Weak entanglement improves quantum communication using only product measurements

We show that weakly entangled states can improve communication over a qubit channel using only separate, interference-free, measurements of individual photons. We introduce a communication task corresponding to the cryptographic primitive known as secret sharing and show that all steerable two-qubit isotropic states provide a quantum advantage in the success rate using only product measurements. Furthermore, we show that such measurements can even reveal communication advantages from noisy partially entangled states that admit no quantum steering. We then go further and consider a stochastic variant of secret sharing based on more sophisticated, yet standard, partial Bell state analysers, and show that this reveals advantages also for a range of unsteerable isotropic states. By preparing polarisation qubits in unsteerable states, we experimentally demonstrate improved success rates of both secret sharing tasks beyond the best entanglement-unassisted qubit protocol. Our results reveal the capability of simple and scalable measurements in entanglement-assisted quantum communication to overcome large amounts of noise.


I. INTRODUCTION
Entanglement is well-known to be a crucial resource for quantum communication applications.However, not all forms of entanglement are always useful.For instance, while entanglement that is strong enough to generate nonlocality has been found to improve noiseless classical communication beyond its conventional limitations (see, e.g., [1][2][3][4][5][6]), weaker entangled states that cannot violate any Bell inequality do not have that ability [7].In contrast, if the system communicated is itself quantum, e.g. a qubit instead of a bit, then some weaker forms of entanglement also become useful.This can be seen in the celebrated dense coding protocol [8] where a maximally entangled state, |ϕ + ⟩ = |00⟩+|11⟩ √ 2 , is exploited to allow a qubit message to transmit two bits instead of one bit.When the state is noisy, the isotropic state where v ∈ [0, 1] is the visibility, achieves an advantage over an unassisted qubit message whenever v > 1 3 [9,10].This coincides with the visibility at which the state becomes separable, v sep = 1  3 and it is considerably lower than the critical visibility for quantum steering, v unsteer = 1 2 [11,12], and far below the even higher visibility needed for Bell nonlocality [13].
However, to harness the dense coding advantage one must measure in the Bell basis {Φ + , Φ − , Ψ + , Ψ − }, where Φ ± = |ϕ ± ⟩⟨ϕ ± | and Ψ ± = |ψ ± ⟩⟨ψ ± | are the projectors onto the states |ϕ ± ⟩ = |00⟩±|11⟩ √ 2 and |ψ ± ⟩ = |01⟩±|10⟩ √ 2 .In optical systems, it is impossible to implement a linear optics Bell basis measurement on separate photons without the use of auxiliary photons [14].While dense coding experiments have been reported [15][16][17][18][19], implementation of the Bell basis is not expected to be scalable in optical systems in the near future.Nevertheless, it has recently been found that entanglement can yield advantages in one-shot quantum communication scenarios by using much more limited, yet much less experimentally demanding, optical measurements that are compatible with passive linear optics or even just separate single-photon mea-surements [20].However, the schemes considered thus far come with greater requirements on the quality of entanglement, in particular needing states that violate a Bell inequality.
Here, we show that the simplest optical measurements are sufficient to reveal communication advantages from highly noisy entangled states.To this end, we introduce a version of secret sharing.In secret sharing, a secret is distributed between two parties in such a way that they must cooperate to reconstruct it [21,22].This is of considerable interest for quantum cryptography.Previous protocols have achieved secret sharing in a device-dependent way, i.e. by assuming that devices perform specific operations (see e.g.[23][24][25][26][27]).In contrast, our protocols are semi-device-independent, in the sense that all operations are uncharacterised up to the assumption that they act on qubit systems.We prove that for our secret sharing task every steerable isotropic state enables an advantage over unassisted qubits using only product measurements on separate photons.Furthermore, we also identify unsteerable states for which an advantage is possible with such measurements.In addition, we prove that, at the cost of using somewhat more sophisticated measurements, namely partial Bell state analysers, which require optical interference but are still feasible with standard passive linear optics [28,29], it is possible to reveal advantages from unsteerable instances of the isotropic state.For this, we use a stochastic modification of the original secret sharing task.Using polarisation qubits generated via spontaneous parametric down-conversion, we use both a maximally entangled state to experimentally demonstrate optimal secret sharing protocols, and unsteerable states to outperform the best possible entanglement-unassisted qubit protocols using both product measurements and partial Bell state analysers.

