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Patterns of Maximally Entangled States Within the Algebra of Biquaternions

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Patterns of Maximally Entangled States Within the Algebra of Biquaternions J. Phys. Commun. 4 (2020) 055018 https://doi.org/10.1088/2399-6528/ab9506 PAPER Patterns of maximally entangled states within the algebra of OPEN ACCESS biquaternions RECEIVED 21 March 2020 Lidia Obojska REVISED 16 May 2020 Siedlce University of Natural Sciences and Humanities, ul. 3 Maja 54, 08-110 Siedlce, Poland ACCEPTED FOR PUBLICATION E-mail: [email protected] 20 May 2020 Keywords: bipartite entanglement, biquaternions, density matrix, mereology PUBLISHED 28 May 2020 Original content from this Abstract work may be used under This paper proposes a description of maximally entangled bipartite states within the algebra of the terms of the Creative Commons Attribution 4.0 biquaternions–Ä . We assume that a bipartite entanglement is created in a process of splitting licence. one particle into a pair of two indiscernible, yet not identical particles. As a result, we describe an Any further distribution of this work must maintain entangled correlation by the use of a division relation and obtain twelve forms of biquaternions, attribution to the author(s) and the title of representing pure maximally entangled states. Additionally, we obtain other patterns, describing the work, journal citation mixed entangled states. Finally, we show that there are no other maximally entangled states in Ä and DOI. than those presented in this work. 1. Introduction In 1935, Einstein, Podolski and Rosen described a strange phenomenon, in which in its pure state, one particle behaves like a pair of two indistinguishable, yet not identical, particles [1]. This phenomenon, called by Einstein ’a spooky action at a distance’, has been investigated by many authors, who tried to explain this strange correlation [2–9]. To describe this relationship and shed more light to this phenomenon, we also proposed a mathematical framework–a non-standard model of a collective set theory, i.e., mereology (NAM)1 [10–17]. Mereology is nothing else than a power set theory without a null element. In NAM sets are considered as aggregates, which can be divided into subsets in different ways, e.g., a decay of a physical particle mesonπ [18].In an abstract way, let us consider an example of the interval I=[0,1]. Mereologically, it can be seen as a set composed of two halves: S1=[0,1/2], S2=[1/2,1], therefore I1={S1, S2} or as a set composed of four =[ / ] =[ / / ] =[ / / ] =[ / ] quarters: IQQQQ21234= {},,, , where: Q1 0,1 4 , Q2 1 4,1 2 , Q3 1 2,3 4 , Q4 3 4,1 , or yet something else. In the case of halves, I1 has two elements: S1, S2;in the case of quarters–I2 has four elements: Q1, Q2, Q3, Q4. Since mereological class is synonymous with the concept of sum, we have I1=I2, i.e. 2 {SS12,,,,}{= QQQQ 1 2 3 4 }. In comparison, in classical, ZFC set theory , we will have: II12¹ , i.e. {SS12,,,,}{¹ QQQQ 1 2 3 4 }because these two sets have different elements, and a set is uniquely determined by its elements. Mereology, as a collective set theory, takes into consideration relations between elements. In comparison, ZFC set theory neglects such connections, i.e., we have abstract wholes in which elements are treated as independent entities. Having in mind entanglement, we incorporated a kind of an entangled correlation and made it a founding relation in NAM. We analyzed properties of the obtained model and incorporated ZFC set theory within NAM [19]. Moreover, we represented NAM in terms of non-classic algebras [20]. Finally, we were able to express a bipartite entanglement by the use of quaternions [17]. The novelty of the proposed model stands in the fact that in an investigated universe we can have classical atoms and indecomposable, but composite atoms represented by entanglement, which are not elements of classical ZFC set theory. 1 NAM=Non-Antisymmetric Mereology. It is a model of mereology in which we assume that the inclusion relation–⊆ is only reflexive and transitive. We deny the antisymmetry of ⊆. 2 The ZFC set theory is a classical model of set theory. The abbreviation–ZFC follows from the names of mathematicians, which contributed to developing this model: E. Zermelo and A.Fraenkel. ‘C’ stands for the Axiom of Choice taken into consideration in ZFC. This model forms a basis of exact sciences. © 2020 The Author(s). Published by IOP Publishing Ltd J. Phys. Commun. 