Multipartite entanglement in fermionic systems via a geometric measure
We study multipartite entanglement in a system consisting of indistinguishable fermions. Specifically, we have proposed a geometric entanglement measure for spin- fermions distributed over modes (single particle states). The measure is defined on the qubit space isomorphic to the Fock space for single particle states. This entanglement measure is defined for a given partition of modes containing subsets. Thus this measure applies to partite fermionic system where is any finite number, giving the number of sites. The Hilbert spaces associated with these subsets may have different dimensions. Further, we have defined the local quantum operations with respect to a given partition of modes. This definition is generic and unifies different ways of dividing a fermionic system into subsystems. We have shown, using a representative case, that the geometric measure is invariant under local unitaries corresponding to a given partition. We explicitly demonstrate the use of the measure to calculate multipartite entanglement in some correlated electron systems. To the best of our knowledge, there is no usable entanglement measure of partite fermionic systems in the literature, so that this is the first measure of multipartite entanglement for fermionic systems going beyond the bipartite and tripartite cases.
pacs:03.65.Ud, 71.10.Fd, 03.67Mn, 05.30Fk
The nonlocal correlations implied by the states of multipartite quantum systems form the basis of quantum information processing technologies like quantum teleportation and quantum computation Dirk (); wooters (); Ye (). Recently the question of understanding and using entanglement in the systems of identical and indistinguishable particles has seen a surge of interest zanardi (); indpar (); wiseman (); amico (). Two or more identical particles become indistinguishable when their wave functions overlap. Such a situation can arise, for example, in a quantum device based on quantum dot technology Loss-Burk (); Petta (). Here qubits are realized by the spins of the electrons in a system of quantum dots. The overlap between the electron wave functions in different dots can be varied by controlling parameters like gate voltages or magnetic field which change the tunneling amplitudes of the electrons from one dot to the other. For non-negligible overlaps, the entanglement between the qubits is then intimately connected to the electron entanglement which is essentially that of indistinguishable fermions. Entanglement is also expected to play a fundamental role in many physical phenomena like quantum phase transitions, quantum Hall effect, etc qpt (); Hal () involving many body quantum systems.
The study of entanglement in many body quantum systems consisting of identical particles has posed challenging fundamental questions about the definition and nature of entanglement in such systems due to the inherent indistinguishability of particles zanardi (); Eck (); Gir (); Fang (). For example, the (anti)symmetrization necessary for indistinguishable (fermions) bosons already leads to quantum entanglement. However, it is known that the correlations due to symmetrization or anti-symmetrization are not by themselves a physically useful resource for quantum information and communication technologies; for example, there is no measurement we can do locally on a fermion in a localized state which is affected by the existence of identical fermions in other parts of the universe peres (). Most studies of entanglement in many body systems consisting of identical particles have focussed on the study of bipartite entanglement. There have been few studies on multipartite entanglement multi (); multi2 () although the study of multipartite entanglement in such systems is also an important and interesting question.
In this work, we study multipartite entanglement in a system consisting of identical particles. We use the idea due to Zanardi zanardi () whereby the Fock space of a system of fermions is mapped to the isomorphic qubit or ‘mode’ space. We then discuss entanglement in this ‘mode’ space via a geometric measure. The idea is to use the Bloch representation of the state of the -partite quantum system kim (). The measure is defined by the Euclidean norm of the -partite correlation tensor in the Bloch representation. This correlation tensor contains all information of genuine -partite entanglement (see section III). Such a measure was proposed earlier by one of the authors (and Hassan) for qubit and qudit pure states ps1 (); ps2 () and shown to satisfy most of the properties expected of a good measure. The Bloch representation of a quantum state has a natural geometric interpretation, which is why we call this measure a geometric measure kim (). Other geometric measures are based on the distance of the given state from the set of separable states in the Hilbert space ved (); wei (). An important question in the context of quantum entanglement is that of locality. For indistinguishable particles distributed over ‘modes’ (which are taken to be single particle states of particles constituting the system), local operations have meaning only in the context of partitions over modes. A quantum operation confined to a single subset in a given partition is then a local operation. We therefore define entanglement in such a system with respect to partitions and require it to be invariant under local unitaries defined with respect to a given partition. We explicitly demonstrate the use of the measure to calculate entanglement in some correlated electron systems.
