# The Interplay between Frustration and Entanglement in Many-Body Systems

###### Abstract

Entanglement (a quantum feature) and frustration (a non quantum feature) are important properties of many particle systems. This paper investigates the link between those two phenomena. To this end, we define a family of Hamiltonians and solve the eigensystem problem for different numbers of particles and different distant-depending interaction strengths. Though the microscopic properties vary extremely we can show that the introduced characteristic functions quantify frustration and entanglement reliably. One main result is a function, termed anisotropy function, sensible simultaneously to frustration and asymmetries of local entanglement. Our toolbox allows for a classification of models with certain entanglement-frustration properties that may be extended to other Hamiltonians.

###### pacs:

03.65.Ud, 03.67.Mn, 05.30.-d, 75.10.Jm## I Introduction

Frustration is a common feature in complex physical systems since it generally characterizes competing needs that are impossible to be fulfilled simultaneously. It is not a phenomenon that, per se, is connected to the quantum nature in our world. Frustration is to all of us a known experience if an obstacle prevents the satisfaction of desire. Even in the jurisdiction definitions of frustration can be found. In strong contrast to entanglement that is lacking in a classical world. It can lead to correlations that are stronger than those that can be generated with classical systems. Two particles can share information that is stored in the whole system but is not available in the separated systems. Even stronger for multipartite systems sharing of information of two particles limits the information with other particles (monogamy) Coffman2000 , there is a trade-off of shared information. Though both phenomena are distinct properties of systems, they are not unrelated, revealing the interplay for many particle systems is the focus of this paper.

In the last years researchers have developed several toolboxes to compute the local and global ground states of physical systems consisting of many particles exposed to different interactions White . With this progress the computation of entanglement, defined onto states given a certain subalgebra structure, has become a possible enterprise. Indeed, for a plethora of models entanglement in many-body systems has been shown to play a fundamental role (for an overview see e.g. Amico2008 ). For instance the area law Holzhey1994 ; Vidal2003 ; Korepin2004 ; Calabrese2004 is one that allows to understand the emergent of topologically ordered phases of matter Chen2010 . Entanglement explains the local thermalization Deutsch1991 ; Srednicki1994 ; Popescu2006 ; Barthel2008 or the selection of a particular subset of states from a huge ground state space Hamma2016 . Even the absence of entanglement gives a strong insight into the type of macroscopic phases that are realized Giampaolofatt . It has been shown that the standard density matrix renormalisation group (DMRG) algorithm, also known as matrix product formalism, allows to relate properties such as polymerisation to entanglement Murg ; Barcza2015 by reordering the particles. However, also more refined quantities can be introduced. In Ref. GabrielHiesmayr macroscopic observables capable of detecting (genuine) multipartite entanglement have been shown to work, including non-zero temperatures. Here, the authors compared the minimal energy of ground states to the minimal energy when optimised over the set of states with particular entanglement features. If both minimal energies do not overlap, then clearly there is entanglement in the game. This entanglement-energy gap witnesses via the macroscopic observable, energy, the presence of multipartite entanglement.

Most works have so far focussed on the analysis of frustration originating from the impossibility for a many-body system to satisfy all the energy constraints simultaneously Toulouse1977 ; Villain1977 ; Kirkpatrick1977 ; Binder1986 ; Mezard1987 ; Lacroix2011 ; Diep2013 . Recent works Giampaolo2011 ; Marzolino2013 have introduced a measure of frustration that allows to tackle both the frustration originating of classical (geometrical) or quantum sources. In particular, extending the classical Toulouse condition a set of inequalities bounded by entanglement was put forward.

If we think of maximally entangled states in the context of quantum information theory they differ only in their local information, in the choice of the reference system. They describe the same resource of information. Therefore, any proper entanglement measure has to assign the same value to each one of those. Complex many bodies are characterized by shaping different orders. Thus frustration marks a difference between the particular realisations of configurations. Immediately, the question arises how frustration and entanglement interplay? Despite growing interest in entanglement in condensed matter systems, up to our best knowledge this question has not been tackled and it presents the main aim of this contribution.

