Scalable entanglement certification via quantum communication

Harnessing the advantages of shared entanglement for sending quantum messages often requires the implementation of complex two-particle entangled measurements. We investigate entanglement advantages in protocols that use only the simplest two-particle measurements, namely product measurements. For experiments in which only the dimension of the message is known, we show that robust entanglement advantages are possible, but that they are fundamentally limited by Einstein-Podolsky-Rosen steering. Subsequently, we propose a natural extension of the standard scenario for these experiments and show that it circumvents this limitation. This leads us to prove entanglement advantages from every entangled two-qubit Werner state, evidence its generalisation to high-dimensional systems and establish a connection to quantum teleportation. Our results reveal the power of product measurements for generating quantum correlations in entanglement-assisted communication and they pave the way for practical semi-device-independent entanglement certification well-beyond the constraints of Einstein-Podolsky-Rosen steering.


I. INTRODUCTION
Shared entanglement between a sender and a receiver that are connected over a quantum channel is the most powerful communication resource in quantum theory.This is famously showcased in the dense coding protocol, where entanglement doubles the classical capacity of a noise-free qubit channel [1].If the channel is used only once, which is the scenario most pertinent for experimental considerations, this entanglement-assisted prepare-and-measure (EAPM) scenario (see illustration in Fig. 1) can equally well be viewed as setting for efficient quantum communication and as a platform for semi-device-independent quantum information protocols.The latter is because the state, the sender and the receiver are uncharacterised devices, and only knowledge of the dimension of the channel is required to deduce the quantum nature of the correlations.In this sense, the EAPM scenario offers an appealing path to certify the advantages of entanglement in experiments with limited characterisation.
A central obstacle for harnessing entanglementadvantages in the EAPM scenario is that protocols commonly need the receiver (Charlie) to measure both the particles, namely the one coming from the entanglement source and the one arriving over the channel, in an entangled basis.In for example optical systems, such measurements are well-known to be impossible without extra photons or nonlinear effects [2][3][4], which for instance can limit experiments to using only single-photon carriers of multiple qubits (see e.g.[5,6]).In the EAPM scenario, for the simplest case of qubit systems, a series of dense coding experiments have over time implemented increasingly sophisticated Bell basis measurements and thereby approached the theoretical limit of the entanglement advantage [7][8][9][10][11][12][13][14].For systems of higher dimension than FIG. 1. Entanglement-assisted prepare-and-measure scenario between a sender (Alice) and a receiver (Charlie).The information, x, is encoded into one share of the entangled state.
qubit, the situation is extra challenging.Even resolving one element of a high-dimensional entangled basis is impossible with ancilla-free linear optics [15].The most high-dimensional optical Bell basis measurement hitherto realised is limited to three-level systems and uses ancillary photons [16,17].In the EAPM scenario, this has led experiments based on high-dimensional entanglement and quantum communication to instead focus on simpler, suboptimal, measurements, compatible with standard linear optics [18].The challenges associated with entangled measurements are broadly relevant in the different correlation tests accommodated by the EAPM scenario [19][20][21].
However, while entangled measurements are provably necessary for the specific task of dense coding [22], this is not true in general.Interestingly, it was recently shown that there exists well-known communication tasks which in the EAPM scenario admit their best implementation in protocols that rely only on the simplest joint measurements [23].These are mere product measurements of the sourceand message-particles; they are therefore classical postprocessings of two completely separate single-particle measurements.In principle, this can greatly simplify the experiments, as the particles do not need to interfere with each other in the measurement device, and can even be measured at separate times.Nevertheless, presently, little is known about how such protocols can be constructed.Moreover, another important aspect concerns the noiserobustness of protocols based on product measurements.While entangled measurements are well-known to reveal the correlation advantages of shared entanglement even from very noisy states [24,25], no counterpart is known for product measurements.That is, even though product measurements can sometimes be optimal under ideal conditions, their performance might deteriorate in the presence of signficant amounts of noise in the entanglement distribution, rendering them unable to certify entanglement that is well-within the reach of schemes that use entangled measurements.Indeed, certifying noisy forms of entanglement is a central matter for correlation experiments.
Here, we investigate entanglement advantages revealed by product measurements in the EAPM scenario and show that they are much more powerful than previously known.In all our scenarios, the source is fully untrusted.The operations of all the parties are also untrusted, up to the bounded dimension of the quantum communication channels.The article is structured as follows.In section II we formalise the EAPM scenario.In section III, we investigate high-dimensional entanglement by introducing concrete certification schemes in the EAPM scenario.We prove that the paradigmatic isotropic state is certified at noise rates well-above the known thresholds for Bell nonlocality, and closely resembling the thresholds known for Einstein-Podolsky-Rosen steering.In section IV, we show that the results from section III are actually close to optimal.This follows from a no-go result, in which we show that steering is a necessary condition for certifying entanglement advantages in any EAPM scenario with product measurements.In section V, we set out to circumevent this fundamental limitation.We do so by considering a natural extension of the standard EAPM scenario, which we name the symmetric EAPM scenario.In the symmetric scenario, classical information is not only encoded in one share of the state (Alice, in Fig 1) but in both shares of the state (see Fig 2).For qubit systems, we prove that every entangled Werner state implies an advantage.Finally, in section VI, we introduce a prime-dimensional generalisation of the scheme in section V. We present evidence, on the basis of which we argue that every state supporting fidelity-based quantum teleportation can be certified.This notably includes every entangled isotropic state.