The outline of the paper is as follows: In Sec.II, we begin by briefly reviewing some details about the isomorphism between the Fock space of a system of indistinguishable particles and the ‘mode’ space zanardi (). This will help us in defining various quantities and also set up notation necessary for the subsequent analysis. In Sec.III, we define and construct the geometric measure. Various properties of the measure are also discussed with reference to a specific example in Sec.IV. In Sec.V, we use the measure to study entanglement in the Hubbard dimer and trimer. Finally, we conclude in Sec.VI.
Ii Mapping between Fock space and qudit space
We deal with spin- fermions on a site lattice. The total number of available (localized) single particle states are then in number. The fermionic Fock space in the occupation number representation has basis states of the form . We further assume that the total number of particles is conserved. This means that we only deal with subspaces of the Fock space corresponding to a fixed eigenvalue for the total number operator. We shall refer to this number super-selection rule as N-SSR. For a -fermion system, we call such a subspace of the Fock space ‘-sector’ and denote it by . The -sector of a mode system is the subspace of dimension of the Fock space with the dimension of the Fock space for single particle states being
Since a qubit Hilbert space has exactly this dimension, it is possible to construct an isomorphism between the Fock space and the qubit Hilbert space halmos (). The particular isomorphism we implement is
where, in qubit space we associate and Note that the Slater rank of the Fock basis states is so that these are separable states. Thus the above isomorphism maps separable basis states in Fock space to the separable basis states in qubit space. Further, the subspace structure of the Fock space namely,
is carried over to the qubit space under mapping (Eq.(2)) because each subspace of the Fock space with conserved fermion number is mapped to a subspace of the qubit space spanned by the basis vectors with ones and zeros. We can write
Next crucial step is to transfer the action of the creation and annihilation operators on Fock space to the qubit space, under the isomorphism given by Eq.(2) Cab1 (). We need the creation and annihilation operators and acting on a single qubit state,
Here () is the annihilation (creation) operator acting on Fock space annihilating (creating) a fermion in th mode. is the identity on single qubit space. The tensor product satisfying the correspondence in Eq.(II) must be consistent with the anti-commutation property of the Fock space creation and annihilation operators,
This requirement leads to the following action of the tensor product operators on the qubit states
Here and is evaluated Using Eq.(II), it is straightforward to see that
Henceforth, in this paper, by ‘fermions’ we mean spin- fermions. Further, we call a single particle state a mode. Thus two spin- fermions on two sites is a four mode system. In general, spin- fermions on sites is equivalent to fermions on modes. For example, two spin- fermions on a two site lattice, say, generate four single particle states or modes . In this work, we deal with entanglement between subsets forming a partition of a -mode fermionic system. We define the entanglement measure for any such partition of a -mode system without any restriction on the number and the size of the subsets forming the partition. These subsystems may involve different degrees of freedom, for example, we can deal with the entanglement between spins and sites or entanglement between two spins on the same site (intra-site entanglement). Or if each of the subsets partitioning the modes comprises modes with common site label we have the entanglement between sites or the ‘site entanglement’. Thus all physically realizable subsystems of fermions over single particle states can be addressed by dividing the modes into suitable partitions, for example the ‘particle entanglement’ defined in wiseman ().
We now define the local and non-local operations on the mode fermionic system zanardi (); indpar (); wiseman (); zoz (); Plas (). We do this by using the corresponding operations on the isomorphic qubit space We note that, due to isomorphism between (Eq.(2)) and (Eq.(4)), partitioning modes is equivalent to the corresponding partitioning of the qubit system into subsystems. Locality is defined with respect to the partition of qubits (or, the corresponding partition of modes) between whose subsets we are seeking entanglement. The operations on the state space of a single subset in a partition of qubits is taken to be local. The entanglement measure defined with respect to a partition must be invariant under a unitary operation which is local with respect to that partition. We will illustrate this point later, using the geometric entanglement measure defined below. Henceforth ‘mode’ and ‘qubit’ are taken to be synonymous and we shall use the expression ‘modes’ instead of ‘qubits’. In other words, the spaces and are taken to be the same.
