Measurements and Results - Distribution and certification of photonic entanglement for quantum

The set of input states employed to certify the entanglement of the Werner states down toλ = 1/3 is{τ_x,y}={|0i±|1i;|0i±i|1i;|0i;|1i}. To set the corresponding phases at the two interferometers, a polariser at 45^o is momentary placed at one output of each interferometer to allow interference between the upper and lower arm, see Figure2.4 (a). The singles of one detector on each side are then recorded while scanning the phase. The voltages applied on the PZTs to obtain the phases

2.5. Measurements and Results 33

W (Witness)

λ (Bell state fraction) β^λ β^λ=0.94

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3

0 0.2 0.4 0.6 0.8 1

Figure 2.5: Witness value as a function of the weight λ of the two-qubit Werner state (black line). The data pointsβ^λ andβ^λ=0.94correspond to the results obtained when β are calculated for each points and only for the first point, respectively.

The uncertainty associated with each point has been calculated using Monte-Carlo algorithm and a Poissonian noise on the detection rate.

θ_j ={0; π/2; π; 3π/2}, are extracted from the two fits of the singles. The polariser is then removed and the coincidence counts from the BSMs are recorded for all thirty-six pairs of input states. Without the added noise, the average coincidence rate is about 16 counts per second, corresponding to a total detection efficiency around 3%. The integration time is set to 10 seconds for each input pair such that, using automatic control of PZTs and shutters, one complete measurement lasts about 6 minutes.

The number of observed events N_λ(abxy) were collected for each visibility pa-rameterλ of the Werner state. The construction of the MDI-EW is insensitive to any rescaling of the probability distributionsP_λ(ab|τ_x, τ_y) induced by losses or low detection efficiency [19]. Thus, the normalization constant can be chosen arbitrarily.

Here, it was chosen such that 0≤P(ab|τ_x, τ_y)≤1:

P_λ(ab|τ_x, τ_y) = N_λ(abxy)

N_λ^∗ , N_λ^∗ = max

N_λ(abxy). (2.18) From the estimated probability distributions, the coefficientsβ_abxycan be extracted to construct the MDIEW, as previously shown in section2.2. The values of the wit-ness were computed via two different methods. We first constructed an MDIEW for eachλ, with coefficients β_abxy^λ tailored to the corresponding P_λ(ab|τ_x, τ_y), such that W_λ = P

abxyβ_abxy^λ P_λ(ab|τ_x, τ_y). The computed witness values are plotted in Fig-ure 2.5 (square data points), starting from λ = 0.94. Note, λ < 1 due to the intrinsic noise of the detectors and the source imperfections. We observe that the witness value saturates at W = 0 when entanglement cannot be certified; this

Bob phase (rad)

Figure 2.6: Fitting of the detection probabilities (not normalised) measured for all the 16 output pairs, as a function of phase θB and for θA= 0.

comes from the minimization present in the SDP 2.5. This first method has the disadvantage of requiring a first set of data dedicated to the construction of the MDI-EW, for every single λ. This can be avoided by constructing the MDI-EW once for one λ and then apply it to the other λ. We extracted the coefficients β_abxy^λ=0.94 from a first set P_λ=0.94(ab|τ_x, τ_y), and constructed the following MDI-EW:

Wˆ_λ =P

abxyβ_abxy^λ=0.94P_λ(ab|τ_x, τ_y). The values of the witness computed this way are also plotted in Figure2.5(circle data points). By linearity of Eq.(2.17), the witness value ˆW_λ becomes positive for separable Werner states.

In both approaches, the entanglement of the Werner states could not be certified all the way down to the separability limit λ = 1/3. The reason for this is that the values of the witness are limited by the imperfections in the BSMs and the residual birefringence inside the interferometers. These induce small phase shifts between the outputs of the BSMs, as observed in Figure 2.6. Here, the joint probabilities were measured for all output pairs while scanning the phase θ_B at Bob and fixing the phase θ_A at Alice.

To understand how sensitive we are to these imperfections, we simulated the effects on the joint probabilities of the birefringence and small errors on the trans-formation done by the HWP in the lower arm of the two interferometers, see Fig-ure 2.7. In the case of birefringence (b), we added a phase difference of π/20 between the |Hi and |Vi states in the upper arm of Alice’s interferometer. This phase difference will, of course, induce similar phase shifts between the different outputs pairs, as observed here. Note that these shifts will depend on the relative phase difference caused by birefringence between each arm and each interferometer.