II. DETERMINISTIC SECRET SHARING
Alice wishes to generate a random secret bit a ∈ {0, 1} that is shared with Bob and Charlie in such a way that they individually have no knowledge of a but can learn its value if they cooperate.To this end, we consider the scenario illustrated in  Bob and Charlie each privately select two uniformly random bits x ≡ (x 0 , x 1 ) ∈ {0, 1} 2 and y ≡ (y 0 , y 1 ) ∈ {0, 1} 2 , respectively, and each communicate a message to Alice.Alice privately selects a binary input z ∈ {0, 1} and accordingly decodes the two incoming messages into an output a ∈ {0, 1}.Here, z, which is later announced publicly, determines which bits of Bob and Charlie hold the shared secret: the success condition is a = x z ⊕ y z .The average success probability of secret sharing becomes Naturally, the parties must have some physical limitations.In a purely classical scheme, Bob and Charlie would only share a classical random variable and they would individually send to Alice classical messages in the form of a bit respectively.Alice would use a decoding function to decide her output based on her input.However, our main focus is on quantum protocols.On the one hand, we are interested in the situation in which Bob and Charlie share no prior entanglement and simply send messages encoded in qubit states β x and γ y , respectively, while Alice can decode using a general quantum measurement {M a|z } a .The observed correlations are then given by p qubit (a|x, y, z) = tr (β x ⊗ γ y )M a|z .On the other hand, we also consider the situation illustrated in Fig. 1, where Bob and Charlie may additionally share a twoqubit entangled state ρ and then encode their qubit messages through local quantum channels Λ B x and Λ C y , respectively.In this entanglement assisted case, the correlations are given by p EAqubit (a|x, y, z) = tr Λ B x ⊗ Λ C y (ρ) M a|z .We now show that there exists an entanglement-assisted quantum protocol that achieves a perfect secret sharing rate while requiring only separate measurements of the incoming quantum messages.

III. IDEAL ENTANGLEMENT-ASSISTED AND UNASSISTED PROTOCOLS
Consider a maximally entangled state ρ = |ϕ + ⟩⟨ϕ + | and let Bob's and Charlie's local channels correspond to implementing the four unitaries respectively, where σ X and σ Z are the Pauli bit-flip and phaseflip operators.This means that the two-qubit state arriving to Alice is one of the four Bell states.When z = 0, Alice measures σ Z ⊗ σ Z , i.e., she discriminates the states |ϕ ± ⟩ from the states |ψ ± ⟩.When z = 1, she measures σ X ⊗ σ X , i.e., she discriminates the states |ϕ + ⟩ and |ψ + ⟩ from the states |ϕ − ⟩ and |ψ − ⟩.This gives S = 1.
In contrast, in the scenario when entanglement is absent, it is no longer possible to succeed deterministically.We have determined the maximum value of S achievable with entanglementunassisted qubits and general measurements for Alice.This is achieved using a straightforward modification of a hierarchy of semidefinite programs for bounding dimensionally-restricted quantum correlation [30].In order to obtain sufficiently tight upper bounds on S qubit , we have considered terms appearing in the first three levels of this hierarchy [31].Up to solver precision, we obtain the upper bound S qubit ≤ 3  4 .This bound holds even if all three parties share some pre-agreed classical randomness.
In fact, a simple classical strategy can saturate S qubit , thus showing both that the bound is tight and that there exists no quantum-over-classical advantage without using entanglement.To this end, consider that Bob and Charlie send x 0 and y 0 respectively to Alice.When z = 0 she correctly outputs a = x 0 ⊕ y 0 and when z = 1 she chooses a at random.This leads to S classical = 3  4 .We conclude that any value in the range 3 4 < S ≤ 1 implies an advantage over both the best classical and the best entanglement-unassisted quantum protocol, and hence is powered by the consumption of entanglement.

