Deterministic controlled enhancement of local quantum coherence

We investigate assisted enhancement of quantum coherence in a bipartite setting with control and target systems, which converts the coherence of the control qubit into the enhanced coherence of the target qubit. We assume that only incoherent operations and measurements can be applied locally and classical information can be exchanged. In addition, the two subsystems are also coupled by a fixed Hamiltonian whose interaction strength can be controlled. This coupling does not generate any local coherence from incoherent input states. We show that in this setting a measurement and feed-forward based protocol can deterministically enhance the coherence of the target system while fully preserving its purity. The protocol can be iterated and several copies of the control state can be consumed to drive the target system arbitrarily close to a maximally coherent state. We experimentally demonstrate this protocol with a photonic setup and observe the enhancement of coherence for up to five iterations of the protocol.

The term (cot 2 α − tan 2 α) is always positive for 0 < α < π/4, cot 4 α > 1, cot 2 β > 0 holds, and therefore also the second inequality is valid. For π/2 > α > π/4 the inequalities (8) read cot α < tanα and tanα < tan α. We again use relation (6) to transform the first inequality into cot 4 α + cot 2 β cot 2 α − tan 2 α < 1 (S7) and the second one into For π/4 < α < π/2, cot 4 α < 1, tan 4 α > 1 and the term cot 2 α − tan 2 α is negative, therefore we see that both inequalities are valid. In the remaining case of α = π/4, where tan α = cot α = 1 we see that tanα = 1 and identify α = π/4 as a fixed point of the contraction mapping performed by the protocol, concluding the proof. We did not discuss the case of negative α but the proof is very similar and straightforward. The case of negative α can be converted to the case of positive α just by applying π phase shift on the first input qubit.

Replacement of conditional σ X operation with conditional choice of coupling strength
In a single step of the protocol, input state |ψ A ⊗ |φ B defined in Eq. (1) transforms into a conditional state |ψ j A = cos(α j )|0 + sin(α j )|1 where j ∈ {0; 1} is the outcome of computational basis measurement on qubit B, and tanα 0 = tan α cot β + cot α tan β 2/5 and α, β are input state parameters. Clearly, the coherence of |ψ 0 A and |ψ 1 A are equal. Instead of using conditional σ X operation, we can keep track of α j and use Eq. (5) to determine the optimal coupling strength for the next iteration, i.e., tan ω j = tanα j − cotα j tan β + cot β . (S11) Here we introduced index j to emphasize the dependence of the next coupling strength on the measurement outcome j and consequently on parameterα j . This choice guarantees the equal coherence of conditional states in the next iteration because we used Eq. (5) that fulfills this condition of equal coherence. The relation between ω 0 and ω 1 reads and is the consequence the relation between α 0 and α 1 described by Eq. (S10).

Derivation of Equation (15)
We consider fixed coupling ω = π/4. Here, we show how to achieve equal coherence of the two conditional output states by tuning the phase of input qubit B. Such tuning allows using the protocol with fixed coupling. Input state is transformed by operation (4)  (S14) The corresponding non-normalized conditional states are The requirement for equal coherence of the conditional states is equivalent to equal population imbalance, as we have discussed in Sec. 1 of this supplement. Condition transforms into tan 2 α + tan 2 β − 2 tan α tan β cos ϕ = cot 2 α + cot 2 β + 2 cot α cot β cos ϕ.
When the condition is met, σ X |ψ 1 A is equivalent to |ψ 0 A in terms of population. The conditional output states |ψ 0 A and |ψ 1 A are locally phase-shifted by δ 0 and δ 1 , respectively. The value of these phase shifts is a non-trivial function of α and β but we can compensate for them by conditional application of phase gates. Because cos ϕ is an even function, both ±|ϕ| satisfies the condition (S17). In the case of negative ϕ, the population ratio remains the same as in the previous solution, but the additional phases have the opposite sign, δ 0− = −δ 0 and δ 1− = −δ 1 .

3/5 5 Equivalence of partial swap gate with operation (4)
Partial swap operator is defined as U PSWAP = Π + + exp(i2ω)Π − , where Π − = |Ψ − Ψ − | is the projector onto the anti-symmetric singlet Bell state |Ψ − = 1 √ 2 (|01 − |10 ), Π + = I − Π − is the projector onto the three-dimensional symmetric subspace of two qubits, and I denotes the identity operator. In the matrix form, the operator reads One can easily check the unitarity of the operator by evaluating U † PSWAP U PSWAP and seeing that this product is equal to the identity matrix. Operator U PSWAP clearly differs from operator (3). Albeit this difference, we will now show that U PSWAP is equivalent to (3) in the context of the presented protocol.
By comparing these states to the original conditional states (S2) and (S3) we see that they differ by phase factors only. We can compensate the phase factor i exp(iω) in conditional state |ψ 0 A by applying a fixed local phase gate V = exp(iσ Z (ω/2 + π/4)). The same gate V also transforms conditional state |ψ 1 A into |ψ 1 A = cos α cos ω sin β + cos β sin α sin ω sin α sin βe −2iω .
The remaining phase factor e −2iω can be compensated using feed-forward operation by a conditional phase gate W = exp(−iσ Z ω). With these corrective local phase gates V and W , the coupling operations U PSWAP and (4) are equivalent in the context of the protocol.

Characterization automated phase alignment
Here we describe details of the procedure for setting the phase ϕ in the displaced Sagnac interferometer, which determines the coupling strength of the partial SWAP gate, ω = ϕ/2. First, we block the input B to the interferometer and set all waveplates in the experiment to 0 degrees. We count the single photons at the output ports of the interferometer and scan the piezo voltage to obtain a reference interferogram. Using this reference, we calculate the set point intensity corresponding to the desired phase. We coarsely set the piezo voltage for the target phases from 10 to 170 degrees using the reference interferogram. We then tune the piezo voltage using feedback proportional to the error signal until the observed intensity reaches the set point within a defined tolerance. We resort to scanning the piezo voltage for phases between 0 and 10 degrees and between 170 and 180 degrees. When the intensity reaches the setpoint, we stop the scan. We have executed the alignment procedure 15 times for the tested desired coupling strengths ω n and recorded the actual coupling strength ω e . We have identified failed alignment attempts as cases where the size of the deviation ∆ = ω e − ω n was greater than 5 degrees. In Fig. S1(a) we compare the desired coupling strength ω n with the strength ω e achieved by successful alignment. Their respective deviation ∆ is plotted in panel (b). The deviation of ω e alignments from the nominal ω n is smaller than 1.1 degrees RMS.
The reliability of our alignment procedure is limited mainly by the hysteresis of a piezo-spring system. Due to hysteresis, the interference fringe generally moves relatively to the previously measured reference fringe. When we execute proportional-feedback loop tuning in flat regions of the fringe near the extremes, we sometimes exceed the allowed voltage on the piezo or exceed the maximal limit of iterations, and the procedure fails. In the main experiment, we have always checked whether the phase alignment succeeded, and in the case of failure, we have repeated the phase alignment.