 2 matrices o, and 1 are given in Eq. (10). Then, the local purity becomes The traditional concept of pure multipartite entanglement in an N-spin-S quantum system refers to quantum states that cannot be written as a product of states of each party. The h- purity distinguishes pure product states from entangled ones if it is calculated relative to the (irreducible) algebra of ocal observables h=@N,su(2S+1), (see Appendix I). By Eq. (6), the measure P,, is invariant under local unitary op- erations as desired. In "articular, the usual concept of en- tanglement in an N- qubit quantum state (N-spin- 5) can be recovered if the purity is calculated relative to the local al- gebra h= =e su(2)i= ={o!, oy ,ol, - oN, oy N oN, where the Pauli operators o! iWA@=X,Y, 2) are now
otice that access to local angular momentum observables suffices to operationally characterize the system as describ- able in terms of two spin-1 particles (by imagining, for in- stance, performing a Stern-Gerlach type of experiment on each particle). Thus, even when a subsystem decomposition can be naturally identified from the beginning in this case, states which are manifestly separable (unentangled) in the standard sense may exhibit GE (see also A ppendix I). On the other hand, this is physically quite natural in the example, since there are no SU(2) x SU(2) group operations (local ro- tations) that are able to transform the state |0,0) into the unentangled product state |1, 1). have zero projection along z becomes generalized entangled relative to the (irreducible) algebra su(2), ®su(2). of local spin rotations, which is generated by {S’,;i:1,2;a=x,y,z}, the spin-1 angular momentum operators S, for each spin being given by
otice that access to local angular momentum observables suffices to operationally characterize the system as describ- able in terms of two spin-1 particles (by imagining, for in- stance, performing a Stern-Gerlach type of experiment on each particle). Thus, even when a subsystem decomposition can be naturally identified from the beginning in this case, states which are manifestly separable (unentangled) in the standard sense may exhibit GE (see also A ppendix I). On the other hand, this is physically quite natural in the example, since there are no SU(2) x SU(2) group operations (local ro- tations) that are able to transform the state |0,0) into the unentangled product state |1, 1). have zero projection along z becomes generalized entangled relative to the (irreducible) algebra su(2), ®su(2). of local spin rotations, which is generated by {S’,;i:1,2;a=x,y,z}, the spin-1 angular momentum operators S, for each spin being given by
FIG. 2. Purity relative to different algebras for an N-spin-S sys- tem. The quantum states |GHZ), |W3), and |FY) are defined in Egs. (9).
FIG. 2. Purity relative to different algebras for an N-spin-S sys- tem. The quantum states |GHZ), |W3), and |FY) are defined in Egs. (9).
Upon introducing the pseudospin operators which satisfy the su(2) commutation relations of the angular momentum algebra, where @=-o, and the fermionic operators c/.,(c,,) create (annihilate) a fermion in the level identified by the quantum numbers (k, o) and satisfy the fermionic commutation rela- tions given in Sec. III C. Thus, the interaction V scatters a pair of particles belonging to one of the levels, and the inter- action W scatters a pair of particles belonging to different levels. Note that the factor 1/N must be present in the inter- action terms for stability reasons, as the energy per particle must be finite in the thermodynamic limit.
Upon introducing the pseudospin operators which satisfy the su(2) commutation relations of the angular momentum algebra, where @=-o, and the fermionic operators c/.,(c,,) create (annihilate) a fermion in the level identified by the quantum numbers (k, o) and satisfy the fermionic commutation rela- tions given in Sec. III C. Thus, the interaction V scatters a pair of particles belonging to one of the levels, and the inter- action W scatters a pair of particles belonging to different levels. Note that the factor 1/N must be present in the inter- action terms for stability reasons, as the energy per particle must be finite in the thermodynamic limit.
FIG. 3. Angular momentum coordinates in the three- dimensional space.
FIG. 3. Angular momentum coordinates in the three- dimensional space.
FIG. 4. Representation of the classical ground state of the LMG model.
FIG. 4. Representation of the classical ground state of the LMG model.
FIG. 5. Ground-state energy per particle in the LMG model.
FIG. 5. Ground-state energy per particle in the LMG model.
FIG. 6. Purity relative to the observable J, in the ground state of the LMG model.
FIG. 6. Purity relative to the observable J, in the ground state of the LMG model.