Thus, slightly different phase shifts will be observed when including birefringence

2.5. Measurements and Results 35 (a) Ideal case,θA= 0 (b) Birefringence, θA= 0

Figure 2.7: Simulations of the joint probability P for all the output pairs as the function of the phase at Bob θB and for a given θA. (a) Ideal case for θA = 0.

(b) Adding a phase difference ofπ/20 between |Hi and |Vi in the arm 1 of Alice’s interferometer due to birefringence. (c) Adding an amplitude error of 10⁻³ in the transformationH(V)→V(H) by the HWP inside both interferometers. (d) Same as (c) but forθ_A=π/2.

in the other paths of the interferometers. These phase shifts can also be induced by phase errors at the HWP in the lower arm of each interferometers. Moreover, a small phase error of the HWP will also introduce a linear error on the amplitude of each polarization component after the transformation. Considering only amplitude errors, the transformation of |Hi to |Vi becomes: |Hi → √

1−|Vi +√ |Hi. With an error of = 10⁻³, we observe either changes in the amplitude of the joint probabilities or small phase shifts, depending onθ_A, as seen in Figure2.7(c-d). For the relatively small phase and amplitude errors considered above, the probabilities can differ noticeably from the ideal ones; up to∼15% for certain phases{θ_A, θ_B}in the case of birefringence. For an easier numerical comparison, the joint probabilities of one output pair are presented in Appendix 5.0.2. Following these simulations,

we can say that the value of the witness violation is rather sensitive to small phase errors, and additional compensation stages must be applied to reach entanglement detection down to the separability limitλ = 1/3.

2.6 Conclusion and Outlook

We have proposed and demonstrated a practical MDI-EW protocol, certifying en-tanglement for a family of polarization Werner states down to a visibility parameter close to 0.4. This protocol does not require a priori knowledge of the state under test. The witness can be reconstructed directly from the measured output probabil-ities. Interestingly, this MDI-EW can be used not only to certify the entanglement but also to quantify it. The value of the witness violation was shown to be related to a lower bound on the amount of entanglement present in the Werner state, as quantified by the negativity [75]. Similar quantitative MDI-EWs were proposed by several authors [87, 88, 89].

Our implementation is resource-efficient as it replaces the need for additional single photons by encoding the input qubits directly on an extra degree of freedom of the entangled photons. Additionally to lowering the experimental complexity, this approach has the advantage that even entangled states encoded on photon pairs that are not spectrally pure can be characterized. The downside of choosing path-encoding onpolarization-entangled photons is that small phase errors, introduced by the birefringence of optical elements or by imperfect HWPs, will effectively reduce the values of the witness. Therefore, one must carefully compensate these phase errors to be able to detect entanglement all the way down to the separation limit.

This implementation could easily be extended to n-partite states, given the optimal scaling of the number of photons required (n vs 3n in the case of qubits encoded on heralded single photons). If we further consider qudits, e.g. encoded in time bins, it only requires that the number of spatial modes in the LOC corresponds to the number of time bins.

Moreover, these MDI schemes are faithful even in the presence of arbitrary losses and classical communication between the systems; they are perfectly robust to the detection and locality loophole. Thereby, it is interesting to ask whether similar approaches could be exploited to characterize entanglement on stationary qubits, and whether or not extra degrees of freedom for the encoding could be accessible.

They are also of interest for other applications such as the certification of quantum steering [90,91] or the quantification of generated randomness [89,92,93]. Finally, there are recent works on how to make these approaches fully device-independent by removing any assumptions on the entangled state or measurements [94].

Chapter 3 Distribution of Single Photon Path Entanglement

Entanglement is a valuable resource for quantum communication and quantum information. Its distribution between two or many parties, over short or long dis-tances, is becoming increasingly important. Indeed, several interesting protocols such as quantum teleportation [21, 95, 96] or device-independent quantum key distribution (DI-QKD) [5, 8, 97] are based on shared entanglement. The most straightforward way to distribute entanglement between two distant parties is by physically sending entangled photons through optical fibres. However, loss in optical fibre quickly sets a limit on the distance and the rates achievable. The main chal-lenge of long-distance quantum communication is to find advantageous resources and protocols to overcome this limitation.