Consequently, we search for a system for which the local ground states have the same entanglement properties. This rules out models with external fields and puts strong restrictions onto the presence of cluster terms (including such as Dzyaloshinskii-Moriya interaction terms , where refers to the magnetic spin vector and and labels the different spins). Since we are interested in very interplay between frustration and entanglement, our system should allow us to distinguish the source of frustration, i.e. geometrical or quantum type. As we show in detail anisotropic Heisenbergs models defined on a one-dimensional ring with an even number of spin- serve our purpose. In detail we introduce a family of Hamiltonians with short- and long-range interactions and two-body Heisenberg-like interactions (Sect. II). We take the interaction to depend on the power of the inverse distance between the spin-. These distance depending forces allow us to obtain a continuous transformation between these two extreme interactions. As we show in detail the local entanglement of ground states is not sensitive neither to the interaction strength nor to any introduced anisotropy between interactions in the or direction. Moreover, our chosen family benefits from the fact that all possible sources of frustration are existent. In this sense we provide a complete, qualitative and qualitative picture of the interplay between entanglement and frustration for an exemplary system that may turn out to be a reference system for systems which can undergo more interactions.

Our paper is organized as follows, after the definition of the family of Hamiltonians (Sect. II), we start to analyse frustration due to geometrical origin (Sect. III. Then we proceed our analysis by turning on the antiferomagnetic interactions in one dimension. We define a proper frustration measure and compute its depending on the interaction strength . In this case the frustration can be said to be only due to geometrical (classical) sources. This changes when we allow also interaction in different spatial directions (Sect. IV). Again we find continuity in the behavior of the short- to long-range interactions though the huge differences in the properties of the local and global states of the system under investigations. In the conclusions (Sect. V) we summarize the rich underlying physics with respect to the interplay of entanglement and frustration.

## Ii Family of many-body systems

Our family of many-body systems undergoes two-body Heisenberg-like interactions on a one-dimensional ring of an even number of spin- particles with periodic boundary conditions. For sake of generality we consider systems invariant under spatial translations and only distant-depending interactions are considered. Given these constraints the Hamiltonian reads

(1) |

where with are the standard Pauli operators defined on the -th spin of the one-dimensional ring with periodic boundary condition (). are the anisotropies in the different spatial directions. Since the different spatial directions are interchangeable, without loss of generality, we may fix and consider the behavior of the system varying only the anisotropies and in the interval .

For this parameter space we observe three different local ground states of any pair of spins . Namely for an antisymmetric maximally entangled state, , and for the opposite case a symmetric maximally entangled state, . Whereas for we have a twofold degenerated local ground state which projector is the sum of the projectors of the two above maximally entangled states (these two maximally entangled states are two out of the four so called Bell states defined explicitly in Eq. IV.1).

The distance between two particles and is defined by . The strength of the distance depending interactions are modelled by . For all spins are interacting with all other ones with the same strength, thus this is the infinite-range interaction case. The other extreme case is obtained for , where we basically have only nearest-neighbour interactions, i.e. a short-range interactions. In-between we have the long-range interactions. We have also for some cases tested an exponential function for the distance rather than the above polynomial. We have not observed any qualitative difference.

The forthcoming analyzes is done by computing the exact ground state of the system and from it the entanglement and frustration properties of all the pair of spins in the systems. When analytical results were not attainable we used the Lanczos algorithm allowing numerical diagonalization. We use an exact diagonalization algorithm via the program Octave 3.6 to obtain the whole set of the global ground states from which we construct the maximally mixed ground state. From this representative state we calculate the corresponding frustration function . Numerical tests are performed for the finite-size models with definite , which represents the (even) number of spins in the system, running from to . The parameter is varied between (the limiting case for infinite range interactions) and . To obtain the frustration function we take into account that and range between and . In this way we take for each anisotropy coefficient (except fixed ) equidistant different values and evaluate for each one of the models with fixed and .

## Iii Frustration due to geometrical sources

We start the analyses of frustration for the family of Hamiltonians (1) by first discussing the case with the choices . This is an antiferromagnetic Ising model with short-range or long-range interactions modeled by the parameter

(2) |

The local ground states, i.e. the ground states of the local terms of the Hamiltonian, and the global ground states can be analytically derived. To compute frustration we explore the quantity introduced in Ref. Marzolino2013 . Independently of the interaction strength the local Hamiltonian of any pair of spins occupies a two-fold degenerated Hilbert space which is spanned by two separable states and . Consequently, the projector onto the local ground space is given by