IV. ADVANTAGE FROM NOISY ENTANGLEMENT
If, in the above ideal entanglement-assisted protocol, we substitute the maximally entangled state for the isotropic state (1), we find that S = 1+v 2 .Hence, when v > 1 2 = v unsteer , product measurements reveal an entanglement-based advantage in the task.We conclude that every steerable isotropic state is a communication resource under product measurements.Furthermore, while unsteerable isotropic states do not provide an advantage, there exist other unsteerable states that are useful in the task.To this end, consider the states , where v ∈ [0, 1] and |ϕ + θ ⟩ = cos θ|00⟩ + sin θ|11⟩ is a pure partially entangled state for θ ∈ (0, π/4].In [32], local hidden state models have been computed for ρ θ v .A particularly sizeable advantage is found at θ * = 0.2356, for which the state is unsteerable when v ≲ 0.7325 but achieves a secret sharing success rate S > S qubit whenenver v ≳ 0.6878.Hence, a significant amount of additional noise can be added to the unsteerable state before it ceases to generate a communication advantage under product measurements.Note also that applying the protocol to an arbitrary pure entangled state |ϕ + θ ⟩, we obtain S = 3+sin(2θ) 4 which exceeds the entanglement-unassisted limit if and only if the state is entangled.

V. STOCHASTIC SECRET SHARING
We have seen that advantages in secret sharing can be propelled by all steerable isotropic states using product measurements.Now we will see that advantages can even be obtained with unsteerable isotropic states, at the cost of switching from product measurements to standard, linear optics, partial Bell state analysers.To this end, we will consider such measurements in a stochastic version of our secret sharing task.
Consider again the scenario in Fig. 1 but now let Alice have three possible outcomes a ∈ {0, 1, ⊥}, where a ∈ {0, 1} corresponds to the secret bit stored in x z ⊕y z , where z = z⊕1, and a =⊥ is associated to a control parameter which indicates that the round is not used for secret sharing.We define that a round should be used for secret sharing exclusively when x z ⊕ y z = 1.Hence, the average success rate of secret sharing in the relevant rounds becomes whereas the average probability of correctly identifying a round not used for secret sharing is This serves as a control parameter.
We are now interested in determining the largest value of R scrt for a given value of the control parameter.Note that an ideal protocol would achieve perfect secret sharing in all relevant rounds and perfectly identify all non-secret sharing rounds, i.e.R scrt = R ctrl = 1.This is possible with entanglement: Bob and Charlie perform the same unitaries as before and Alice does a three-outcome measurement corresponding to a projection onto {Ψ + , Ψ − , Φ + + Φ − } for z = 0 and onto {Φ − , Ψ − , Φ + + Ψ + } for z = 1.These measurements may be seen as partial Bell state analysers, resolving two of the four Bell states, as compatible with passive linear optics [33].
In contrast, in the scenario when entanglement is absent, the trade-off between R scrt and R ctrl is non-trivial.Numerical exploration suggests that it is characterised by Exploiting symmetries [34] to make the problem more tractable, we used semidefinite relaxations [35] to prove the upper bound 5  8 up to solver precision.As in the deterministic case, this bound can be saturated classically: Bob and Charlie relay the bits x 0 and y 0 , respectively, to Alice.If z = 1, she outputs a = x 0 ⊕ y 0 .If z = 0, Alice outputs a =⊥ if x 0 ⊕ y 0 = 0 and a = 0 otherwise.This gives R scrt = 3  4 and R ctrl = 1 2 , thus saturating the entanglement-unassisted bound.
If the above entanglement-assisted protocol is performed using the isotropic state (1), we find that R = 3+5v 8 .Thus, we observe an advantage ascribed to entanglement whenever R > 5  8 , which occurs whenever v > 2 5 .Consequently, when v ∈ 2 5 , 1 2 , the isotropic state is both unsteerable and a communication resource with the employed decoding resources.The unitaries of Bob and Charlie are implemented using half wave plates (HWP) and phase shifters (PS).Alice's projective measurements are implemented using HWP, polarising beam splitters (PBS) and then detected by single photon detectors (DET).For stochastic secret sharing experiment (Box II), isotropic states are prepared by randomly transforming the maximally entangled |ϕ + ⟩ state into one of the other Bell states using one half wave plate (HWP) and a phase shifters (PS).The unitaries of Bob and Charlie are implemented using half wave plates (HWP) and phase shifters (PS).Alice's partial Bell state measurements are implemented using HWP, polarising beam splitters (PBS), beam splitters (BS) and then detected by single photon detectors (DET).