Iii Geometric Measure for Entanglement.
We define a geometric measure for the partitions of the mode fermion systems in pure states. Although the definition of the measure is quite general, the fermion number super-selection rule restricts the pure states to the appropriate subspace corresponding to fermions, namely (Eq.(4)). Thus a state may be partitioned as or as etc where the definition of the tensor product is consistent with Eq.(II). We use the Bloch representation of partite states (drawn from ) to get this measure ps1 (); ps2 ().
First we assume that a partition equally divides modes into subsets, ie all subsets in the partition contain equal number of modes, say This corresponds to the case of divided into subspaces of dimension being some divisor of the number of modes To get the entanglement measure we expand the state of the system, supported in the appropriate in its Bloch representation.
In order to give the Bloch representation of a density operator acting on the Hilbert space of an -qudit quantum system, we introduce following notation ps2 (). We use , to denote a qudit chosen from qudits, so that , take values in the set . The variables for a given or span the set of generators of group for the th or th qudit, namely the set for the th qudit. For two qudits and we define
where and occur at the th and th places (corresponding to th and th qudits respectively) in the tensor product and are the th and th generators of . Then we can write
where is a Bloch vector corresponding to th qudit, which is a tensor of order one defined by
where is the reduced density matrix for the th qudit. Here is a subset of and can be chosen in ways, contributing terms in the sum in Eq.(12), each containing a tensor of order . The total number of terms in the Bloch representation of is . We denote the tensors occurring in the sum by which are defined by
where is the reduced density matrix for the subsystem . Each of the tensors of order , occurring in the Bloch representation of , contains all information about entanglement of the corresponding set of subsystems. All information on the entanglement contained in is coded in the tensors occurring in the Bloch representation of . The tensor in last term in Eq.(12), we call it , contains all the information of genuine -partite entanglement. This follows from the observation that all other terms in the Bloch representation of (Eq.(12)) correspond to subsystems comprising qudits and the density operator contains all possible information about the state of the system.
The operators are given by Mahler ()
Note that each of the generators of the group acts on a single qudit space and hence is local (see Sec.II), apart from the phase factor contributed by their action, as given by Eq.(II). We assume these phase factors to be absorbed in the coefficients in the expansion of the density operator
Let a mode fermion system be partitioned by subsets, each containing n modes. Then for this partition, we define the entanglement measure for a state by ps2 ()
and is for separable (product) qudit state
iii.2 Entanglement in partitions with unequal subsets
We can also generalize the definition of the entanglement to the case where the corresponding qubit subsystems have unequal dimensions. We discuss the simplest case of bi-partite entanglement with partitions having unequal dimensions, say and . In this case, the definition of the geometric entanglement measure generalizes to
Here and are the generators of and respectively. is given by
Here and are the norms of the Bloch vectors of the reduced density operators for each subsystem.
It is straightforward to extend these definitions to partitions containing more than two subsets.
Iv Entanglement in a four mode system
With the entanglement measure defined as above, we give an example wherein we can compute the entanglement for different partitions and illustrate our comments on local and non-local operations. We also compute the upper bounds on the entanglement.
iv.1 Local and Non-local operations
Consider a four mode system and the normalized state defined as
where are real. Note that can be treated as a member of the Fock space with the kets appearing in it being its basis states. Consider the evolution of the system in state via the Hamiltonian
acting on Here term is the interaction between two modes on different sites (inter-site interaction), term is the interaction between two modes on the same site (intra-site interaction). and correspond to single mode on site and respectively. q term involves number operators for first two modes, on site. We have included all the different kinds of typical interactions encountered in condensed matter systems, respecting number super-selection rule. After an infinitesimal unitary evolution via this Hamiltonian, the state evolves to
Now we find the entanglement for different partitions of this four mode system, using the geometric entanglement measure, Eq.(15). We first partition four modes into four subsets, each containing one mode. This case gives genuine entanglement between four modes, which is more general than only the bipartite entanglement considered in the literature. For this case so that and we get, for the genuine four mode entanglement,
where are the generators of the group (Pauli operators). The resulting entanglement in is
where the first term gives the entanglement for the state as defined in Eq.(22). For this partition, the operations on a single mode are the only local operations, while all others are non-local. Therefore, the terms and are the only local interactions. Therefore, we expect that the four mode genuine entanglement should not depend on or to the first order in which is the case, as seen from Eq.(IV.1).