(3) |

The global ground state is always degenerated. In contrast to the local ground states, however, the global ground states depend on the strength of the interacting spin pairs. In the computation of the frustration the degeneracy of the Hilbert space is taken into account by a kind of thermodynamic mixtures, i.e. all possible ground states are mixed with equal probabilities. As shown in Ref. Marzolino2013 this obtained maximally mixed ground state and all substates conserve all the symmetries of the Hamiltonian under investigation. Thus they represent the system’s properties accordingly Giampaolo2015 . The quantity measuring frustration of a pair of spins is then defined by

(4) |

where is the reduced density matrix obtained by tracing out from the maximally mixed ground state all the spins except and . In our case we have

(5) |

where the last equality holds because our model is invariant under spin inversions ( are the elements corresponding to the projections onto ).

Since the global ground state is alternating between up and down states –so-called Néel states– the number of states in the state depends on the distance between the spins (strictly speaking only for ). The reduced density matrix obtained from the maximally mixed global ground state for any two spins tracing out any other spins is for odd distances with

(6) |

or for even distances with

(7) |

Frustration does therefore have a dependence on the distance between the spins: in the odd case the number of states is zero () and in the even case the number of states is maximal (). Consequently, independent of the strength the frustration quantity attains the two dichotomic values or .

From the quantum information theoretic point of view frustration quantifies the difference between the local and the global ground states independently of interaction strength. From the statistical mechanical point of view a high value of frustration for weakly coupling spins should be less relevant for the global behavior of the system than a comparable low value of frustration for highly coupled spins. Consequently, we are looking for a quantity that is sensitive to the distance’s strength and the relevance of the frustration. Therefore, we define the following function

(8) |

This introduced function is not a new measure of the frustration per se, rather it is an average taking into account the relative strength of the interactions. The following results prove its potentiality.

In case the number of spins in the chain is an integer multiple of , we find

(9) |

whereas for with we obtain

(10) |

In the thermodynamic limit both functions should coincide which is indeed the case, the characteristic frustration of the system becomes

(11) |

where is the Riemann zeta function. Only in the case of the Riemann zeta function is converting to a finite value. In Fig. 1 we have plotted our characteristic frustration function for different and . As can be observed for the frustration has a finite value and with increasing it converges to zero. In contrast to the case of where the value diverges in the thermodynamic limit. Let us further discuss this case by looking closer to the limiting case of , i.e. all spins are equally interacting with any spin independently of the distance. In this case the spins can be reordered arbitrarily as long half of all spins are taking to be spin up and the other half is spin down. In the ground space we have therefore

(12) |

possibilities, thus the degeneracy of the ground state space grows exponentially. At first sight one would assume that the reduced density matrices become inaccessible, however, since all possible global ground states are mixed equally, tracing out leads to an analytic density matrix. The number of states for which any two spins are in the state is given by fixing these two spins and a variation over all possibilities for the remaining spins obeying the fact that the total magnetization along is zero

(13) |

Dividing this number by all possible global ground states and multiplying by we obtain the frustration of any pair of spins

(14) |

which is dependent on the length of the chain . Notably this function does not exceed the value . Now computing our characteristic frustration function (8) we find

(15) |

It is extremely notable that this function can be also obtained from Eq. (9) or Eq. (10) by performing the limit . This fact implies a consistency and a continuity of even if the behavior of the degree of degeneracy of the ground state space is exponential.

## Iv Frustration due to geometrical and quantum sources

Now we proceed by considering also non-zero values of the - and -interaction in addition to the one in (fixed with ). The Hamiltonian is obviously no longer decomposable into local commuting terms. These models are no longer analytically solvable, thus we developed a numerical approach for finite size models (Lanczos algorithm) from which we extrapolate to the thermodynamical case.

In general we expect that frustration can have contributions from a classical, geometrical source (competing interactions as in the first case analyzed), however, also contributions from the quantum feature “entanglement”. Our aim is to understand this origin. We start by discussing the general properties of the local and global quantum states and proceed then to discuss frustration.

