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 . The density operator of the above 10 composite system rAB =YñáY∣ ∣ , in a matrix representation, will take the following form: (rAB)ij, kl = ccij* kl. For a pure state ∣Yñ, the density operator is a projection of the rank one [40, 41]. Alice’s knowledge about A without looking at B (we assume that Alice acts by the use of an operator, which belongs to A and acts trivially on ) fi B can be expressed in terms of a partial trace, de ned by the use of a density matrix rAB as follows:
rrA ==trB AB å ccij* kj∣∣ i ñá A k A . () 11 j
In the case when Alice has only a single spin (which can be a superposition of two states: ∣ñ = 1 or (0) ∣ñ = 0 , the density matrix is the following: (1)
ryyA =ñá∣∣AA.12 ()
22 In other words if ∣yaAñ =∣∣ ñ + b ñ (ab, Î and ∣ab∣∣∣+=1) then:
⎛aa** ab⎞ r = ⎜ ⎟.13() A ⎝ba** bb⎠
For a bipartite entangled system we have a 44x density matrix; hence, if ⎛1⎞ ⎛0⎞ ⎛0⎞ ⎜0⎟ ⎜ ⎟ ⎜ ⎟ ∣Yñ =cccc∣∣∣∣ ñ + ñ + ñ + ñ (ccÎ ,1å ∣∣2 = ); ∣ñ = ⎜ ⎟ ∣ñ = ⎜1⎟, ∣ñ = ⎜0⎟, 123 4i i i ⎜0⎟ ⎜0⎟ ⎜1⎟ ⎝0⎠ ⎝0⎠ ⎝0⎠ ⎛0⎞ ⎜ ⎟ ∣ñ = ⎜0⎟, then: ⎜0⎟ ⎝ ⎠ 7 1 In the algebra of biquaternions it is possible to define more than one norm. 8 We limit ourselves only to a bipartite system. 9 ∣ijññAB, ∣ are basis vectors of ()2 . 10 where áY∣ is a dual vector to ∣Yñ.
3 J. Phys. Commun. 4 (2020) 055018 L Obojska
⎛ ****⎞ ⎜cc1 1 cc1 2 cc1 3 cc1 4⎟ ⎜cc****1 cc2 cc3 cc4⎟ r = ⎜ 2 2 2 2 ⎟.14() AB cc**** cc cc cc ⎜ 3 1 3 2 3 3 3 4⎟ ⎝cc4****1 cc4 2 cc4 3 cc4 4⎠ ’ – Alice s density matrix rA obtained from the full density matrix rAB is the following: ⎛ ****⎞ ⎜cc1 1 ++ cc2 2 cc1 3 cc2 4⎟ rA = ⎜ ⎟.15() ⎝cc3****1 ++ cc4 2 cc3 3 cc4 4⎠
A mixed state of one spin particle is of the form: rM = åi pi∣yyiiñá ∣, where åi pi = 1, 0 pi 1. For – – every mixed state rM there exists a pure state ∣Yñcomposed of two spin particles and:
rrrM ==trB AB A .() 16 Moreover, for a mixed state we will have:
22 rrM ¹ Therefore, a subsystem A of a pure entangled state is a mixed state. Now, since rM is Hermitian, it can be expressed in a diagonal form and none of its elements is equal to one, otherwise rM would represent a pure state. By the use of a partial trace, we can also calculate the von Neumann entropy, which is considered the best measure of pure entangled states and is defined as follows: 2 Eplogp=- 2 2.18() rM å ii2 i=1 We can also take into consideration the concurrence–C expressed in the following way [42]: CccccY =-2;∣∣14 23 () 19 or entanglement of formation, which quantify mixed entanglement; however, it will not be used in the presented manuscript. 4. Bipartite entanglement within the algebra of biquaternions Considering a singlet state as a collective whole, we can assume that it has been created in a process of splitting (=dividing) one particle into a pair of two different, yet indistinguishable particles, i.e. one particle is annihilated and in its place, two new particles appear11. Taking into consideration features of a division relation and the above property of an entangled state, we get that the entangled correlation (EC) is reflexive, non-antisymmetric and transitive. In an abstract way, if | denoted entanglement, it would fulfill the