Single photon path entanglement is a promising resource for long distance com-munication. It consists of one photon delocalized over two or more paths. Its single photon nature makes it simpler to generate and more robust to loss and detec-tor inefficiency than two-photon entanglement. In this chapter, I first introduce a displacement-based method to verify path entanglement in distributed scenar-ios. I then present the implementations of a photon amplifier and an entanglement swapping adapted for the distribution and detection of path entanglement.

3.1 Single Photon Path Entanglement

Single photon path entangled states refers to a single photon delocalized over two or more spatial modes. They present several advantages over two-particle entangled states. First of all, it is one of the simplest forms of entanglement to generate in a heralded manner. In SPDC processes, previously introduced in Chapter 2 (section2.3), the photons are generated in pairs with a certain probability. Hence, the detection of one of the twin photons heralds the presence of the other one, as illustrated in Figure 3.1. The heralded single photon can then be sent through a beam splitter (BS), creating the path entangled state. For a 50:50 splitting ratio,

b SPDC

Figure 3.1: Heralding of path entangled states

the resulting state is of the following form in the Fock basis representation:

|ψi= 1

√2(|0ia|1ib+|1ia|0ib), (3.1) wherea andb refer to the two output modes of the BS. The generation of the path entangled states depends then on the heralding rate of the single photons, which can be up to MHz with SPDC sources [98]. The generation of heralded two-photon entangled states will be orders of magnitude lower as it requires the generation of photon triplets, for example by cascading two entangled down-conversion pro-cesses [99, 100].

Secondly, bipartite path entangled states can be easily scaled up to multipartite by adding extra beam-splitters on each spatial mode. Importantly, the generation rate of then-partite entangled states will be the same as in the bipartite case, as it will also depend on the heralding rate of the single photon.

Moreover, with only one photon delocalized over two spatial modes, path entan-glement decreases linearly with losses, instead of quadratically as it would be for two-photon entanglement. After propagation through patha andb, with respective transmissionsη_a and η_b, the path entangled state becomes:

|Ψi hΨ|−−−→^η^a^,η^b ρ=







1−^η₂^a − ^η₂^b 0 0 0

0 ^η₂^a

√ηaηb

2 0

√ηaηb

η_b

2 0

0 0 0 0





 (3.2)

One can see that the coherence terms h01|ρ|10i and h10|ρ|01i are proportional to √η_aη_b. While in the case of two photons entanglement, these terms would be proportional to η_aη_b. Note that we can include the detector efficiencies in η_a and ηb. As previously mentioned, this robustness against losses is a great asset for long-distance quantum communication. Several important protocols such as teleporta-tion [101,102], swapping [103], entanglement purification [104], EPR steering [105]

and quantum storage [106, 107] were already experimentally demonstrated with path entangled states.

While the generation of path entanglement is very simple, its detection, on the other hand, demands more resources than usually needed for polarization entangle-ment and brings additional experientangle-mental challenges.

3.1. Single Photon Path Entanglement 39

3.1.1 Displacement-Based Measurement

To verify and characterise any entangled states, one must perform measurements in complementary base. This is not straightforward for path entanglement in a distributed scenario and using only local measurements. The Z-basis, {0,1} basis is simple to measure. One only needs to place a single photon detector (SPD) in each of the spatial modesa and b. To access other measurement base, we need different approaches. Some methods use continuous variable measurements via homodyne detection [108, 109]. The one I will present here, is based on a small displacement operation followed by a single photon detector. It was first introduced in 1991 [110]

in effort to prove that single photon path entanglement also demonstrates non-local correlations, which were at the time thought to result only from quantum states with at least two particles. This protocol was further discussed in [111,112], while its implementation came much later [113, 114, 105].