### iv.1 Entanglement properties of the involved states

Due to the symmetries of the interaction terms the local ground states of two spins are generally maximally entangled states, i.e. one out of the four so-called Bell states

(16) |

With the antiferromagnetic interaction turned on () the antisymmetric Bell state with total spin zero or the symmetric Bell state with total spin one is favoured (reference basis is the -basis). For entanglement theory those four (two) Bell states do not differ in their entanglement property since they are connected by local unitaries. In particular any basis of maximally entangled states may be constructed out of one seed state by then applying in one subsystem unitaries (Pauli matrices or generally Weyl operators). This general property has been utilized in discussing separability properties extensively or are relevant in experimental realiations of mixed states GHZOAMPOL . For the discussion here it is important to note that the amount of entanglement (measured by any proper entanglement measure) is the same for both Bell states under investigation since they differ only by a local unitary operation, e.g. or . The aim is now to show how the local-to-global properties of the system under investigation influence each other.

Rotating locally the Hamiltonian (1) along the -axis leads to local rotations of the Pauli matrices and . In particular for the case , if we apply to spin particles such a -rotation we can change the sign of or , respectively. Thus there exists a bijective map between models above and below the symmetry line . Since we considered only local unitaries the amount of entanglement cannot change under those transformations. This symmetry is no longer given if the value becomes finite.

### iv.2 Frustration properties

We start our analysis with the case of , thus basically only short-range interactions (nearest-neighbour interactions) are present. We choose an even number of spins such that the frustration of classical origin can be assumed to be zero (see previous section), consequently being only sensitive to frustration originating from entanglement. Since there is no long-range interaction, we have no competing forces, but since frustration is still finite (see Fig. 4) we conclude it is of quantum origin.

In strong contrast to the other limiting case of infinite-range interactions (). Here, frustration becomes sensitive to the values of and and is plotted in Fig. 2. Interestingly, the models fall into two families classified by the local ground states. One for which the local ground state of any two spins is the symmetric Bell state ; this is the case if the values of are above the line . And another family for which the local ground state is the antisymmetric Bell state (below the line ).

Above the line (white area in Fig. 2) the global ground state is always non-degenerated. The value of the frustration we have computed numerically for the choices and found that it depends on generally. However, for fixed vales of we found that the value of frustration does not change if we move along lines given by the equation

(17) |

Surprisingly, for , where the frustration reaches the minimum for any fixed , this family of models has the same frustration as in the case with the separable ground states () given in Eq. (14). This unexpected identity is most likely due to the fact that in this case the Hamiltonian commutes with the total magnetization along the -axis (). This implies that the eigenstates of the Hamiltonian can be decomposed into non-interacting subspaces, exactly like in the case . However, note that the degeneracy in the this case of the global ground state scales exponentially with , whereas in the case () we have no degeneracy.

On the contrary in the cases the characteristic frustration functions tends to the maximum possible value in this region, , independently of the values of and . Taking into account that in the thermodynamic limit also the minimum of the frustration, Eq. 14), tends to , we can conclude that in the thermodynamic limit the value of frustration of any pair of spins tends to for the whole family under consideration.

In Ref. Giampaolo2015 it was generally shown that for models with a non-degenerated local ground state there is a striking relation (monogamy) between frustration and the value of the local entanglement. Applying it to our case the concurrence Hill1997 ; Wootters1998 , a computable measure of bipartite entanglement, is bounded from above by the value , thus approaches zero in the thermodynamic limit.

### iv.3 The case of the antisymmetric maximally entangled local ground states

The picture for the case of the local ground states corresponding to the antisymmetric Bell state (below the line ) is more refined. We find two different regions which depend on the value of (see Fig. 2). In the thermodynamic limit these two families are distinguished by having same or opposite signs in the and interactions. More precisely, the family (type family) is given for values of below the line and above the lines

(18) | |||||

This family has non-degenerated global ground states. The remaining values form the type family which in the thermodynamic limit corresponds to . Here the global ground states are degenerated.

In contrast to the symmetric Bell case the type family has maximal frustration. In this case the conditions for the monogamy between frustration and concurrence found in Ref. Giampaolo2015 do not hold. Our numerical results show that the values of concurrence are continuous extensions from the values obtained for the one above the line .

The type family shows degeneracy that scales exponentially with . Indeed, for the degeneracy evaluates to times those of Eq. (12), which is exactly equivalent to the case . Surprisingly, we find that the values of our characteristic frustration function are depending on but not on the specific values of and is given by

(19) |

and thus frustration takes values from () to (). The high degeneracy leads to strongly mixed local ground states, all entanglement disappears though the presence of frustration.