following conditions: a∣∣bbaa,,¹ b. This statement is trivial in division algebras, e.g., in ; however, in nature, there are no fractions during decays of particles; therefore, the range of values of quotients will be limited to integer values; in the case of it will be as follows i,1}. It is known that the wave function–∣Fñ, describing a composite bipartite state, is an element of a HHÄ@()4 @: H ==()2;however, if ∣Fñis entangled then it forms an indecomposable whole. ()b We assumed that ∣Fñis created from ∣yñÎ(2) in a process of splitting. The most natural way to decompose a, b is the following: a =+aiaaa1212 =(), , bb=+1212 ibbb =(), , where a1212,,,abbÎ . This decomposition is not unique but it seems to be the feature of real phenomena. Since a composite bipartite state–∣Fñis an element of , we create biquaternions as pairs of complex numbers as follows: (aabb1212,,,) or (bbaa1212,,,). a and b are related by (EC); hence, the quotients are elements of {--ii,, 1}. In this way we create a subset of in which we hope to find all entangled states. The rest of the presented work will be dedicated to prove that the obtained elements of are indeed entangled. Let us begin with a simple case, i.e.: ba22==0 and a=bi. We will get the following two biquaternions: Qbib= ((,0 )( 0, ), P = ((0,bi )( b , 0 )). For Q, we will have12: 11 By analogy, in mathematics, we find a famous Banach-Tarski theorem on the paradoxical division of a ball, which can correspond to entanglement. 12 In each step of investigation, we assume the normalization. 4 J. Phys. Commun. 4 (2020) 055018 L Obojska ⎛bi⎞ ⎜ ⎟ ∣()Yñ =Q = ⎜ 0 ⎟,20 ⎜ 0 ⎟ ⎝ b ⎠ ⎛ **- ⎞ ⎜ bb00 ibb⎟ r ()Q = ⎜ 0000⎟.21 ( ) AB ⎜ 0000⎟ ⎝ib** b00 b b ⎠ The rank of rAB (Q) is one; therefore, ∣Yñrepresents a pure state. Moreover, ⎛ * ⎞ r ()Q = ⎜⎟bb 0 ;22 ( ) A ⎝ 0 bb* ⎠ hence, A is maximally mixed and ∣Yñis maximally entangled (EA(Q)=1). Similar outcomes we get for a =-bi, a =-b and for P, a =bi or a =-b. As a result, we get eight different biquaternionic patterns: ⎛bi⎞ ⎛-bi⎞ ⎛ 0 ⎞ ⎛ 0 ⎞ ⎛-b⎞ ⎛ 0 ⎞ ⎛b⎞ ⎛0⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 ⎟, ⎜ 0 ⎟, ⎜ b ⎟, ⎜ b ⎟, ⎜ 0 ⎟, ⎜ b ⎟, ⎜0⎟, ⎜b⎟.23() ⎜ 0 ⎟ ⎜ 0 ⎟ ⎜bi⎟ ⎜-bi⎟ ⎜ 0 ⎟ ⎜-b⎟ ⎜0⎟ ⎜b⎟ ⎝ b ⎠ ⎝ b ⎠ ⎝ 0 ⎠ ⎝ 0 ⎠ ⎝ b ⎠ ⎝ 0 ⎠ ⎝b⎠ ⎝0⎠ For example, the last case in (23) corresponds to: a ==b 0 and a = bi. We can observe that for b =i 12 2 we get Bell states defined in : ⎛1⎞ 1 ⎜0⎟ 1 1 Bi1 ==+⎜ ⎟ ()⟼()1 k 1, -sz () 24 2 ⎜0⎟ 2 2 ⎝1⎠ ⎛0⎞ ⎜ ⎟ 1 1 1 1 Bii2 ==+--⎜ ⎟ ()⟼(ij ssxy ),25 () 2 ⎜1⎟ 2 2 ⎝0⎠ ⎛ 0 ⎞ ⎜ ⎟ 1 1 1 1 Bii3 = ⎜ ⎟ =-()⟼(ij -+ssxy ),26 () 2 ⎜-1⎟ 2 2 ⎝ 0 ⎠ ⎛ 1 ⎞ 1 ⎜ 0 ⎟ 1 1 Bi4 = ⎜ ⎟ =-()⟼()1 k 1. +sz () 27 2 ⎜ 0 ⎟ 2 2 ⎝-1⎠ Let us study a general case, now. For any bb=+1212 ibbb =(), : bb12,,,0ι bb 12 and a = bi, we will have: a =-bibbb21 + =() - 21,;therefore, ∣Yñ1 = ()ab, will take the following form: ⎛-b ⎞ ⎜ 2⎟ b ∣()Yñ =⎜ 1 ⎟,28 1 ⎜ ⎟ ⎜ b1 ⎟ ⎝ b2 ⎠ ⎛ * ***⎞ ⎜ bb2 2 --- bb2 1 bb2 1 bb2 2⎟ ⎜-bb1****2 bb1 1 bb1 1 bb1 2 ⎟ r ()Y=1 ⎜ ⎟.29 () AB -bb**** bb bb bb ⎜ 1 2 1 1 1 1 1 2 ⎟ ⎝-bb2****2 bb2 1 bb2 1 bb2 2 ⎠ Since rAB ()Y1 is hermitian, if we denote: A = bb2* 2, Bbb=- 2* 1, Cbb=- 1* 2, D = bb1* 1, then: ⎛ ABB- A⎞ ⎜ - ⎟ r ()Y=⎜ CDD C⎟.30 () AB 1 ⎜ CDD- C⎟ ⎝---ABBA⎠ 5 J. Phys. Commun. 