VI. EXPERIMENT
We report here on experimental implementations of the deterministic and stochastic secret sharing protocols.We implement both the ideal quantum protocols using maximally entangled states encoded in the polarisation of photons, and the protocols using weakly entangled, unsteerable, states.We begin by presenting the deterministic secret sharing experiment using partially entangled states ρ θ v , before explaining how this is modified for the stochastic protocol using the simpler isotropic states ρ v .
To generate the entangled states, ultraviolet light centered at a wavelength of 390 nm is focused onto two 2 mm thick β barium borate (BBO) nonlinear crystals placed in an interferometric configuration to produce photon pairs emitted into two spatial modes (a) and (b) through the second order degenerate type-I spontaneous parametric down-conversion process (SPDC).The spectral, and temporal distinguishability between the down-converted photons is carefully removed by passing through narrow-bandwidth interference filters and quartz wedges respectively (see Fig. 2).For the deterministic secret sharing protocol, we prepare the pure partially entangled state |ϕ + θ ⟩ = cos θ|HH⟩ + sin θ|V V ⟩ by setting the polarisation of the pump laser (cos θ|H⟩ + sin θ|V ⟩), where H and V are, respectively, the horizontal and vertical photonic polarisation modes used to encode the basis states in the standard way as |0⟩ := |H⟩ and |1⟩ := |V ⟩.We then prepare a depolarisation ρ θ v of the the state |ϕ + θ ⟩ = cos θ|HH⟩+sin θ|V V ⟩.To achieve the mixture with the maximally mixed state, we randomly transform, with probability 1 − v, the pure partially entangled state into one of these four orthogonal states: |ϕ + θ ⟩, |ϕ − θ ⟩ = sin θ|HH⟩−cos θ|V V ⟩, |ψ + θ ⟩ = cos θ|HV ⟩+sin θ|V H⟩ and |ψ − θ ⟩ = sin θ|HV ⟩ − cos θ|V H⟩.These transformations were experimentally realised by a motorised rotation of two half wave plates and a phase shifter (see Fig. 2).Details and experimental settings are provided in Appendix B. Alice performs product measurements in the H-V and diagonal bases using two HWPs placed before the input of two PBS distributed on each of the two modes (a) and (b), see Fig. 2, Box I.
Bob's and Charlie's transformations U B x and U C y , respectively, are performed using two HWPs and a PS to change the relative phase between the two modes (a) and (b) (see Fig. 2 and Appendix B for the settings).
We chose the visibility v = 0.72 (with θ = θ * = 0.2356), for which the ideal state ρ θ * v is unsteerable.Performing state tomography, we obtain a state fidelity of 0.9965 ± 0.0007 (see Appendix E), and we verified that the experimentally realised state is indeed unsteerable.In the implementation of the protocol the two-photon coincidences rate was one per second.The total number of events recorded is approximately 8 × 10 5 , leading to a parameter value of S = 0.757 ± 0.001 wich is significantly above the entanglement-unassisted limit S = 3 4 (see Appendix D).We also performed the protocol using the pure state |ϕ + θ * ⟩ = cos θ * |HH⟩ + sin θ * |V V ⟩, with a total of 7 × 10 7 two-photon coincidence events.We measured the state fidelity to be 0.990 ± 0.0001 and found a success probability of S = 0.8519 ± 0.0002 which can be compared to the theoretical prediction of S ≈ 0.8635.
For the stochastic secret sharing protocol, we prepared the maximally entangled state |ϕ + ⟩ and isotropic states ρ v in a similar way (recalling that |ϕ + ⟩ = |ϕ + π/4 ⟩).In this case the mixing with white noise, obtained by randomly transforming, with probability 1 − v, |ϕ + ⟩ into one of the four Bell states |ϕ ± ⟩, |ψ ± ⟩ is simplified, and was realised by a motorised rotation of one half wave plate (HWP) and a phase shifter (PS) placed on the mode (a); see Fig. 2. Bob's and Charlie's transformations remain the same as in the deterministic secret sharing protocol.
The polarisation measurements are performed using HWP and polarising beam splitters (PBS), beam splitters (BS) placed at the two output ports of the PBS, and then by single photon detectors (actively quenched Si-avalanche photodiodes).The partial Bell analyser (Fig. 2, Box II) is implemented through two-photon interference, using PBS and HWPs set at 22.5 • .The two-photon Hong-Ou-Mandel dip visibility is 99.2% ± 0.6, where the substantial statistical error is due to the low two-photon coincidence rate used in the experiment (one per second) and a measurement time of 2400 seconds per point (see Appendix H).To switch from a Bell measurement discriminating between {Ψ + , Ψ − , Φ + + Φ − } and {Φ − , Ψ − , Φ + +Ψ + }, Alice uses three HWPs placed before the first PBS (see Appendix F for the HWP settings).All singledetection events were registered using a VHDL-programmed multichannel coincidence logic unit, with a time coincidence window of 1.7 ns.
To perform state tomography of isotropic states ρ v for dif-ferent v, we performed tomography of the four Bell states generated in the randomisation procedure.Measurements were made at a rate of one two-coincidence per second over 1400 seconds for each of the nine settings needed to perform the state tomography for each Bell state.These results were then recombined a posteriori at different ratios to establish the density matrices ρ v .We considered the reconstructed states ρ v for v = 0.4 to v = 0.5, with a step-size of 0.01, corresponding to the resourceful but unsteerable range.Naturally, the reconstructed density matrices are not exactly isotropic states.To ensure the unsteerability of the experimentally realised states, we used the linear programming method of Ref. [36] which allows one to obtain a certificate of unsteerability for an arbitrary two-qubit state.We chose to proceed during the experiment with v = 0.47, as this represented a good balance between being below the steering threshold of v unsteer = 1 2 while allowing for good enough statistics to show a significant quantum advantage (see Appendix A).The fidelity of the reconstructed state with the target isotropic state for v = 0.47 is 0.9983 ± 0.0004.Detailed tomography results for the density matrices, the state fidelities, and the certificates of unsteerability are presented in Appendix I.
The protocol was then carried out with the isotropic state at v = 0.47 with the noise added by randomly changing the Bell state between each two-photon coincidence while maintaining the necessary ratio between these states.To obtain at most one event per change of Bell state and thus ensure the randomness and therefore unpredictability of each event, we chose to work at a rate one two-photon detection coincidence per second.The effective measurement time per setting was 2.8 hours.We obtained a success probability of R = 0.6524 ± 0.0004 which significantly goes beyond the theoretical entanglementunassisted limit of R qubit ≤ 5  8 , hence showing an advantage from unsteerable states (see Appendix G).
The same experiment was also performed for a maximally entangled state |ϕ + ⟩, i.e. without any randomisation over Bell states.This increased the rate to 800 two-photon detection coincidences per second, with a measurement time per setting of 2 hours.The fidelity of the state was measured to be 0.9947 ± 0.0009.The observed success probability is R = 0.9748±0.0001,which is close to the ideal maximum value of R = 1, thus showing a large advantage due to entanglement.

