The Conal representation of Quantum States and Non Trace-Preserving Quantum Operations Pablo Arrighi, Christophe Patricot To cite this version: Pablo Arrighi, Christophe Patricot. The Conal representation of Quantum States and Non Trace- Preserving Quantum Operations. Physical Review A, American Physical Society, 2003, 68, pp.042310. hal-00944124 HAL Id: hal-00944124 https://hal.archives-ouvertes.fr/hal-00944124 Submitted on 7 May 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The Conal representation of Quantum States and Non Trace-Preserving Quantum Operations 1, 2, Pablo Arrighi ∗ and Christophe Patricot ∗∗ 1Computer Laboratory, University of Cambridge, 15 JJ Thomson Avenue, Cambridge CB3 0FD, U.K. 2 DAMTP, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, U.K. d We represent generalized density matrices of2 a -complex dimensional quantum system2 as a sub- Rd E1,d −1 cone of a real pointed cone of revolution in , or indeed a Minkowskian cone2 in . Gener- alized pure states correspond to certain future-directed light-like vectors of E1,d −1. This extension of the Generalized Bloch Sphere enables us to cater for non-trace-preserving quantum operations, and in particluar to view the per-outcome effects of generalized measurements. We show that these consist of the product of an orthogonal transform about the axis of the cone of revolution and a positive real linear transform. We give detailed formulae for the one qubit case and express the post-measurement states in terms of the initial state vectors and measurement vectors. We apply these results in order to find the information gain versus disturbance tradeoff in the case of two equiprobable pure states. Thus we recover Fuchs and Peres’ formula in an elegant manner. PACS numbers: 03.65, 03.67, 03.67.d Keywords: Cones, Bloch sphere, Generalized measurements, Information gain versus disturbance I. INTRODUCTION inite) positive hermitian operators, thereafter denoted + C Hermd ( ), is a closed convex cone, and that all ad- missible quantum operations should be a subset of the The space of pure states of finite d-dimensional Quan- transformations of this cone. d tum Mechanics CP , set of rays in the complex Hilbert In spite of being so central in Quantum Information d space C , is, as most complex spaces, not easy to vi- Theory, the tradeoff between how much Shannon Infor- sualize. Physical motions, let alone unitary time evolu- mation one may gain about a quantum system versus tions, have no clear geometric interpretation. However, how much Disturbance the observation must necessar- Cd C the set of hermitian operators on , Hermd( ), is a ily cause to the system, remains extremely difficult to (d2 1)-dimensional real vector space, and as such is cer- − quantify. Quantum cryptographists tend to circumvene tainly easier to represent geometrically. States, or more the problem: most of their proofs are a witty blend of generally density matrices, form of course a subset of the particular symmetries of the protocol in question, C Hermd( ). The Generalized Bloch Sphere Representa- together with a convoluted machinery. A few attempts tion ([1]-[3]) is a famous application of this fact which have been made to solve the tradeoff [4][5], but only one has proved to be popular and elucidating: a given den- [6] deals with discrete ensembles - namely the case of two sity matrix can be represented as a real vector inside a equiprobable pure states, and this is already something. (hyper) sphere. Unfortunately the approach involves lengthy algebra and It turns out that this representation defined for density a number of assumptions. One should be able to find a arXiv:quant-ph/0212062v3 25 Jan 2003 C matrices, or unit trace positive elements of Hermd( ), is method which gives a glimpse of intuition about the ge- only good at handling unitary or trace preserving quan- ometry of optimal measurements, and for this purpose, tum operations on density matrices: the former induce we think that our approach is useful. rotations of the Bloch vector, the latter affine transfor- In section II we consider general quantum systems of d mations [2]. Individual outcomes of generalized mea- complex dimensions. We give a representation of the set surements, for example, are not directly representable. + C of positive hermitian matrices Hermd ( ) as a subcone of Considering the insight the Bloch Sphere representation 2 Rd gave to unitary and trace-preserving operations, it seems a real Minkowskian cone in , and analyse geometrical