For we found

(20) | |||||

which corresponds to the lower bound of for any . It supports our intuition that when quantum effects become more and more relevant, also frustration increases. However, as discussed above there are distinct differences in the behavior due to different relation of the maximal entanglement (symmetric or anitsymmetric).

When assuming higher values of (Fig. 3) the local effects get more and more washed out and the values of the frustration decrease to a fixed value. In Fig. 4 we made that more precise by computing the maximal and minimal value of the characteristic frustration function optimized over all values that lead to non-degenerated local ground states for . We compared these optima with the maxima for .

Summarizing, for the long-range cases () the models fall in three different subfamilies, each with different local properties and different frustration values (e.g. drawn in Fig. 3 (a)). In the thermodynamic limit these models show generally high values of frustration preventing the system from building up strong local quantum correlations. In the limit to the short-range case the local properties of the ground states are washed out restoring the symmetry between families below and above the line . The frustration values are equal and are generally lower than the ones for smaller (e.g. pictured in Fig. 3 (a)). Thus in the region between the two extreme cases frustration is due to quantum and geometrical effects. No bijective map between the models below and above the line exists.

### iv.4 A frustration quantity sensitive to the local ground states

To quantitatively understand how the behavior of frustration changes when going from low to high values of we need a quantity that is sensitive to the difference of the frustration between the different subregions (above and below the line ). Starting from our definition of the weighed frustration we introduce the anisotropy parameter between the symmetric and antisymmetric local ground state space

where is the parameter space of . Thus is defined by Eq. (4).

The computations of are displayed in Fig. 5. For the extremal case (symmetry between below and above the line restored) we have and thus it follows for all . In the other extremal case we were not able to find any analytical solution, but we can extract from our numerical evaluations for that the asymmetry parameter goes with . For we find an exponential decay of the asymmetry function which is proportional to independently of .

## V Summary and conclusions

The aim of this contribution is to study the role of (local) entanglement, a quantum phenomenon, in frustrated many-body systems in order to understand how entanglement relates to frustration, a non-quantum phenomenon. For that we defined a family of Hamiltonians describing systems of spin- particles with distance-dependent two-body interactions. By deriving the eigenstates and eigenvalues for these multipartite systems we were able to study the properties related to frustration and entanglement. Only in few cases there exist analytical results, thus for the most cases we have performed numerical computations for a finite number of particles. This, however, turned to be very consistent and fast converging.

In general we find that our results show a continuity of the two quantum–non-quantum features between the two extremal cases of very short-range and very long-range interactions. Though, however, the microscopic properties differ strongly in their behaviour such as, e.g., exhibiting degenerated or non-degenerated ground states. From the continuity of the characteristic functions varied by several input parameters we conclude that our results reflect in some limit the realistic systems.

Our main result is a function, the anisotropy function , capable to characterize the frustration and simultaneously being sensitive to the local asymmetry in the entanglement. With its help we can understand the different sources of frustration and its interplay.

With this work of analyzing a well-defined family of multipartite spin systems by designed functions we provide a framework that allows in future to turn to more realistic (involved) Hamiltonians and contributed to the understanding how the microscopic properties of shared information of parties relates to global properties observed in condensed matter systems.

Acknowledgements: S.M.G. acknowledges financial support from the Ministry of Science, Technology and Innovation of Brazil, K.S. acknowledges the Austrian Science Fund (FWF-P26783), A.C. acknowledges partial financial support from MIUR and INFN and B.C.H. Austrian Science Fund (FWF-P23627).