VII. DISCUSSION
In this letter we demonstrated theoretically, and confirmed experimentally, that one can obtain quantum communication advantages in a secret sharing task using weakly entangled unsteerable states while restricting only to product measurements or entangled measurements compatible with passive linear optics.Figure 3 summarises the ranges of noise v for which quantum communication advantages are achievable with twoqubit isotropic states and the relation to its nonlocal properties.It remains an open question whether the visibilities can be further reduced both for standard linear optics measurements and product measurements.Of particular relevance is to investigate whether product measurements also can gen- erate advantages from highly noisy multi-qubit states, as this would pave the way for experiments that go beyond small-scale demonstrations.
state, we use an HWP to change the power distribution between the H and V crystals.When the angle of this HWP is 0, only the H-polarised crystal is pumped and when the HWP is π 2 , only the V-polarised crystal is pumped.When the angle is π 4 , the prepared state is |ϕ + ⟩.We have to select the right angle (θ) value to prepare the desired state.To create the maximally mixed state 1
We can thereby generate ρ θ v for any v by choosing correctly the weights with which we randomly change |ϕ + θ ⟩ to one of the other states.The switching between these different states is done with the help of two half-wave plates and a phase plate; see Fig. 2 of the main text.
These plates are motorised and switched randomly (with the desired probability) from one state to another every second.The number of coincidences being less than one per second, the result is to switch from one state to another randomly and independently between each pair of successive coincidences.The values of the angles allowing one to pass from one state to another are given in Table I.Phase plate