4 (2020) 055018 L Obojska and: A* = A, D* = D, BC* = ; hence A, D Î . As a result: ⎛ ABB- A⎞ ⎜ ⎟ BDDB**- r ()Y=1 ⎜ ⎟.31 () AB ⎜ BDDB**- ⎟ ⎝---ABBA⎠ Alice’s density matrix is the following: ⎛ADBC+-⎞ r ()Y=⎜⎟.32 () A 1 ⎝ CBAD-+⎠ if if For a maximally entangled state we will have: A=D; hence, ∣bb12∣∣∣= . Let bbe11= ∣∣ 1, bbe22= ∣∣ 2, then BCBB-=-* =2Im i() B = 2sin i (ff12 -).Iff12-=Îfpkk; , then we will get maximally entangled states. Since a ¹ b then kn=+21, n Î0. As a result we get four different patterns of pure maximally entangled states: ⎛-b ⎞ ⎛ b ⎞ ⎛ b ⎞ ⎛ b ⎞ ⎜ 2⎟ ⎜ 2 ⎟ ⎜ 2 ⎟ ⎜ 2 ⎟ b -b b b ⎜ 2 ⎟,,,.⎜ 2⎟ ⎜ 2 ⎟ ⎜ 2 ⎟ () 33 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ b2 ⎟ ⎜ b2 ⎟ ⎜-b2⎟ ⎜ b2 ⎟ ⎝ b2 ⎠ ⎝ b2 ⎠ ⎝ b2 ⎠ ⎝-b2⎠ If A ¹ D then bb12¹ and bb1 ¹ 2* and the rank of rAB ()Y1 is two; therefore, ∣Yñ1 represents a mixed state. If we denote by: M =+AD, N =-B C then rA ()Y1 takes the following form: r ()Y= MN.34 () A 1 ()-NM It can be verified that if the the rank of r ()Y is one; hence, ∣Yñis separable, e.g. bb==1 , i , A 1 1 1 2 2 2 2 M =-iiii +1 =1 , N =-- + = . If the rank of r ()Y is two, we get a mixed entangled state, e.g.: 4 4 2442 A 1 bb==1 ,;1 + i hence, M ==-+1 ,;N 1 -+iii1 1 1 =therefore, ∣Fñ=()ab,, a = bi 1 3 2 23 2 23 3 3 23 3 1 represents a separable state and ∣Fñ=2 ()ab,, a = birepresents a mixed entangled state: ⎛ --⎞ ⎛ ⎞ 1 i -i ⎜ 2 ⎟ ⎜ ⎟ 1 1 1 1 -i 1 ⎜ 1 ⎟ ∣∣Fñ=12⎜ ⎟ =Ä, Fñ= ⎜ ⎟ ()35 2 ⎜ 1 ⎟ 2 ()i (1 ) 3 ⎜ 1 ⎟ ⎝ i ⎠ ⎜ 1 + i ⎟ ⎝ 2 ⎠ and ⎛ ⎞ ⎛ ⎞⎛ ⎞ 1 i 1 0 i 1 ⎜ 23⎟ -ii⎜ 6 ⎟⎜ 2 2 ⎟ rA ()F=2 ⎜ ⎟ = ⎜ ⎟⎜ ⎟,36 () - i 1 ()11 0 5 - i 1 ⎝ 3 2 ⎠ ⎝ 6 ⎠⎝ 2 2 ⎠ EA ()F=2 0, 508. Similar results we obtain for a =-bi; hence, we can get other patterns of biquaternionic ( ) mixed entangled states bb12¹ and bb1 ¹ 2* in spite of the following one: ⎛-b ⎞ ⎜ 2⎟ b ∣()Fñ = ⎜ 1 ⎟.37 ⎜ ⎟ ⎜ b1 ⎟ ⎝ b2 ⎠ In conclusion: Corollary 1. In the algebra of biquaternions there are no other maximally entangled states than those described by patterns (23), (33). For maximal entanglement, the concurrence (or von Neumann entropy) is equal to 1; therefore, the proof consists of finding the solutions of the following equation: ∣cc-= cc∣ 1 , where å ∣∣c 2 = 1. This means that 14 23 2 i i either ∣cc∣∣∣==1 and ∣cc∣∣∣==0 or ∣cc∣∣∣==1 and ∣cc∣∣∣==0 or ∣cccc∣∣∣∣∣∣∣====1 142 23 232 14 123 42 and the phase difference between products cc14and cc23equals (21,kk+Î)p . 6 J. Phys. Commun. 4 (2020) 055018 L Obojska For example the state: ⎛1 - i ⎞ ⎜ 2 ⎟ ⎜ 0 ⎟ ∣()Yñ = ⎜ ⎟ 38 ⎜ 0 ⎟ ⎜1 + i ⎟ ⎝ 2 ⎠ is of the form described by (23), for b = 1 + i . We can observe, that the above example corresponds to the case in 2 which one of the coefficients is a complex conjugation of the other. 5. Conclusions In the following work we showed that within the biquaternion algebra– =Ä, by the use of a division relation, applied for description of bipartite entanglement, we can figure out twelve patterns of pure maximally entangled states, i.e.: ⎛bi⎞ ⎛-bi⎞ ⎛ 0 ⎞ ⎛ 0 ⎞ ⎛-b⎞ ⎛ 0 ⎞ ⎛b⎞ ⎛0⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 ⎟, ⎜ 0 ⎟, ⎜ b ⎟, ⎜ b ⎟, ⎜ 0 ⎟, ⎜ b ⎟, ⎜0⎟, ⎜b⎟.39() ⎜ 0 ⎟ ⎜ 0 ⎟ ⎜bi⎟ ⎜-bi⎟ ⎜ 0 ⎟ ⎜-b⎟ ⎜0⎟ ⎜b⎟ ⎝ b ⎠ ⎝ b ⎠ ⎝ 0 ⎠ ⎝ 0 ⎠ ⎝ b ⎠ ⎝ 0 ⎠ ⎝b⎠ ⎝0⎠ for b = 1 