Let us first consider the effect of a small displacement on a single photon state of the form:

ρ₁ =

1−η_a k k η_a

(3.3) The displacement D(α) is made by interfering a coherent state |αi of small am-plitude and the single photon state. The important point is that we will observe both displaced vacuum (D(α)|0i=|αi) when the state is empty, and single-photon displacement when it is not. This will give us access to information about the off-diagonal terms|0i h1|and|1i h0|, denotedk in eq.3.3. If we place a binary detector right after the displacement operation, the probability of no-detection or no-click events is given by

p^α₀ = tr{|0i h0|D(α)ρ₁D(α)^†}= tr{D(α)^†|0i h0|D(α)ρ₁}

= tr{|αi hα|ρ₁}, (3.4)

where we have used the cyclic property of the trace¹. From the equation above, we can write the no-click operator ˆP₀^α:

Pˆ₀^α =|αi hα| ≈

e^−|α|² |α|e^−|α|²^−iθ

|α|e^−|α|²^+iθ |α|²e^−|α|²

, (3.5)

where |α| and θ are the amplitude and the phase of the coherent state, respec-tively. The matrix form above is in the {0,1} subspace and holds for|α|<1. The probability of a no-click event is then:

p^α₀ = (1−η_a)e^−|α|² +η_a|α|²e^−|α|² + 2k|α|e^−|α|²cos(θ), (3.6) fork real. Finding the projection operator for click events in a similar way is more complicated, as detections from higher number states must be taken into account

1Re-writing |−αi h−α|as |αi hα|is equivalent to a redefinition of the phase.

0 0.2 0.4 0.6 0.8 1

-0.2 0 0.2 0.4 0.6 0.8 1

Figure 3.2: Projective part of the measurement ˆσαin the Bloch sphere while increas-ing the amplitude of the displacement. Each circle corresponds to an increment of

|α| of 0.1, starting with|α|= 0 at point (x=0, z=1).

when using non photon-number resolving detectors (non-PNRD). One way around this is to use the fact that ˆP₀^α+ ˆP_c^α =1. Thus:

Pˆ_c^α=1−Pˆ₀^α =1− |αi hα| (3.7) This expression includes contributions from higher photon number states. In the {0,1} subspace, the probability of a click event is:

p^α₁ = (1−η_a)(1−e^−|α|²) +η_a(1− |α|²e^−|α|²) + 2k|α|e^−|α|²cos(θ) (3.8) From eq. 3.6 and eq. 3.8, we see that these probabilities give a measure of the diagonal terms of ρ₁ when |α| = 0 but also information about the off-diagonal terms (∝k) when|α| 6= 0.

We will now look at how well this displacement-based method approximates measurements in the X-basis. If we associate an outcome +1 to a no-click event and -1 to a click event, the measurement operator corresponding to a small displacement followed by a binary detection is given by:

σ_α =D(α)^†( ˆP₀−Pˆ_c)D(α)≈Pˆ₀^α−Pˆ_c^α, (3.9) where ˆP₀ = P

n=0(1−η)ⁿ|ni hn| and ˆPc = 1−Pˆ₀. The detection efficiency η can be included in the losses before the displacement. The matrix form of ˆσ_α can then be written in the {0,1} subspace as:

ˆ σ_α =

2e^−|α|² −1 2|α|e^−|α|²^−iθ 2|α|e^−|α|²^+iθ 2|α|²e^−|α|² −1

, (3.10)

By varying the amplitude of the displacement, we can go from a ˆσ_z measurement when |α| = 0, to an approximation of ˆσ_x measurement, as shown in Figure 3.2.

3.1. Single Photon Path Entanglement 41 Here, the x and z component of ˆσ_α when decomposed in the Pauli matrices basis are plotted for increasing values of|α|. For|α|= 0, marked by the point (x=0, z=1), the vector is along the z-direction. As |α| is increased, the z-component decreases and we approach the x-axis. The maximum x-component value is achieved for

|α|= 1/√

2 = 0.707.

3.1.2 Measure of Path Entanglement

In this section, we will see how to verify path-entanglement using displacement-based measurements. We consider now the path-entangled state of eq. 3.2 and we apply displacements α_a =|α|eîθâ on modea and α_b =|α|eîθ^b on mode b. The joint probability of no-click events in both modes is given by:

p^α₀₀= tr{Pˆ₀₀ρ}

=e^−2|α|²(1 + ηa+ηb

2 (|α|²−1) +√

ηaηb|α|²cos(θ)), (3.11) where ˆP₀₀ = ˆP₀ ⊗Pˆ₀ and θ = θb −θa is now the relative phase between the two displacements. Similarly to the one-mode case of the previous section, we see that the joint probabilities will give information about both the diagonal term ofρ and the off-diagonal terms (∝ √ηaηb). Interestingly, we notice that p^α₀₀ will oscillate with θ, the relative phase between the two displacements. This interference effect is due to the entanglement contained in ρ and will also be observed for p^α_0c, p^α_c0 and p^α_cc. A good control of θ is necessary to observe the entanglement. If the relative phase fluctuates between measurements, the √η_aη_b term will vanish. A simple way to solve this issue is to ensure that the coherent state and the path entangled state co-propagate in each path, such that they accumulate the same phase, as shown in Figure 3.3. The single photon and the coherent state enter the BS with orthogonal polarizations. The displacement can later be adjusted with a half-wave plate (HWP) followed by a PBS, setting the HWP to obtain a 99:1 ratio between the photon state² and coherent state, to reduce the losses on the photon state. This simple method was applied in [114] to witness bipartite and tripartite path-entanglement, and to demonstrate EPR steering [105]. However, we will see that this method cannot be directly applied when introducing a photon amplifier or for swapping protocols, and active methods must be employed to control the relative phase between the displacements.

A witness adapted to the displacement-based measurement was developed by V. Vivoli, J.-D. Bancal and N. Sangouard [114] and takes the form:

Z₂ = ˆσx⊗ˆσx+ ˆσy⊗σˆy −σˆz⊗σˆz

≈2ˆσ_α⊗σˆ_α−σˆ₀⊗σˆ₀ (3.12)

2By photon state, I refer here to the de-localized photon in the mode in question. I will use this simplification of expression throughout this chapter.

HWP

BS a

b c d

PBS

Figure 3.3: Simple implementation of displacement-based measure of path entan-glement where the coherent state and the photon co-propagate in the two paths.

where hσˆ_α⊗σˆ_αi=p^α₀₀+p^α_cc−p^α_0c−p^α_c0 and hσˆ₀⊗ˆσ₀i=p₀₀+p_cc−p_0c−p_c0. The joint probabilities with upper index α are measured for a displacement α, while the ones without upper index are measured for |α|= 0. A separable threshold can be found by computing the maximum value of the witness for states which stay positive under partial transpose (Z_PPT^max). If the value of the witness obtained from measurements on the stateρ is larger than Z_PPT^max, we can affirm that the measured state is entangled. This entanglement witness represents an efficient way to verify entanglement but does not give a measure of the amount of entanglement. This can be achieved by computing a fidelity³ of the formF =hψ|ρ|ψi, where|ψiis the maximally path-entangled state. This fidelity is given by:

F =|d|+ 0.5(p_0c+p_c0), (3.13) where d = h01|ρ|10i is the off-diagonal coherence term. The density matrix of ρ can be written in terms of joint probabilities and coherence termd as:

ρ=







p₀₀ 0 0 0 0 p_0c d^∗ 0 0 d p_c0 0 0 0 0 p_cc





 (3.14)

We can extract the coherence termd from the correlator hσˆα⊗σˆαias follows:

hσˆ_α⊗σˆ_αi= tr{σˆ_α⊗σˆ_αρ}

= 8|d|e^−2|α|²|α|²+p₀₀(2e^−|α|² −1)²

+ (p_0c+p_c0)(2e^−|α|² −1)(2|α|²e^−|α|² −1)−p_cc(2|α|²e^−|α|² −1)²

(3.15)

We have assumed here that relative phase between the two displacements is zero.

Usinghσˆ_α⊗σˆ_αi=p^α₀₀+p^α_cc−p^α_0c−p^α_c0, we can write|d|in terms of joint probabilities

3We have used here the fidelity as defined by R. Jozsa [115]. Note that the square root fidelity F=p

hψ|ρ|ψidefined in [2] is sometimes used but has no interpretation as a probability.

3.1. Single Photon Path Entanglement 43 measured with displacement and without displacement:

|d|= e^2|α|²

8|α|²[p^α₀₀+p^α_cc−p^α_0c−p^α_c0 −p₀₀(2e^−|α|² −1)²

−(p_0c+p_c0)(2e^−|α|² −1)(2|α|²e^−|α|² −1)−p_cc(2|α|²e^−|α|² −1)²] (3.16) Displacement-based measurements allow us to verify and quantify path entan-glement. A quantitative measure is necessary to assess the amplification of path entangled states, as we will see in Section 3.2. To perform amplification and swap-ping experiments with displacement based measurements, one needs to generate photon pairs and coherent states. The generation of these resources is presented in

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