4 (2020) 055018 L Obojska In quantum mechanics, the space of bipartite systems–(4)3, which is a complex Hilbert space, can also be thought of as a space of complex quaternions (known as biquaternions)–Ä 4. Real quaternions form a non- commutative division algebra, while complex quaternions have divisors of zero; therefore, nothing is simple any more5. Biquaternions have been widely applied as the Pauli matrices6: [21–25]; some authors investigated the relationship between rotations, biquaternions and the Pauli matrices, projective representations of the Lorentz group, etc [26–29]. In the following work, we will focus onmaximal entanglement within the algebra of biquaternions–Ä . We will construct representations of Bell states and create patterns of pure maximally entangled states. Finally, we will prove that there are no other maximally entangled states than those described in this paper. The work is organized as follows: section 2 presents some preliminary notions regarding the algebra of biquaternions. Section 3 describes different measures of entanglement. Section 4 presents a description of maximal bipartite entanglement within Ä and define patters of entangled states. Section 5 contains concluding remarks. 2. Biquaternions The algebra of biquaternions, was elaborated by W R Hamilton in the nineteenth century [30]. It is a non- commutative, non-division algebra, which is a tensor product of complex numbers and Hamilton quaternions–Ä . Any element of Ä can be described as a linear combination: ccicici01+++ijk 2 3,1() where: cccc0123,,, Î ; ijk,, are anti-commuting operators, i.e., ij=- ji, jkkj=- , ki=- ik.The anti- community leads to identities: ijkijk22== 2 = =-1. On the other hand, the complex imaginary unit–i commutes with every element of Ä and i2 =-1. If we denote by =Ä, VW, Î , V =+vV: v=c0, V =++ccc12ijk 3, |–the inner product, ×–the cross product, then V is a three dimensional vector: ⎛c1⎞ ⎜ ⎟ V = ⎜c2⎟ ()2 ⎝c3⎠ and v+V forms a biquaternion: ⎛c0⎞ ⎜c ⎟ V = ⎜ 1 ⎟.3() ⎜c2 ⎟ ⎝c3⎠ It can be easily shown that the space of biquaternions is linear: aVW+=+++=+++ b a()()()( v V b w W av bw aV bW ) for a,, b Î () 4 with quaternionic multiplication: VW·()·()=+v V w + W = vw -áñ++ V ∣ W vW wV +´ V W.5 () For any VWZ,,Î , the biquaternion algebra is associative: (·VW )· Z= V ·(·) WZ,6 () and non-commutative: []VW,2.7=-=´ V · W W · V VW () what means that Ä is a representation of a Lie algebra–sl ()2, [31]. We can observe that is obtained as a result of complexification of . This means that we change the basis from {1,ijk , , }to {1,iiiijk , , }. In this way the biquaternions ii, ij, ik, viewed in M2() representation become the Pauli matrices: ii = sx, ij = sy, ik = sz. Therefore, is isomorphic to 2×2 complex matrices; it is an algebra over with 4 dimensions or, equivalently, it is an algebra over with 8 dimensions [27, 32, 33]; the matrix representation is not unique. 3 ()4 is a four-dimensional complex space equipped with an inner product. 4 indicates real quaternions; i.e., quaternions with real coefficients. 5 A division algebra is an algebra, in which every, non zero element, has its inverse. In a ring P, an element a ¹ 0 is called a divisor of zero if there exists such b ¹ 0, bPÎ : ab=0. 6 s = 01, s = 0 - i , s = 10 . x (10) y ( i 0 ) z (01- ) 2 J. Phys. Commun. 4 (2020) 055018 L Obojska 2 can be seen as a Clifford algebra–Cl2, which contains both a complex plane– and a vector plane– . This – + - + - 2 means that Cl2 has an even odd grading, i.e. Cl2 =Å Cl22 Cl , where Cl2 = , Cl2 = . Cl2 has three involutions similar to complex conjugation in ; only a grade involution is an automorphism, while others are + anti-automorphisms. is isomorphic to the even subalgebra Cl3 ; the center of Cl3 is isomorphic to ; hence, 3 =Äcan be seen as a Clifford algebra–Cl3 of the Euclidean space . Cl3 has three different involutions likewise Cl2 [34]: grade ()* , quaternion (Clifford-conjugation) ()- and Hermitian (in Clifford algebras known as a reversion)(†). Under grade conjugation i -i, ii, j j, kk ; under quaternionic conjugation i i, ii- , j -j, kk - and under Hermitian conjugation: i -i, ii - , j -j, kk - . For a biquaternion V its norm can be defined in the following way7: ∣∣VVV ∣∣2 = tr († ) ; ( 8 ) however, since the outcome is a complex number, we rather speak about a semi-norm than a norm [35–38]. In the inner product can be introduced in the following way [38]: áñVW∣≔()S VW† ,9 () where SvwVW()VW =-áñ ∣. As a result: ∣∣VVV ∣∣2 ≔áñ ∣.10 ( ) 3. Measures of bipartite entanglement To state whether a composite system is separable or not it is enough to examine its tensor product, but we can also do it by the use of a density operator. It is known that in the case of entanglement8 , even if a composite system is absolutely pure, each of its constituents must be described by a mixed state [39]. Let us adopt a classical example of Alice and Bob. Let us assume that Alice has a complete knowledge of a combined bipartite system, – 9 composed of two parts A, B, described by a state ∣Yñ = åij, ciij∣∣ñÄ A j ñ B .
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