or b = i or b = 1 i or b = 13i , etc. 2 2 2 22 ⎛-b⎞ ⎛ b ⎞ ⎛ b ⎞ ⎛ b ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ - ⎜ b ⎟,,,,⎜ b⎟ ⎜ b ⎟ ⎜ b ⎟ () 40 ⎜ b ⎟ ⎜ b ⎟ ⎜-b⎟ ⎜ b ⎟ ⎝ b ⎠ ⎝ b ⎠ ⎝ b ⎠ ⎝-b⎠ for b =1 or b =i or b = 1 i , etc or 2 2 22 Moreover, we proved that they are the only patterns describing such states within (Bell states derive from them); hence, we have uncountably many pure maximally entangled states defined by (23), (33) in . Finally, for bb12,:ι¹ bbbb 1 2 , 1 2*, we can get patterns, describing mixed entangled states, e.g.: ⎛-b ⎞ ⎜ 2⎟ b ∣()Fñ = ⎜ 1 ⎟.41 ⎜ ⎟ ⎜ b1 ⎟ ⎝ b2 ⎠ One can ask whether it is possible to prove the presented theory, experimentally. We do not exclude such a possibility but, first of all, we have to understand the nature of rotations in (4) since it seems they influence entanglement. We know, that each rotation in splits in two rotations acting within two orthogonal planes [38]. We have to connect these rotations with a process of making objects entangled what means that we have to create a special unitary operation acting on biquaternions. Having invented such an operation, we will be able to use it, for example, to reproduce the protocol of teleportation. Such investigations are in progress. Another question that can arise is how quantum decoherence is related to entanglement. It is known that quantum decoherence treats a local quantum system as something interacting with a much larger open system. This interaction is expressed in the information loss, i.e., the loss of quantum coherence what is described by the use of a density matrix. Its diagonal elements represent the probability to find the system in a given state and the off diagonal elements are the amplitudes between states (coherences); therefore, a quantum decoherence can be seen in the temporal decay of these terms However, in the eigenspaces of the observable Λ that commutes with the interaction Hamiltonian, when the interaction with environment dominates, the reduced density matrix ends up being diagonal. This commutation relation guarantees that the observable Λ is the constant of motion of the interaction Hamiltonian. Thus when a system is in the eigenstate of Λ, interaction with the environment will leave it unperturbed [43, 44]. Now, coming back to entanglement. We know that in the case of the information loss, any quantum state evolves from a pure state to a mixed state. In the case of maximally entangled states, we observe the evolution from pure states (canonical basis) to pure states (entangled basis). Algebraically, we simply change the basis by the use of a unitary operation. To determine whether a system is entangled or not, we often observe its spin angular momentum–Λ, which is an intrinsic property of a particle, a constant unrelated to any sort of motion. Therefore, taking into consideration the above facts, it seems that our entangled system is in the eigenspace of Λ what causes that it remains immune to decoherence. 7 J. Phys. Commun. 4 (2020) 055018 L Obojska Acknowledgments I would like to thank the Referee for his stimulating questions, which helped in improving this paper. 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