Acta Polytechnica doi:10.14311/AP.2013.53.0470 Acta Polytechnica 53(5):470–472, 2013 © Czech Technical University in Prague, 2013 available online at http://ojs.cvut.cz/ojs/index.php/ap THE NUMBER OF ORTHOGONAL CONJUGATIONS Armin Uhlmann∗ University of Leipzig, Institute for Theoretical Physics, PB: 100920, D-04009 Leipzig, Germany. ∗ corresponding author: armin.uhlmann@t-online.de Abstract. After a short introduction to anti-linearity, bounds for the number of orthogonal (skew) conjugations are proved. They are saturated if the dimension of the Hilbert space is a power of two. For other dimensions this is an open problem. Keywords: anti- (conjugate) linearity, canonical Hermitian form, (skew) conjugations. Submitted: 31 March 2013. Accepted: 1 May 2013. 1. Introduction The use of anti-linear or, as mathematicians call it, conjugate linear operators in physics goes back to E. P. Wigner, [11]. Wigner also discovered the struc- ture of anti-unitary operators, [12], in finite dimen- sional Hilbert spaces. The essential difference from the linear case is the existence of 2-dimensional irreducible subspaces so that the Hilbert space decomposes into a direct sum of 1- and 2-dimensional invariant spaces in general. Later on, F. Herbut and M. Vujičić were able to clarify the structure of anti-linear normal oper- ators, [4], by proving that such a decomposition also exists for anti-linear normal operators. While any linear operator allows for a Jordan decomposition, I do not know a similar decomposition of an arbitrary anti-linear operator. In the main part of the paper there is no discussion of what is happening in case of an infinite dimensional Hilbert space. There are, however, several impor- tant issues both in Physics and in Mathematics: A motivation of Wigner was in the prominent applica- tion of (skew) conjugations, (see the next section for definitions), to time reversal symmetry and related in- executable symmetries. It is impossible to give credit to the many beautiful results in Elementary Parti- cle Physics and in Minkowski Quantum Field Theory in this domain. However, it is perhaps worthwhile to note the following: The CPT-operator, the com- bination of particle conjugation C, parity operator P, and time-reversal T, is an anti-unitary operator acting on bosons as a conjugation and on fermions as a skew conjugation. It is a genuine symmetry of any relativistic quantum field theory in Minkowski space. The proof is a masterpiece of R. Jost, [7]. A further remarkable feature of anti-linearity is shown by CPT. This operator is defined up to the choice of the point x in Minkowski space on which PT acts as x →−x. Calling this specific form CPTx, one quite forwardly shows that the linear operator CPTxCPTy is representing the translation by the vector 2(x − y). The particular feature in the example at hand is the splitting of an executable symmetry operation into the product of two anti-linear ones. This feature can be observed also in some completely different situations. An example is the possibility to write the output of quantum teleportation, as introduced by Bennett et al. [1], [2, 8], as the action of the product of two anti-linear ones on the input state vector, see [3, 9, 10]. These few sketched examples may hopefully con- vince the reader that studying anti-linearity is quite reasonable — though the topic of the present paper is by far not so spectacular. The next two sections provide a mini-introduction to anti-linearity. In the last one it is proved that the number of mutually orthogonal (skew) conjugations is maximal if the dimension of the Hilbert space is a power of two. It is conjectured that there are no other dimensions for which this number reaches its natural upper bound. 2. Anti- (or conjugate) linearity Let H be a complex Hilbert space of dimension d < ∞. Its scalar product is denoted by 〈φb,φa〉 for all φa,φb ∈H. The scalar product is assumed linear in φa. This is the “physical” convention going back to E. Schrödinger. 