Next, we consider the partition consisting of two subsets, each containing two modes on each site, and (site partition). Thus we have two subsystems with corresponding to a qudit system. Further, giving so that the geometric entanglement is
where are the generators of The entanglement of in Eq.(IV.1) is then given by
According to the ‘site partition’, in addition to the operations on single modes, the operations on the pair of modes having the same site label are also local. Therefore, the resulting entanglement cannot change under the intra-site operations in the Hamiltonian, namely the term, the term and as before, and terms. Thus, to the first order in the entanglement is expected to depend only on the non-local part of the Hamiltonian, that is, on the parameter. From Eq.(IV.1) we see that this is the case. The inter-site entanglement quantified using Von-Neumann entropy has been reported earlier zanardi ().
Thus we see that the geometric measure has the capability to quantify the genuine multi-mode fermionic entanglement as against the mainly bipartite entanglement reported in the literature. Also, the geometric entanglement measure, for the given partition of modes, shows the correct behavior under local and non-local unitary operations.
iv.2 Upper bound for the geometric measure in a four mode system.
We discuss here some upper bounds that one can find for the entanglement for the four mode system and compare with existing results obtained from other measures. We find that the general state which leads to a maximum inter-site entanglement computed via the geometric measure also leads to a maximum for the von-Neumann entropy.
We treat the entanglement for the given partition to be the function of the coefficients of the general state namely,
where each ket is labeled by the (four bit) binary representation of We then maximize with respect to the coefficients .
For the site partition (see above) the entanglement (Eq.(30) and two equations following it) is a function of the coefficients in state given in Eq.(32). We find that the the maximum value of the entanglement is given by
The Schmidt canonical form (with respect to the basis) of corresponding to this is
where each ket is labeled by the four bit binary representation of Also, both these forms lead to the von-Neumann entropy which is the maximum possible Gir ().
For the four mode entanglement (Eq.(26)) we find that entanglement is maximum for the state given by
and the maximum value is found to be
However, we do not have any entanglement measure to compare with the geometric measure. We also note that, for four modes, no canonical form like the Schmidt or Acin canonical form (for two and three modes respectively) is available.
We now consider some correlated fermionic lattice models and discuss multi-mode entanglement in these models using the geometric measure.
v.1 Hubbard dimer.
The Hubbard dimer model is a simple model for a number of physical systems, including the electrons in a molecule, double quantum dots, etc zanardi (); wiseman (); hdimer (). The Hamiltonian can be written as
where are the site labels and and are spin labels. is the hopping coefficient measuring hopping between two sites while conserving spin and quantifies Coulomb interaction between fermions on the same site. By varying we can vary the relative contributions of hopping and Coulomb mechanisms.
where is the normalization factor and
with By mapping to via Eq.(2) we get,
The normalized ground state can be expressed in the qubit space as
The ground state entanglement can be then calculated to be
We plot the four-partite entanglement as a function of and (Fig.1(a)) and as a function of (Fig.1(b)). The entanglement is seen to monotonically increase as a function of , saturating at large values of to the maximum value . The saturation to the maximum value can be obtained either for very large values of or very small values of . We can interpret this result in the following way: since the total particle number is fixed to be , the four mode entanglement essentially measures the correlations between the spins. The entanglement increases as a function of because the spin correlations increase with .