interesting, for the mere sake of geometry at first, but properties of generalized measurements in this setting. mainly to give a useful picture to tackle Quantum In- We find that our approach is particularly useful to rep- formation problems, to extend it to cater for non-trace- resent per-outcome post-measurement states, and that preserving quantum operations. This is further moti- pure states correspond to certain light-like vectors of the vated by the fact that the space of (semi-definite or def- cone. Unitary operators on the complex system become real orthogonal transforms, while positive operators be- come real positive transforms. Section III should be of special interest for quantum information theorists: we ∗Electronic address: [email protected] treat the d = 2 one qubit case in full detail. We find ∗∗Electronic address: [email protected] further geometrical relations between measurement vec- 2 tors, state vectors and post-measurement state vectors Equation (3) says that φ is an isometric isomorphism of 2 d and give explicit formulae. In section IV we apply our (Hermd(C), Tr( )) onto (R , (1/d)( . )). Therefore any 1 results to a typical Information gain versus Disturbance linear operator L on Hermd(C) defines via φ and φ− an 2 tradeoff scenario in which Alice gives Eve two equiprob- d 1 operator on R , M(L) = φ L φ− . This definition able pure states. Thus we recover Fuchs and Peres’ for- yields the following “morphism”◦ property◦ : mula in an elegant and geometrical manner. Lemma 1 If L1,L2 are linear operators on Hermd(C), 1 then M(Li)= φ Li φ− for i=1, 2 are endomorphisms 2 ◦ ◦ II. CONAL REPRESENTATION OF of Rd and satisfy d-DIMENSIONAL QUANTUM SYSTEMS M(L1 L2)= M(L1)M(L2) (4) ◦ The state of such a system is described by a d d × In particular, any complex d d matrix A defines via density matrix. We shall express hermitian matrices as × AdA : ρ AρA† a linear operator on Hermd(C) which real linear combinations of Hilbert-Schmidt-orthogonal 7→ corresponds to a real endomorphism M(AdA): ρ hermitian matrices, and then restrict this representation 7→ + C M(AdA)ρ ; and AdAB = AdA AdB implies M(AdAB)= to elements of Hermd ( ), or generalized density matri- ◦ + C M(AdA)M(AdB ). As a direct consequence of this and ces. Hermd ( ) turns out to be “isomorphic” to a con- 2 the previous definitions , calling GLn(K) the group of vex subcone of a cone of revolution in Rd , or indeed a K 2 invertible n n matrices on the field , we get : 1,d 1 Minkowskian future cone in E − . We then analyse × the effects of quantum operations on density matrices in Lemma 2 For any subgroup G of GLd(C), the following this representation. mapping ψ : G ψ(G) GL 2 (R) → ⊂ d 1 A. Hermitian matrices A M(AdA)= φ− AdA φ (5) 7→ ◦ ◦ 2 is a group homomorphism. ψ(G) is a subgroup of Let τi , i 1,...,d 1 , be a Hilbert-Schmidt GL 2 (R). orthogonal{ } basis∈ (as{ in (1))− of d} d traceless hermitian d × I matrices, and let τ0 be the identity matrix . Throughout Note that since ψ(I) = ψ( I) = I, ψ is not necesseraly 2 this article latin indices will run from 1 to d 1, greek injective. Moreover ψ is certainly− not linear. An inter- indices from 0 to d2 1, and repeated indices are− summed − esting subgroup is the Special Unitary group SU(d) = unless specified. We take the τµ’s to satisfy by definition: U GLd(C) / UU † = I, det U = 1 . We call { ∈ T } SO(n) = O GLn(R) / OO = I, det O = 1 the special orthogonal{ ∈ group in n-dimensions. } µ,ν Tr(τµτν )= dδµν (1) ∀ Lemma 3 Special Unitary transformations on with δ the Kronecker delta. τµ µ is a basis of Hermd(C), Hermd(C), AdU : ρ UρU † with U SU(d), in- { } 2 7→ ∈ and any hermitian matrix A Hermd(C) decomposes on duce rotations of Rd about the I-axis. In fact the linear ∈ this basis as transforms ψ(U): ρ ψ(U)ρ are special orthogonal and 7→ ψ(SU(d)) is a subgroup of SO(d2 1). It is a proper 1 − A = Tr(A)I + Tr(Aτi)τi subgroup when d 3. Moreover, ψ(U(d)) = ψ(SU(d)). d ≥ 1 Proof: Let ρ = (1/d)(Tr(ρ)I + ρ τ ) a hermitian matrix. = Tr(Aτµ)τµ (2) i i d Using (3) and the fact that AdU is trace-preserving for 2 U unitary: Rd Letting A = (Aµ) with Aµ = Tr(Aτµ) be the component vector of∈A in this particular basis, we have 2 UρU †.UρU † = (Trρ) + ψ(U)ρ i ψ(U)ρ i = dTr(UρUUρU ) C 1 † † A, B Hermd( ), AB = AµBν τµτν ∀ ∈ d2 = dTrρ2 = ρ.ρ 1 1 hence TrAB = A.B A B (3) = (Trρ)2 + ρ ρ d ≡ d µ µ i i 2 In addition to preserving the first component ρ = Trρ, Rd 0 We shall call A the vector in , −→A = (Ai) the restricted 2 2 d 1 d 1 ψ(U) preserves the R − scalar product ρ .ρ .