1 is the identity operator. Definition 1. An operator ϑ acting on a complex linear space is called anti-linear or, equivalently, con- jugate linear if it obeys the relation ϑ( c1φ1 + c2φ2 ) = c∗1ϑφ1 + c ∗ 2ϑφ2, cj ∈ C. (1) As is common use, B(H) denotes the set (algebra) of all linear operators from H into itself. The set (linear space) of all anti-linear operators is called B(H)anti. Anti-linearity requires a special definition of the Hermitian adjoint. Definition 2 (Wigner). The Hermitian adjoint, ϑ†, of ϑ ∈B(H)anti is defined by 〈φ1,ϑ†φ2〉 = 〈φ2,ϑφ1〉, φ1,φ2 ∈H. (2) 470 http://dx.doi.org/10.14311/AP.2013.53.0470 http://ojs.cvut.cz/ojs/index.php/ap vol. 53 no. 5/2013 The Number of Orthogonal Conjugations A simple but important fact is seen by commuting ϑ and A = c1. One obtains (cϑ)† = cϑ†, saying: ϑ → ϑ† is a complex linear operation,(∑ cjϑj )† = ∑ cjϑ † j. (3) This is an essential difference from the linear case: Taking the Hermitian adjoint is a linear operation. A similar argument shows, that the eigenvalues of an anti-linear ϑ form circles around the null vector. If there is at least one eigenvalue and d > 1, let r be the radius of the largest such circle. The set of all values 〈φ,ϑφ〉, φ running through all unit vectors, is the disk with radius r. See [6] for the more sophisticated real case. We need some further definitions. Definition 3. An anti-linear operator ϑ is said to be Hermitian or self-adjoint if ϑ† = ϑ. ϑ is said to be skew Hermitian or skew self-adjoint if ϑ† = −ϑ. The linear space of all Hermitian (skew Hermitian) anti-linear operators are denoted by B(H)+anti respectively B(H) − anti. Rank-one linear operators are as usually written( |φ′〉〈φ′′| ) φ := 〈φ′′,φ〉φ′, and we define similarly( |φ′〉〈φ′′| ) anti φ := 〈φ,φ ′′〉φ′, (4) projecting any vector φ onto a multiple of φ′. Note that we do not use 〈φ′′| decoupled from its other part. We do not attach any meaning to 〈φ′′|anti as an expression standing alone!1 An anti-linear operator θ is called a unitary operator or, as Wigner used to say, an anti-unitary, if θ† = θ−1. A conjugation is an anti-unitary operator which is Hermitian, hence fulfilling θ2 = 1. The anti-unitary θ will be called a skew conjugation if it is skew Hermitian, hence satisfying θ2 = −1. 3. The invariant Hermitian form While the trace of an anti-linear operator is undefined, the product of two anti-linear operators is linear. The trace (ϑ1,ϑ2) := Tr ϑ2ϑ1 (5) will be called the canonical Hermitian form, or just the canonical form on the the space of anti-linear operators. An anti-linear ϑ can be written uniquely as a sum ϑ = ϑ+ + ϑ− of an Hermitian and a skew Hermitian operator with ϑ → ϑ+ := ϑ + ϑ† 2 , ϑ → ϑ− := ϑ−ϑ† 2 . (6) 1Though one could do so as a conjugate linear form. Relying on (5) and (6) one concludes (ϑ+,ϑ+) ≥ 0, (ϑ−,ϑ−) ≤ 0, (ϑ+,ϑ−) = 0. (7) In particular, equipped with the canonical form, B(H)+anti becomes an Hilbert space. Completely ana- logue, −(·, ·) is a positive definite scalar product on B(H)−anti. Bases of these two Hilbert spaces can be obtained as follows: Let φ1,φ2, . . . be a basis of H. Then( |φj〉〈φj| ) anti, 1 √ 2 (( |φj〉〈φk| ) anti + ( |φk〉〈φj| ) anti ) , (8) where j,k = 1, . . . ,d and k < j, is a basis of B(H)+anti with respect to the canonical form. As a basis of B(H)−anti one can use the anti-linear operators 1 √ 2 (( |φj〉〈φk| ) anti − ( |φk〉〈φj| ) anti ) . (9) By counting basis lengths one gets dimB(H)±anti = d(d± 1) 2 . (10) It follows: The signature of the canonical Hermitian form is equal to d = dimH. Indeed, dimB(H)+anti − dimB(H) − anti = dimH. (11) 4. Orthogonal (skew) conjugations The anti-linear (skew) Hermitian operators are the elements of Hilbert spaces. Their scalar products are restrictions of the canonical form (up to a sign in the skew case). It is therefore a