and . Here s are the generators of there are partitions (same as the number of sites) and each partition has dimension This leads to an inter-site entanglement of the form
The bi-partite entanglement between sites and was calculated earlier using the von-Neumann entropy zanardi ()
We plot the inter-site entanglement (the von-Neumann entropy is also plotted for comparison) as a function of in Fig.2. It is seen that both measures show qualitatively similar behavior, i.e, a monotonically decreasing entanglement as a function of saturating at very large values of . The entanglement between the sites and decreases as a function of because with increasing on-site repulsion , the four dimensional local state space at each site gets reduced to a two dimensional local state space zanardi () due to a suppression of charge fluctuations or in other words, as the partition goes over to a partition. We have explicitly checked that the entanglement obtained in the limit matches with that obtained for the partition.
We can also discuss bi-partite entanglement with unequal partitioning. Consider four modes partitioned into two subsets containing one and three modes respectively. This would correspond for example to one observer controlling a register which measures the occupancy of the spin up at site and the other observer controlling a register which measures the occupancy of the spin down at site as well as the occupancy at site . The bi-partite entanglement in this can be obtained as discussed in Sec.(III.2). In the present case, the two partitions have dimension as and respectively. This gives, for the entanglement,
or, using in Eq.(43),
Interestingly, the entanglement is independent of . It turns out that this is the maximum value possible for the entanglement (Eq.(52))( we checked this by maximizing the entanglement given by Eq.(52) as a function of the coefficients in the general state Eq.(32). We have also checked that von-Neumann entropy in this case is also independent of and has the maximum possible value, i.e. .
v.2 Three electrons on three sites.
We next consider the Hubbard trimer, i.e, electrons on three sites with the sites with periodic boundary conditions. Ignoring the chemical potential , the Hamiltonian is
where is the hopping parameter and . We begin our analysis as earlier by mapping from the fermionic to the six qubit space corresponding to . In the qubit space, the basis respecting number super-selection rule is given by
where the twenty basis states are labeled by the six bit binary representation of the values. We numerically diagonalize the Hamiltonian in Eq.(54) in the basis given by Eq.(55) at different values of as increases from zero. The total spin quantum number and the component of the total spin commute with the Hamiltonian and can therefore be used as good quantum numbers to characterize the states. The ground state has total value . The triangular geometry of the three site model (with periodic boundary conditions) also leads to an additional symmetry under reflection about one of the medians of the triangle. This leads to a two fold degeneracy for the ground state (for a fixed value). Since these symmetries are preserved even in the presence of the interaction , the ground state remains two-fold degenerate for all values.
The entanglement in any state can be calculated in a similar manner as shown previously (Sec.V.1). We now show the results of our calculations for the entanglement in one of the ground states.
The six-mode entanglement as a function of the interaction parameter is shown in Fig.3a while the tripartite entanglement (between the sites ) is shown in Fig.3b as a function of . The tripartite entanglement between the sites is seen to decrease with increasing - we interpret this result in a similar way as that for the dimer as due to the fact that the local state space at each site decreases with increasing . We find that the six-mode entanglement increases with , saturating at large values, however, the behavior is not monotonic. Such non- monotonic behavior as a function of is also shown by the bipartite entanglement between sites and calculated using the geometric measure as well as the von-Neumann entropy (Figs.3c and 3d).
We can understand the non-monotonic behavior in the following way: the bi-partite entanglement measures the entanglement between the the sites and the sites . With small increase in from zero, there is an increase in the spin fluctuations between and which shows up as an initial increase in the entanglement between and . However, with further increase in , the charge fluctuations get completely suppressed leading to an asymptotic behavior similar to that for the dimer. One can understand the non-monotonic behavior of the six-mode entanglement in a similar way. For small , there is a larger correlation between two sites which leads to a decrease in the overall entanglement, however with further increase in , the contribution to the entanglement is solely due to spin fluctuations which increases with leading to the observed increase in the entanglement as well.