Appendix C: Bob's and Charlie's rotations
As described in the main text, on inputs x = (x 0 , x 1 ) ∈ {0, 1} 2 and y = (y 0 , y 1 ) ∈ {0, 1} 2 , Bob and Charlie apply the unitaries U B x = σ y0 X σ y1 Z and U C y = σ y0 X σ y1 Z , respectively.Table II summarises explicitly the operations that Bob and Charlie must perform in inputs x and y, as well as the Bell state expected to be received by Alice in the ideal case (assuming Bob and Charlie share a Φ + state).The settings to be applied to achieve each of the four different possible rotations are given in Table III.

Alice's measurements
Depending of her setting z ∈ {0, 1}, Alice performs a projective measurement in the diagonal or in the H-V basis.To make these measurements, she uses two HWPs λ A1 2 and λ A2 2 placed before the input ports of each PBS (see Fig. 2 of the main text).The values of the angles of these HWPs are given in Table IV.Tables V and VI list our experimental results alongside the theoretical success rates S.
The errors were calculated using the same method as for the stochastic secret sharing experiment.

Appendix E: Density matrix tomography for the deterministic secret sharing experiment
To ensure that the prepared state is the desired one, a tomography measurement is performed for a perfect |ϕ + θ * ⟩ state and for the noisy states ρ θ * v with a visibility of v = 0.72 (and with θ * = 0.2356), which are the two states for which the experimental demonstration of the protocol was carried out.The reconstructed density matrices are shown in Figures 4 and 5, respectively.
The fidelity of each state obtained during the tomography  The setup used generates an entangled Φ + state with a visibility close to 100%.In order to generate an isotropic state ρ v (A1), this state needs to be mixed with probability (1 − v) with the maximally mixed state 1  4 .We implement the maximally mixed state by randomly switching between the four Bell states, Φ ± , Ψ ± , exploiting the fact that 1 1 4 = 1 4 (Φ + +Φ − +Ψ + +Ψ − ).We can thereby generate ρ v for any v by choosing correctly the weights with which we randomly change Φ + to one of the other Bell states.
The switching between these different states is done with the help of one half-wave plates and a phase plate; see Fig. 2 of the main text.
These plates are motorised and switched randomly (with the desired probability) from one state to another every second.The number of coincidences being less than one per second, the result is to switch from one Bell state to another randomly and independently between each pair of successive coincidences.The values of the angles allowing one to pass from one state to another are given in Table VIII.

Bob's and Charlie's rotations
Bob's and Charlie's rotations are the same as in the stochastic secret sharing experiment; see Section 2.B above.