FIG. 7. Anisotropic one-dimensional XY model in an extemal transverse magnetic field B. VI. ANISOTROPIC XY MODEL IN A TRANSVERSE MAGNETIC FIELD
FIG. 7. Anisotropic one-dimensional XY model in an extemal transverse magnetic field B. VI. ANISOTROPIC XY MODEL IN A TRANSVERSE MAGNETIC FIELD
FIG. 8. (Color) Order parameter M2 in the thermodynamic limit asa function of g for different anisotropies y. The critical point is at Jo=:
FIG. 8. (Color) Order parameter M2 in the thermodynamic limit asa function of g for different anisotropies y. The critical point is at Jo=:
Although this functi tion is continuous, its derivative is not and has a drastic change at g= a where the QPT occurs. More- over, P,, is minimum for g > 5 ;, implying maximum entangle- ment at the transi tion point “and in the ordered (ferromag- netic) phase. Remarkably, for g>4 and N-+co, where the ground state of the anisotropic XY model in a transverse magnetic field is twofold degenerate, P, remains invariant for arbitrary linear combinations of the two degenerate states. where the coefficients v;, can be obtained from Eqs. (43) and (44). In particular, for g=0 the spins are aligned with the magnetic field and the fully polarized |BCS), |,J,°**, |) state is generalized unentangled in this limit [a GCS of u(N) with P,,=1]. In the thermodynamic limit, the purity relative to the u(N) algebra can be obtained by inte- grating Eq. (48),
Although this functi tion is continuous, its derivative is not and has a drastic change at g= a where the QPT occurs. More- over, P,, is minimum for g > 5 ;, implying maximum entangle- ment at the transi tion point “and in the ordered (ferromag- netic) phase. Remarkably, for g>4 and N-+co, where the ground state of the anisotropic XY model in a transverse magnetic field is twofold degenerate, P, remains invariant for arbitrary linear combinations of the two degenerate states. where the coefficients v;, can be obtained from Eqs. (43) and (44). In particular, for g=0 the spins are aligned with the magnetic field and the fully polarized |BCS), |,J,°**, |) state is generalized unentangled in this limit [a GCS of u(N) with P,,=1]. In the thermodynamic limit, the purity relative to the u(N) algebra can be obtained by inte- grating Eq. (48),
FIG. 9. (Color) Shifted purity P |.) of the |BCS) as a function of g for different anisotropies y, Eq. (51). P {iy behaves like a Bisonien parameter for this model, sharply identifying the QPT at g,= “The critical behavior of the system is characterized by a power-law divergence of the correlation length «, which is defined such that for g<3, limj4)..(ojo))| ~exp(-|i—j|/¢). Thus, e—~ signals the emergence of long- range correlations in the ordered region g > . Near the criti- cal point (g—g,), the correlation length behaves as e~ (g, -g)~”, where v is a critical exponent and the value v=1 corresponds to the Ising universality class. Let the parameter d,=e"/€, The fact that the purity contains information about the critical properties of the model follows from the possi-
FIG. 9. (Color) Shifted purity P |.) of the |BCS) as a function of g for different anisotropies y, Eq. (51). P {iy behaves like a Bisonien parameter for this model, sharply identifying the QPT at g,= “The critical behavior of the system is characterized by a power-law divergence of the correlation length «, which is defined such that for g<3, limj4)..(ojo))| ~exp(-|i—j|/¢). Thus, e—~ signals the emergence of long- range correlations in the ordered region g > . Near the criti- cal point (g—g,), the correlation length behaves as e~ (g, -g)~”, where v is a critical exponent and the value v=1 corresponds to the Ising universality class. Let the parameter d,=e"/€, The fact that the purity contains information about the critical properties of the model follows from the possi-
As defined, for large g the purity P,, approaches a constant value which depends on y. It is convenient to remove such a dependence in the ordered phase by introducing a new quan- tity P/ =P,-[1/(1+y)] (shifted purity). We thus obtain The new function P;, behaves like a disorder parameter for the system, being zero in the ferromagnetic (ordered) phase and different from zero in the paramagnetic (ordered) one. The behavior of P,, as a function of g in the thermodynamic imit is depicted in Fig. 9 for different values of y. In the special case of the Ising model in a transverse magnetic field y=1), one has the simple behavior P/ => +-2g? for g <4 and P,=0 if g>5 3
As defined, for large g the purity P,, approaches a constant value which depends on y. It is convenient to remove such a dependence in the ordered phase by introducing a new quan- tity P/ =P,-[1/(1+y)] (shifted purity). We thus obtain The new function P;, behaves like a disorder parameter for the system, being zero in the ferromagnetic (ordered) phase and different from zero in the paramagnetic (ordered) one. The behavior of P,, as a function of g in the thermodynamic imit is depicted in Fig. 9 for different values of y. In the special case of the Ising model in a transverse magnetic field y=1), one has the simple behavior P/ => +-2g? for g <4 and P,=0 if g>5 3
FIG. 10. Scaling properties of the disorder parameter for aniso- tropy y=1. The exponent v=1 belongs to the Ising universality class.
FIG. 10. Scaling properties of the disorder parameter for aniso- tropy y=1. The exponent v=1 belongs to the Ising universality class.
FIG. 11. Distribution of the fermion number in the |BCS) state for a chain of N=400 sites and anisotropy y=1.
FIG. 11. Distribution of the fermion number in the |BCS) state for a chain of N=400 sites and anisotropy y=1.
FIG. 12. Nearest-neighbor concurrence and its derivative for the |BCS) state as a function of g in the isotropic XY model, y=0. Both curves correspond to the exact solution in the thermodynamic limit. The value of dC (1)/0g below g, is also zero as C(1) (not shown).
FIG. 12. Nearest-neighbor concurrence and its derivative for the |BCS) state as a function of g in the isotropic XY model, y=0. Both curves correspond to the exact solution in the thermodynamic limit. The value of dC (1)/0g below g, is also zero as C(1) (not shown).
FIG. 14. (Color) Derivative of the purity of the |BCS) state relative to the local algebra as a function of g for y=0.5 and dif- ferent lattice sizes. In this paper, we have explored the usefulness of general- ized entanglement (GE) for characterizing the broken (and
FIG. 14. (Color) Derivative of the purity of the |BCS) state relative to the local algebra as a function of g for y=0.5 and dif- ferent lattice sizes. In this paper, we have explored the usefulness of general- ized entanglement (GE) for characterizing the broken (and
FIG. 13. (Color) Purity of the |BCS) state relative to the local aneer an ,;su(2);, as a function of g for different anisotropies g.=4 ). The number of sites N=400 as in Fig. 11.
FIG. 13. (Color) Purity of the |BCS) state relative to the local aneer an ,;su(2);, as a function of g for different anisotropies g.=4 ). The number of sites N=400 as in Fig. 11.
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Lie AlgebraQuantum nonlocalityQuantum and classical statistical mechanicsCoherent StatesQuantum correlationsFermi liquidQuantum Phase TransitionConvex Combination
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