Competition between decoherence and purification: quaternionic representation and quaternionic fractals David Viennot Institut UTINAM (CNRS UMR 6213, Universit´ede Bourgogne-Franche-Comt´e, Observatoire de Besan¸con), 41bis Avenue de l’Observatoire, BP1615, 25010 Besan¸con cedex, France. We consider the competition between decoherence processes and an iterated quantum purification protocol. We show that this competition can be modelized by a nonlinear map onto the quaternion space. This nonlinear map has complicated behaviours, inducing a fractal border between the aera of the quantum states dominated by the effects of the purification and the area of the quantum states dominated by the effects of the decoherence. The states on the border are unstable. The embedding in a 3D space of this border is like a quaternionic Julia set or a Mandelbulb with a fractal inner structure. PACS numbers: 03.65.Yz, 03.67.Ac, 05.45.Df INTRODUCTION having a bounded orbit (zn)n N (zn+1 = fp(zn)) from the Julia set of the values with∈ unbounded orbits. Recip- Decoherence is a physical process consisting to the lost rocally, another fractal, the Mandelbrot set, is the border between the values p C for which the orbit of z = 0 is of the quantum properties due to the environment ef- ∈ 0 fects. Under decoherence, the purity (which measures bounded from the values for which it is unbounded. The the pure quantum behaviour of a state, see [1]) decreases. maps studied in [6–10, 12] and in this paper to represent Decoherence can result from entanglement between the the competition between decoherence and purification, quantum system and its environment [2], from chaotic or belong to the family of the Julia map fp. stochastic noises induced by the environment [3] or from Since mixed state space is larger than pure state space, thermal fields emitted by the environment [2]. Some re- the associated map describing the competition between searches hope to use the quantum laws for practical ap- decoherence and purification on a qubit has a phase space C plications, as quantum teleportation [4], quantum com- and a parameter space larger than . We will see that H puting [4] and quantum control [5]. For these goals, the the map can be represented into the quaternion space . decoherence processes are hampers ruining the attempts We can then think that the borders between the different to reach the desired targets. Rather than trying to nar- behaviours are not simple fractal curves but more dimen- row the decoherence processes (as in usual strategies), we sional objects as Mandelbulbs (see [13, 14]) or quater- could try to fight them by using a purification protocol. nionic Julia sets [15]. Such a one, as for example in [6] for a qubit (quantum This paper is organized as follows. Firstly, we present the bit), consists to manipulate the quantum system in order purification protocol. Second section presents the quater- to increase the purity of its state. Formally, the purifica- nionic representation of the qubit mixed states. Third tion is a nonlinear map of the state space, which is phys- section presents the quaternionic representation of the ically realized by entanglement, quantum measurement competition between decoherence and purification. Fifth and post-selection (see [6, 7] for details). By repeating a section shows the results of this competition (with the purification protocol, we want to fight against the deco- fractal borders between the purification dominated area herence. The question is then: Is the purification or the and the decoherence dominated area). Finally, we drawn arXiv:2003.02608v1 [quant-ph] 5 Mar 2020 decoherence which wins the competition? We can imag- the quaternionic fractal sets resulting from the competi- ine that the answer depends on the initial mixed state ρ. tion. A second question is then: what is the behaviour of the states at the border between the coherence dominated THE PURIFICATION PROTOCOL area and the purification dominated area? The nonlin- earity of the purification protocol induces some compli- Let z C be the complex parametrization of a pure cated behaviours. As shown in [8–10], if we repeat the ∈ purification protocol onto pure states, some of them are state of a qubit: stable (the pure state orbit reach cyclic points) but some z 0 + 1 ψ = | i | i (1) other states are