We have also calculated the upper bounds for the entanglements in various partitions in the same way as in Sec.IV. These are
We have proposed a multipartite entanglement measure for fermions distributed over modes (single particle states). The measure is defined on the qubit space isomorphic to the Fock space for single particle states. The entanglement measure is defined for a given partition of modes containing subsets, using the Euclidean norm of the -partite correlation tensor in the Bloch representation of the corresponding multi-mode state, viewed as a -partite state (see sec. IV and V). The Hilbert spaces associated with these subsets may have different dimensions. The quantum operations confined to a subset of a given partition are local operations. This way of defining entanglement and local operations gives us the flexibility to deal with entanglement and its dynamics in various physical situations governed by different Hamiltonians. We note that the concept of locality for indistinguishable fermions is distinct from that for distinguishable particles. In the latter case, locality applies to spatially separated subsystems. However, spatially separated fermions become distinguishable. We have shown, using a representative case, that the geometric measure is invariant under local unitaries corresponding to a given partition. As an application, we have also considered some correlated electron systems and demonstrated the use of the multipartite measure in these systems. In particular, we have calculated the multipartite entanglement in the Hubbard dimer and trimer (at half filling). We find that the bipartite entanglement between sites as computed with the geometric measure has a qualitatively similar behavior as a function of the interaction as that of the conventional von-Neumann entropy. We have also calculated the four(six)- partite and the two (three) site entanglement for the Hubbard dimer(trimer). We find that the multi-partite entanglement gives complementary information to that of the site entanglement in both the cases.
Although the entanglement measures given in this paper have been mainly applied to the study of the multipartite entanglement structure in Hubbard dimers and trimers, these measures are completely general and can be applied to other fermionic systems. We have also shown ps1 (); ps2 () that, viewed as a measure on qubit space, this measure has most of the properties required of a good entanglement measure, including monotonicity. To the best of our knowledge, this is the first measure of multipartite entanglement in fermionic systems going beyond the bipartite and even the tripartite case. Further, in this paper we have restricted to applications involving ground states of mode systems with . It is straightforward to extend the calculations for large number of modes () except for the length of the computation. We have shown earlier that ps1 (); ps2 () for antisymmetric states, the computational complexity of computing the measure goes polynomially with the number of parts of the system (here, the number of partitions of modes). It would be interesting to extend these calculations to larger system sizes where new and interesting results might be expected - we plan to do this in a separate study.
Acknowledgements.It is a pleasure to thank R.Shankar, V.Subramanian, Sibasish Ghosh , Sandeep Goyal, Ali Saif M. Hassan and Ali Ahanj for useful discussions. We thank Guruprasad Kar and Prof. R.Simon for encouragement. PD and PSJ also acknowledge support from their respective BCUD, Pune University grants.
- (1) Dirk Bouwmeester, A. Ekert, A. Zeilinger (Eds.), The Physics of Quantum Information, ( Springer-Verlag Berlin Heidelberg 2000)
- (2) C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. 70, 1895 (1993).
- (3) Ye Yeo and Wee Kang Chua, Phys. Rev. Lett. 96, 060502 (2006).
- (4) P.Zanardi, Phys. Rev. A 65,042101(2002); P.Zanardi and X.Wang, J. Phys. A 35, 7947 (2002).
- (5) J.R.Gittings and A.J.Fisher, Phys. Rev. A 66, 032305(2001); Mari-Carmen Banuls,J.I.Cirac and M.M.Wolf Phys.Rev. A 76, 022311(2007) ; X.G.Wang and B.C.Sanders, J. Phys. A 38, L67 (2005)
- (6) H.M.Wiseman and J.A.Vaccaro, Phys. Rev. Lett.91 097902 (2003); M.R.Dowling, A.C.Doherty and H.M.Wiseman, Phys.Rev. A 73, 052323 (2006)
- (7) L.Amico, R.Fazio, A.Osterloh and V.Vedral, Rev. Mod. Phys. 80, 517-576 (2008).
- (8) D.Loss and D.P.DiVincenzo, Phys. Rev. A57, 120(1998) ; G.Burkard, D.Loss and D.P.DiVincenzo, Phys. Rev. B 59 2070 (1999).