## References

- (1) Coffman V, Kundu J and Wootters W K, Distributed entanglement, 2000 Phys. Rev. A 61 052306.
- (2) White S R, Density matrix formulation for quantum renormalization groups, 1992 Phys. Rev. Lett. 69 2863.
- (3) Amico L, Fazio R, Osterloh A and Vedral V, Entanglement in many-body systems, 2008 Rev. Mod. Phys. 80 517.
- (4) Holzhey C, Larsen F and Wilczek F, Geometric and renormalized entropy in conformal field theory, 1994 Nucl. Phys. B 424 443.
- (5) Vidal G, Latorre J I, Rico E and Kitaev A, Entanglement in Quantum Critical Phenomena, 2003 Phys. Rev. Lett. 90 227902.
- (6) Korepin V E, Universality of Entropy Scaling in One Dimensional Gapless Models, 2004 Phys. Rev. Lett. 92 096402.
- (7) Calabrese P and Cardy J, Entanglement entropy and quantum field theory, 2004 J. Stat. Mech.: Theor. Exp. P06002.
- (8) Chen X, Gu Z-C and Wen X-G, Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order, 2010 Phys. Rev. B 82 155138.
- (9) Deutsch J M, Quantum statistical mechanics in a closed system, 1991 Phys. Rev. A 43 2046.
- (10) Srednicki M, Chaos and quantum thermalization, 1994 Phys. Rev. E 50 888.
- (11) Popescu S, Short A J and Winter A, Entanglement and the foundations of statistical mechanics, 2006 Nature Phys. 2 754.
- (12) Barthel T and Schollwöck U, Dephasing and the Steady State in Quantum Many-Particle Systems, 2008 Phys. Rev. Lett. 100 100601.
- (13) Hamma A, Giampaolo S M and Illuminati F, Mutual information and spontaneous symmetry breaking, 2016 Phys. Rev. A 93 012303.
- (14) Giampaolo S M, Adesso G and Illuminati F, Theory of ground state factorization in quantum cooperative systems, 2008 Phys. Rev. Lett. 100 197201; Giampaolo S M, Adesso G and Illuminati F, Separability and ground-state factorization in quantum spin systems, 2009 Phys. Rev. B 79 224434; Giampaolo S M, Adesso G and Illuminati F, Probing quantum frustrated systems via factorization of the ground state 2010 Phys. Rev. Lett. 104 207202.
- (15) Gabriel A, Hiesmayr B C, Macroscopic Observables Detecting Genuine Multipartite Entanglement and Partial Inseparability in Many-Body Systems, 2013 Eur. Phys. Lett. 101 30003.
- (16) Gabriel A, Murg V, Hiesmayr B C, Partial Multipartite Entanglement in the Matrix Product State Formalism, 2013 Phys. Rev. A 88 052334.
- (17) Barcza G, Noack R M, Solyom J and Legeza Ö, Entanglement patterns and generalized correlation functions in quantum many-body systems, 2015 Phys. Rev. B 92 125140.
- (18) Toulouse G, Theory of frustration effect in spin glasses: I, 1977 Commun. Phys. 2 115; Vannimenus J and Toulouse G, Theory of frustration effect: II. Ising spins on a square lattice, 1977 J. Phys. C 10 L537.
- (19) Villain J, Spin glass with non-random interactions, 1977 J. Phys. C 10 1717.
- (20) Kirkpatrick S, Frustration and ground-state degeneracy in spin glasses, 1977 Phys. Rev. B 16 4630.
- (21) Binder K and Young A P, Spin glasses: Experimental facts, theoretical concepts, and open questions, 1986 Rev. Mod. Phys. 58 801.
- (22) Mezard M, Parisi G and Virasoro M A, Spin Glass Theory and Beyond, 1987 (Singapore: World Scientific).
- (23) Lacroix C, Mendels P and Mila F, Introduction to Frustrated Magnetism, 2011 (Heidelberg-Berlin: Springer-Verlag).
- (24) Diep H T (ed.), Frustrated Spin Systems, 2013 (Singapore: World Scientific).
- (25) Giampaolo S M, Gualdi G, Monras A, Illuminati F, Characterizing and quantifying frustration in quantum many-body systems, 2011 Phys. Rev. Lett. 107 260602.
- (26) Marzolino U, Giampaolo S. M and Illuminati F, Frustration, Entanglement, and Correlations in Quantum Many Body Systems, 2013 Phys. Rev. A 88 020301(R).
- (27) Carvacho G, Graffitti F, D’Ambrosio V, Hiesmayr B C and Sciarrino F, Twin GHZ-states behave differently, 2015 arXiv:1512.06262.
- (28) Giampaolo S M, Hiesmayr B C and Illuminati F, Global-to-local incompatibility, monogamy of entanglement, and ground-state dimerization: Theory and observability of quantum frustration in systems with competing interactions, 2015 Phys. Rev. B 92 144406.
- (29) Hill S and Wootters W K, Entanglement of a Pair of Quantum Bits, 1997 Phys. Rev. Lett. 78 5022.
- (30) Wootters W K, Entanglement of Formation of an Arbitrary State of Two Qubits, 1998 Phys. Rev. Lett. 80 2245.