legitimate question to ask for the maximal number of mutually orthogonal conjugation or skew conjugations. These two numbers depend on the dimension d = dimH of the Hilbert space only. Let us denote by N+(d) the maximal number of orthogonal conjuga- tions and by N−(d) the maximal number of skew conjugations. By (10) it is N±(d) ≤ d(d± 1) 2 . (12) To get an estimation from below, one observes that the tensor products of two conjugations and the tensor product of two skew conjugations are conjugations. Therefore N+(d1d2) ≥ N+(d1)N+(d2) + N−(d1)N−(d2) (13) and, similarly, N−(d1d2) ≥ N+(d1)N−(d2) + N−(d1)N+(d2) (14) because the direct product of two orthogonal (skew) conjugations is orthogonal. Now consider the case 471 Armin Uhlmann Acta Polytechnica that equality holds in (12) for d1 and d2. Then one gets the inequality N+(d1d2) ≥ 1 4 ( d1(d1 + 1)d2(d2 + 1) + d1(d1 − 1)d2(d2 − 1) ) and its right hand side yields d(d+ 1)/2 with d = d1d2. Hence there holds equality in (13). A similar reasoning shows equality in (14) if equality holds in (12). Hence: The set of dimensions for which equality takes place in (12) is closed under multiplication. To rephrase this result we call Nanti the set of di- mensions for which equality holds in (12): If d1 ∈ Nanti and d2 ∈ Nanti then d1d2 ∈ Nanti. 2 ∈ Nanti will be shown by explicit calculations below. Hence every power of two is contained in Nanti. Let us briefly look at dimH = 1. It is N+(1) = 1 and N−(1) = 0. Indeed, any anti-linear operator in C is of the form ϑaz = az∗. This is a conjugation if |a| = 1. There are no skew conjugations. The canonical form reads (ϑa,ϑb) = a∗b. Conjecture 1. Nanti consists of the numbers 2n, n = 0, 1, 2, . . . . Skew Hermitian invertible operators exist in even dimensional Hilbert spaces only. Therefore, no odd number except 1 is contained in Nanti. This, however, is a rather trivial case. Already for dimH = 3 the maximal number N+(3) of orthogonal conjugations seems not to be known. 4.1. The case dimH = 2 To show that 2 ∈ Nanti one chooses a basis φ1,φ2 of the 2-dimensional Hilbert space H and defines τ0(c1φ1 + c2φ2) = c∗1φ2 − c ∗ 2φ1, τ1(c1φ1 + c2φ2) = c∗1φ1 − c ∗ 2φ2, τ2(c1φ1 + c2φ2) = ic∗1φ1 + ic ∗ 2φ2, τ3(c1φ1 + c2φ2) = c∗1φ2 + c ∗ 2φ1. (15) For j,k ∈{1, 2} and m ∈{1, 2, 3} one gets 〈φj,τmφk〉 = 〈φk,τmφj〉 saying that these anti-linear operators are Hermitian. One also has τ2m = 1 for m ∈ {1, 2, 3}. Altogether, τ1,τ2,τ3 are conjugations. To see that they are or- thogonal one to another we compute τ1τ2τ3 = iτ0, τ2τ1 = iσ3, (16) τ1τ3 = iσ2, τ2τ3 = iσ1, (17) and τ2τ0 = σ2, τ0τ1 = σ1, τ3τ0 = σ3. (18) The trace of any σj is zero. Because of (16) and (18) we see, that τ1,τ2,τ3 is an orthogonal set of conjugations while τ0 is a skew conjugation. Now N+(2) = 3 and N−(2) = 1 as was asserted above. Acknowledgements I would like to thank B. Crell for helpful support. References [1] C. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W. Wootters. Teleporting an unknown quantum state via dual classical and Einstein-Podolsky- Rosen channels, Phys. Rev. Lett., 70: 1895-1898, 1993. [2] I. Bengtsson and K. Życzkowski, Geometry of Quantum States. Cambridge University Press, Cambridge 2006 [3] R. A. Bertlmann, H. Narnhofer, W. Thirring. Time-ordering Dependence of Measurements in Teleportation. arXiv:1210.5646v1 [quant-ph] [4] F. Herbut, M. Vujičić. Basic Algebra of Antilinear Operators and some Applications. J. Math. 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Wigner: Über die Operation der Zeitumkehr in der Quantenmechanik. Nachr. Ges. Wiss. Göttingen, Math.-Physikal. Klasse 1932, 31, 546–559. [12] E. P. Wigner: Normal form of anitunitary operators. J. Math. Phys. 1960, 1, 409–413. 472 Acta Polytechnica 53(5):470–472, 2013 1 Introduction 2 Anti- (or conjugate) linearity 3 The invariant Hermitian form 4 Orthogonal (skew) conjugations 4.1 The case dim H=2 Acknowledgements References