- (9) J.R.Petta,A.C.Johnson,J.M.Taylor,E.A.Laird,A.Yacoby,M.D.Lukin,C.M.Marcus,M.P.Hanson and A.C.Gossard, Science, 1126, 1-10(2005).
- (10) T.J.Osborne and M.A.Nielsen, Phys. Rev. A 66, 032110 (2002) ; A.Osterloh, L.Amico, G.Falsi and R.Fazio, Nature 416, 608 (2002) ; G.Vidal, J.I.Latorre, E.Rico and A. Kitaev, Phys. Rev. Lett., 90, 227902 (2003) ; J.I.Latorre, E.Rico and G.Vidal, Quant. Inf. Comp.4, 48 (2004) ; F.Verstraete, M.A.Martin-Delgado, J.I.Cirac, Phys. Rev. Lett. 92, 087201 (2004) ; M.Popp, F.Verstraete, M.A.Martin-Delgado, J.I.Cirac, Phys. Rev. A 71, 042306 (2005).
- (11) Hui Li and F.D.M.Haldane Phys. Rev. Lett. 101,010504 (2008).
- (12) K.Eckert, J.Schliemann, D.Bruss, M.Lewenstein Annals of Physics 299, 88-127 (2002).
- (13) GianCarlo Ghirardi, Luca Marinatto and Tullio Weber, J.Stat.Phys. 108, 49 (2002).
- (14) A.Fang and Y.C.Chang Phys. Lett. A 311, 443 (2003).
- (15) A.Peres, Quantum Theory : Concepts and Methods (Kluwer Academic, Dordrecht, 1995).
- (16) P. Levay and P. Vrana, Phys. Rev. A 78, 022329 (2008), P. Vrana and P. Levay, J. Phys. A: Math and Theor.42, 285303 (2009), C. Lunkes, C. Brukner, V. Vedral, Phys. Rev. Lett, 95, 030503 (2005)
- (17) T. Vertesi, Phys. Rev. A 75, 042330(2007), H. Habibian, J. W. Clark, N.Behbood and K. Hinger, Phys. Rev. A 81, 032302 (2010)
- (18) G.Kimura and A.Kossakowski, Open Syst. Inf. Dyn. 12, 207 (2005), G.Kimura, Phys. lett. A 314, 339 (2003).
- (19) Ali Saif M. Hassan and Pramod S. Joag, Phys. Rev. A 77, 062334 (2008).
- (20) Ali Saif M. Hassan and Pramod S. Joag, Phys. Rev. A 80, 042302 (2008).
- (21) V.Vedral, M.B.Plenio, M.A.Rippin, P.L.Knight Phys. Rev. Lett., 78, 2275, (1997), V.Vedral, M.B.Plenio, Phys. Rev. A 57, 1619 (1998).
- (22) T.C.Wei, P.M.Goldbart, Phys. Rev. A 68, 042307 (2003), M.Hayashi, D.Markham, M.Murao, M.Owair, S.Virmani, Phys. Rev. A 77, 012104 (2008), L.Tamarayan, D.K.Park and S.Tamarayan Phys. Rev. A 77, 022325(2008), S.Tamarayan, A.Sudbery, L.Tamarayan Phys. Rev. A 81, 052319(2010).
- (23) P. R. Halmos, ‘Finite Dimensional Vector spaces’ (D. von Nostrand Company Inc. , 1969)
- (24) P.Caban and K.Podlaski J. Phys. A 38, L79 (2005).
- (25) Mari-Carmen Banuls,J.I.Cirac and M.M.Wolf Phys. Rev. A 76, 022311(2007).
- (26) O.S.Zozulya, M.Haque and K.Schoutens Phys. Rev. A 78, 042326 (2008)
- (27) A.R.Plastino quant-ph/10020465 (2010).
- (28) M. A. Nielsen, I. L. Chuang, Quantum computation and quantum information (Cambridge University Press), 2000
- (29) G.Mahler and V.A.Weberruss, Quantum Networks (Springer 1995)
- (30) see for example, N. W, Ashcroft and N. D.Mermin, Solid State Physics (Saunders College, Philadelphia, 1976).