The Depolarizing Channel can be Worse Than the Dephasing Channel When Replacing the Identity map in a Problem of Quantum Control

The Depolarizing Channel can be Worse than the Dephasing Channel when replacing the Identity map in a Problem of Quantum Control Aman Chawla November 7, 2011 There are several types of quantum channels mentioned in the text [2]. One way of comparing quantum channels is to look at their maximum input-output fidelities. Another way, which we employ in this paper, is to use the channel as part of a feedback quantum control system and compute the maximum fidelity achievable. In this paper we look at a feedback quantum control system described in [1]. The system is as follows: •A qubit is prepared in a certain state. •The state is passed through the dephasing channel. •A weak measurement is applied on the state. •Based on the measurement result a correction or feedback control is applied on the qubit. The end-to-end fidelity is then computed and found to be optimal. We will be interested in making a more realistic model of quantum control where the feedback quantum control (weak measurement followed by control) is itself mediated by a noisy quantum channel. That is, after the weak measurement, the qubit undergoes a CPTP quantum operation before being subjected to the quantum control operation. In a sense, this is already implemented in [1] because one can view the absence of the CPTP map as the presence of the Identity map. But we will be interested in putting explicit (non-Identity) CPTP maps in between the weak measurement and quantum control steps. We will study the dephasing map and the depolarizing map, find their optimal fidelities and compare them. As a first baby step, let us consider inserting first the dephasing and then the depolarizing channel where the Identity map is at present - after the weak measurement and before the quantum control - and compute the fidelity of the scheme in each case. Note that this is not the optimal fidelity for each case since 1 one could further optimize the quantum controller specficially for each CPTP map considered. Nevertheless, the calculations are given below, and may be instructive. 1 Dephasing map The initial state of the qubit is: |ψ1i= cos θ 2(|0i+|1i √2) + sin θ 2(|0i−|1i √2) (1) =α|0i+β|1i where α≡1 √2(cos θ 2+ sin θ 2) (2) and β≡1 √2(cos θ 2−sin θ 2) (3) Thus, the corresponding density operator |ψ1ihψ1|=1 2I+αβX +α2−β2 2Z(4) where we have used the fact that α2+β2= 1. Here we introduce a notation that will help in simpler expressions for density matrices. We rely on the fact that the Pauli matrices form a basis for complex 2×2 matrices and we express a general matrix as a vector in this fourdimensional space, with each coordinate specifying the coefficient multiplying the Pauli matrices I, X, Y and Zin that order. Thus the density operator above becomes |ψ1ihψ1|={1 2, αβ, 0,α2−β2 2}(5) This then goes through the dephasing channel of [1]. The corresponding output is E(|ψ1ihψ1|) = p(Z|ψ1ihψ1|Z) + (1 −p)|ψ1ihψ1|(6) This simplifies to E(|ψ1ihψ1|) = {1 2,(1 −2p)αβ, 0,α2−β2 2}(7) Next the weak measurement given by measurement operators M0and M1 is applied. We first simplify the expressions for the measurement operator M0. We obtain, 2 M0=γI + ∆Y(8) where γ≡(1 2cos χ 2+1 2sin χ 2) (9) and ∆=(1 2cos χ 2−1 2sin χ 2) (10) Thus M† 0=γI + ∆Y(11) where we have used the Hermitian-property of the Pauli matrix Y. The non-normalized (by the probability of its occurrence) post-measurement state is given by ρ(0) out =M0Ep(|ψi1hψ|1)M0.(12) This simplifies to ρ(0) out ={1 2(γ2+ ∆2),(1 −2p)αβ(γ2−∆2), γ∆,α2−β2 2(γ2−∆2)}(13) corresponding to the measurement result 0. This state enters the dephasing feedback channel. The corresponding output is Ep0(ρ(0) out) = (1 −p0)ρ(0) out +p 0(Zρ(0) outZ) (14) ={1 2(γ2+ ∆2),(1 −2p 0)(1 −2p)αβ(γ2−∆2),(1 −2p0)γ∆,α2−β2 2(γ2−∆2)} After this the sub-optimal (because not optimized for this channel) control Zηis applied. To compute the corresponding controlled output state, we first compute the effect of this sub-optimal control on the basis states. We find ZηIZ† η=I(15) ZηXZ† η=X(16) ZηY Z† η=Y−sin ηX (17) and ZηZZ† η=Z(18) Thus the controlled output state is easily written down as 3 ρ”={1 2(γ2+∆2),(1−2p0)(1−2p)αβ(γ2−∆2)−sin η, −(1−2p0)γ∆,α2−β2 2(γ2−∆2)} (19) We now compute the fidelity between this output state and the original input state. It is given by F(|ψ1i, ρ”) = hψ1|ρ”|ψ1i(20) To compute this we first compute the components. Thus hψ1|I|ψ1i= 1,(21) hψ1|X|ψ1i= 2αβ, (22) hψ1|Y|ψ1i= 0,(23) and hψ1|Z|ψ1i=α2−β2.(24) Thus, hψ1|ρ”|ψ1i=1 2+2(1−2p0)(1−2p)α2β2(γ2−∆2)−2αβ sin η+α2−β2 2(γ2−∆2) (25) where we used the fact that γ2+ ∆2= 1. 2 Depolarizing map The input is ρ(0) out ={1 2(γ2+ ∆2),(1 −2p)αβ(γ2−∆2), γ∆,α2−β2 2(γ2−∆2)}(26) which corresponds to measurement result “0”. This enters a depolarizing channel with parameter p”. The corresponding output is Ep”(ρ(0) out) = (1 −p”)ρ(0) out +p” 2I(27) ={1 2(1 −p”) + p” 2,(1 −p”)(1 −2p)αβ(γ2−∆2),(1 −p”)γ∆,(1 −p”)α2−β2 2(γ2−∆2)} Thereafter the sub-optimal control is performed. The controlled state is found to be 4 ρ”={1−p” 2+p” 2,(1−p”)(1−2p)αβ(γ2−∆2)−(1−p”)γ∆ sin η, (1−p”)γ∆,(1−p”)α2−β2 2(γ2−∆2)} (28) The fidelity between ρ”and |ψ1ihψ1|is found to be F(|ψi, ρ”) = 1−p” 2+p” 2+2(1−p”)(1−2p)α2β2(γ2−∆2)−2αβγ∆ sin η(1−p”)+(1−p”)(α2−β2)2 2(γ2−∆2) (29) 3 Comparison The difference between the fidelities in the presence of the dephasing and depolarizing channels is as follows. difference =p” 2+p” 2(α2−β2)2(γ2−∆2)+2(p”−2p0)(1−2p)α2β2(γ2−∆2)+2αβ sin η(γ∆(1−p”)−1) (30) which is clearly non-zero. Suppose γ2= ∆2. This happens when χ= 0, π, 2π, . . . , nπ. Then, diff =p” 2+ 2αβ sin η(γ∆(1 −p”)−1) (31) This is positive when p”>2αβ sin η(1 −γ∆) 1 2−2αβ sin ηγ∆(32) Thus the dephasing channel provides greater fidelity as a feedback channel when the above relation holds and the weak measurement is such that χ=nπ. This is the first comparative result. References [1] Agata M. Branczyk, Paulo E. M. F. Mendonca, Alexei Gilchrist, Andrew C. Doherty, and Stephen D. Bartlett. Quantum control of a single qubit. arxiv:quant-ph/0608037v2, 2006. [2] Michael Nielsen and Issac Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2002. 5