unstable. The border between the two | i 1+ z 2 behaviours is a fractal set. | | 1 z 2 z A simple map of the complex plane inducing a compli- p ψ ψ = 2 | | (2) 2 | ih | 1+ z z¯ 1 cated behaviour is for example fp(z) = z + p (with | | p C) [11]. It is associated with a fractal curve which z is the complex coordinates onto the Bloch sphere of the is∈ the border between the Fatou set of the values z C qubit states (the complex plane is the stereographic pro- 0 ∈ 2 jection of the Bloch sphere). The purification protocol The mixed state of the first qubit (the mixing resulting S studied in [6–10, 12] induces the squaring of the pure from the entanglement with the second one) is then state ψ ψ : | ih | ρ = tr2 Ψ Ψ (9) | ih | z2 0 + 1 1 z 2 z cos(λ λ ) S ψ = | i | i (3) = | | 0 − 1 (10) | i 1+ z 4 1+ z 2 z¯cos(λ0 λ1) 1 | | | | − 2 (λ0 λ1) ıα ıϕ 1 z ze − ~−1 epcos x e sin x C ı H∆t = 2 o (|λ0| λ1) (11) Let U = e− = ıϕ ıα be the 1+ z ze¯ − 1 e− sin x e− cos x − | | evolution operator of the qubit during a short time dura- where tr2 is the partial trace onto the state space of the ~ω tion ∆t (with H = 2 σz + e(b)σx + m(b)σy, we have second qubit. The density matrix can then be repre- b sin(r∆t/2) ℜ ℑ ω λ tan x = | | , tan α = tan(r∆t/2), sented by the quaternionic number ζ = ze H (with √ b 2 cos2(r∆t/2)+ω2 − r ∈ | | λ = λ0 λ1 for the entanglement case) with ϕ = arg b π and r = ω2 + b 2). The succession of the − 4 2 purification− protocol and of the| Hamiltonian| evolution in- 1 ζ ζ p ρ = Co | | (12) duces on a pure qubit state the following transformation: 1+ ζ 2 ζ¯ 1 | | f (z) 0 + 1 Note that ζ = zeλ = z cos λ + z¯sin λ = eıφ(C 1 ) US ψ = α,p | i | i (4) − 2 C 2 where φ = arg z is the phase, C = z cos(λ) is the coher- | i 1+ fα,p(z) | | | | ence of the first qubit, and = 2 z sin(λ0 λ1) is the with the complex map: p concurrence of the entanglementC − between| | the− two qubits z2eıα + p [1]. f (z)= (5) We adopt the quaternionic representation of the density α,p e ıα pz¯ 2 − − matrix eq. 12 also for mixed states resulting from a de- ıϕ p = e tan x. fα,p is similar to a “renormalized” Ju- coherence process (note that any qubit mixed state can lia map. The succession of purification protocols with be represented by an entangled state of the qubit with interval ∆t is then represented by the dynamical sys- an ancilla qubit by the Schmidt purification procedure 2 ζ tem zn+1 = fα,p(zn). It as been studied in [8–10] (with H [1]). For ζ , 1+| |ζ 2 is the population of the state α 2πZ) and in [12] (with α 2πZ). C ∈ | | ∈ 6∈ 0 and o(ζ) is the coherence of the mixed state. With these| i interprations,| | several ζ in H correspond to the same QUATERNIONIC REPRESENTATION mixed state, it can be then interesting to transform any ζ in the form zeλ: C ζ Co(ζ) C λ C We want consider mixed states of the qubit: o(ζ)+ | −Co(ζ) | o(ζ) = ze if o(ζ) =0 p(ζ)= π | | 6 2 C 1 z 2 z cos λ (ζ = ze if o(ζ)=0 ρ = | | (6) 1+ z 2 z¯cos λ 1 (13) | | C Co(ζ) C with z = o(ζ) and cos λ = | ζ | for the case o(ζ) = 0. ρ is a density matrix. λ is the mixing angle, for λ = 0 | | 6 π ρ is pure state and for λ = 2 the coherence of the qubit is zero (maximal mixing). It needs to take into account DYNAMICS IN THE QUATERNIONIC this new parameter in the representation. REPRESENTATION In [16] the authors introduce a quaternionic representa- tion of the qubit pair states in order to study the entan- We want consider transformations DUS(ρ) where S is H glement phenomenon. The quaternion space is the set the purification protocol, U is the Hamiltonian evolution of noncommutative numbers ζ = a + ıb + c + kd, with of the qubit, and D is a decoherence process (DU can a,b,c,d R, ı2 = 2 = k2 = 1 and ı = k, ı = k, ∈ − − come from the integration of a Lindblad equation dur- k = ı, k = ı, kı = , ık = . We denote: e(ζ)= a, ing ∆t, see [2]). The purification protocol induces the − − C ℜ m1(ζ) = b, m2(ζ) = c, m3(ζ) = d, o(ζ) = a + ıb squaring of the density matrix: ℑ 2 ℑ 2 2 2ℑ 2 1 ζ¯ and ζ = ζζ¯ = a + b + c + d . Note that ζ− = 2 . | | ζ 1 z 4 z2 cos2 λ For a state of two qubits: | | S(ρ)= | | (14) 1+ z 4 z¯2 cos2 λ 1 z cos λ 00 + z sin λ 01 + cos λ 10 + sin λ 11 | | Ψ = 0| i 0| i 1| i 1| i We see why it is a purification protocol: with- | i 1+ z 2 out Hamiltonian evolution and decoherence process, | | n (7) limn + S (ρ) = 0 0 if z > 1 or 1 1 if z < 1.