Alice's measurements
Depending of her setting z ∈ {0, 1}, Alice performs one of the three-outcome partial Bell measurement {Ψ + , Ψ − , Φ + + Φ − } or {Φ − , Ψ − , Φ + + Ψ + } and obtains an outcome c ∈ {0, 1, ⊥}.To make these measurements, she uses three HWPs 2 and λ A3 2 , with the first two (A1 and A2) placed before one of the input ports of the PBS, and last one (A3) placed before the other input port (see Fig. 2 of the main text).The values of the angles of these HWPs are given in Table IX.Phase plate Charlie and the generation of Werner states, to set the orientation of the wave-plates with a repeatably high precision of 0.02 • .The use of different settings (x, y) induces a systematic error, which we estimate using Monte Carlo simulation.We assume that the wave plates' setting errors are normally distributed with a standard deviation of 0.02 • .Alice's wave plates are not motorised.In order to reduce systematic errors in the angle settings, all settings for z = 0 are measured, then all settings for z = 1, so the results are initially presented separately for each case.We estimate the accuracy of the angles of these WPs to be half a degree.This, together with the Poissonian error in photon counting statistics, comprises the final error reported here.Due to inefficiency in the single photon detectors, the photons are detected randomly and their counting is Poissonian.To decrease Poissonian counting error, we have chosen for v = 1 a measurement time of two hours for every setting and collected approximately 93 million events.To guarantee that both parties receive single qubits, we worked at a low rate (≈ 800 coincidence per sec) to suppress higher order coincidences to almost 0.9 per sec.For v = 0.47, we have collected around one million two-photon coincidences and the low rate renders the higher order coincidences completely negligible.

Appendix H: Two-fold Hong-Ou-Mandel dip visibility
Bell state measurements are implemented through twophoton interference, using PBS and HWP plates set at 22.5°.The photons are detected by Si avalanche photodiodes and the coincidences are registered with an eight channel multifold coincidence counting unit.This Bell analyser consists of coherent interference at a polarising beam splitter (PBS).To achieve the necessary indistinguishability of the photons, due to their arrival times, we adjusted the path length of one of the photons using a delay line [39].In Figure 6, the coincidences between the detectors versus the delay path length is shown.The zero delay corresponds to a maximal overlap (maximum indistinguishability).The interfering photons bunch (i.e., they both exit in the same output arm of the PBS) causing the coincidences to vanish.The measured visibility of the two-fold Hong-Ou-Mandel dip is 99.2% ± 0.6, with a coincidence rate outside the dip around 0.9224 coincidences per second, identical to the rate at which we performed the experiment for v = 0.47.

Appendix I: Density matrix tomography for stochastic secret sharing
To ensure that the prepared state is the desired one, a tomography measurement is performed for a perfect Φ + state and for isotropic states ρ v with a visibility between v = 0.4 and v = 0.5.The reconstructed density matrices obtained for the cases of v = 1 (i.e., the ideal case of ρ 1 = Φ + ) and v = 0.47, which are the two states for which the experimental demonstration of the protocol was carried out, are presented in Figures 7 and 8, respectively.
The fidelity of each state obtained during the tomography with the ideal isotropic state ρ v was found to be between 0.9947 and 0.9983.Table XII gives the fidelity for each of these states.

Fig 1 .
Fig 1.Bob and Charlie each privately select two uniformly random bits x ≡ (x 0 , x 1 ) ∈ {0, 1} 2 and y ≡ (y 0 , y 1 ) ∈ {0, 1} 2 , respectively, and each communicate a message to Alice.Alice privately selects a binary input z ∈ {0, 1} and accordingly decodes the two incoming messages into an output a ∈ {0, 1}.Here, z, which is later announced publicly, determines which bits of Bob and Charlie hold the shared secret: the success condition is a = x z ⊕ y z .The average success probability of secret sharing becomes

FIG. 2 .
FIG.2.Experimental setup.Entangled photon pairs in spatial modes (a) and (b) are generated through the SPDC process.For the deterministic secret sharing experiment (Box I), entangled photon pairs are generated by the same way but with a different ratio between the two modes.Isotropic states are prepared by randomly transforming the maximally entangled |ϕ + ⟩ state into one of three other Bell states using two half wave plate (HWP) and a phase shifter (PS).The unitaries of Bob and Charlie are implemented using half wave plates (HWP) and phase shifters (PS).Alice's projective measurements are implemented using HWP, polarising beam splitters (PBS) and then detected by single photon detectors (DET).For stochastic secret sharing experiment (Box II), isotropic states are prepared by randomly transforming the maximally entangled |ϕ + ⟩ state into one of the other Bell states using one half wave plate (HWP) and a phase shifters (PS).The unitaries of Bob and Charlie are implemented using half wave plates (HWP) and phase shifters (PS).Alice's partial Bell state measurements are implemented using HWP, polarising beam splitters (PBS), beam splitters (BS) and then detected by single photon detectors (DET).

FIG. 3 .
FIG.3.Nonlocal properties (red) and quantum communication advantages (blue) of the two-qubit isotropic state.The range for quantum communication advantages from general entangled measurements follows from dense coding-like protocols[9,10].The ranges shown for quantum communication advantages with passive linear optics measurements and product measurements are established in this work and are upper bounds on the critical visibilities.

FIG. 4 .FIG. 5 .
FIG. 4. State tomography of ρ θ * v for v = 1; the real part of the density matrix is shown on the left, and the imaginary part on the right.

FIG. 6 .
FIG. 6. Two-fold Hong-Ou-Mandel dip.The plot displays the twofold photon counting coincidences versus the delay (the path difference between the two arms).The error bars indicate the Poissonian photon counting error statistics.The data is fitted with a Gaussian curve.

FIG. 7 .
FIG. 7. State tomography of ρv for v = 1; the real part of the density matrix is shown on the left, and the imaginary part on the right.

FIG. 8 .
FIG. 8. State tomography of ρv for v = 0.47; the real part of the density matrix is shown on the left, and the imaginary part on the right.
The secret scharing scenario.Bob and Charlie select private inputs and perform separate transformations on a shared two-qubit entangled state.They relay their respective output qubits to Alice who measures them.Her output is the secret and her input determines how it can be reconstructed collectively by Bob and Charlie.

TABLE I .
Half wave plate rotation angles (in degrees) and phase for the preparation of the differents states.

TABLE II .
Transformation table: the operations σ x 0 X σ x 1 Z and σ y 0 X σ y 1 Z that need to be performed by Bob and Charlie, respectively.The expected Bell state that Alice will receive in the ideal case that Bob and Charlie share a noiseless maximally entangled state is shown in the final column.

TABLE III .
Rotation angles (in degrees) for the unitaries for Bob's and Charlie's operations.

TABLE IV .
Rotation angles for the HWPs used to implement Alice's projective measurements, for z = 0 and z = 1.

TABLE V .
Theoretical and experimental success probabilities for the partially entangled state |ϕ + θ * ⟩ (with θ * = 0.2356), conditioned on z and then averaged to give the overall success probability S. the ideal state ρ θ * v was found to be above 0.99.Table VII gives the fidelity for each of these states. with

TABLE VII .
Fidelity of measured states with the ideal state ρ θ * v for v = 0.72 and v = 1.

TABLE VIII .
Half wave plate rotation angles (in • ) and phase for the preparation of the different Bell states.

TABLE IX .
Rotation angles for the HWPs used to implement Alice's partial Bell state measurements, for z = 0 and z = 1.

TABLE X .
Theoretical and experimental success rate for the maximally entangled state Φ + , conditioned on z and then averaged to give the overall success rate R.

TABLE XI .
Theoretical and experimental success rates for the isotropic state ρv with v = 0.47, conditioned on z and then averaged to give the overall success rate R.

TABLE XII .
Fidelity of each measured state with the ideal isotropic state ρv.