II. THE EAPM SCENARIO
The EAPM scenario is illustrated in Fig 1 .Alice and Charlie share a state ρ AC , which can have an arbitrary local dimension.Alice selects a classical input x and encodes it on her share of the state via a completely positive tracepreserving (CPTP) map, Λ A→R x , whose output system (R), which we call the message, has a known dimension d.The total state arriving to Charlie becomes Finally, Charlie selects a classical input z and performs a joint quantum measurement {M RC c|z } with outcome c.The Born-rule gives the quantum correlations, p(c|x, z) = tr τ RC x M c|z .Notice that the set of states {τ RC x } realisable via arbitrary CPTP maps for Alice and arbitrary initial entangled state can be completely characterised as τ RC x being a d × D dimensional bipartite state with tr R (τ RC x ) = τ C , where τ C is the reduced state which is notably independent of x.Note that D is the dimension of the source particle, which can be arbitrary.
Our focus is on protocols where Charlie's measurements act separately on systems R and C.This can be a product measurement followed by a classical post-processing of the respective outcomes, i.e.
where {N R c 1 |z } and {N C c 2 |z } are single-system measurements and p(c|c 1 , c 2 ) is some (perhaps stochastic) rule for deciding the final outcome c from the local outcomes (c 1 , c 2 ).More generally, the measurements can also be adaptive [22], i.e.Charlie could use the outcome on system R to inform his measurement on system C, and vice versa.These adaptive product measurements take the form respectively.We are interested in comparing the correlations, p(c|x, z), obtained from shared entanglement and product measurements, with those obtained without shared entanglement.
The latters correspond to standard (entanglement-unassisted) quantum prepare-and-measure scenarios, i.e.Alice can send any d-dimensional state α x to Charlie who can perform an arbitrary quantum measurement on it, p(c|x, z) = tr α x M c|z . ( Shared classical randomness is additionally permitted between the parties.

III. CERTIFYING HIGH-DIMENSIONAL ENTANGLEMENT IN THE EAPM SCENARIO
We begin by identifying a scheme in the EAPM scenario that enables us to certify entanglement under substantial and dimension-scalable noise rates.To this end, consider the following scheme.Let Alice have an input x ≡ (x 0 , x 1 ) ∈ {0, . . ., d − 1} 2 and Charlie have an input z ∈ {0, . . ., d}, where d is prime number.The parties have the objective to compute (via the output c), for each z, a specific binary function of x.These functions are The average success probability of computing the functions is therefore Next, we will analyse S d in a quantum setting with and without entanglement and prove that it certifies entanglement under product measurements.
Note that for the special case of d = 2, X and Z are simply two of the Pauli operators and U x is effectively the four Pauli rotations.Finally, we must select Charlie's product measurements.For the special case of d = 2, we choose them as products of the three Pauli observables, namely with Beyond dimension two, following Eq.(2), we define the measurements as postprocessings of the outcomes obtained in two separate basis measurements of systems R and C, where * denotes complex conjugation and the addition in  [26].The unbiasedness property is not de-facto necessary for the success of the protocol, but is a convenient choice.
In order to evaluate the average success probability (5), it is handy to first identify the following relations, which can be straightforwardly verified, valid for z ̸ = d and integer t.Using these, one straightforwardly finds that each of the functions is computed deterministically, that is p(c|x, z) = δ c,w z , leading to S d = 1.

B. Bounding protocols without shared entanglement
Next, we must determine a useful bound S d ≤ L d valid for any quantum strategy without shared entanglement.Since this corresponds to bounding the expression (5) in a standard quantum prepare-and-measure scenario, the correlations are given by Eq. (3).The relevant quantity becomes max where α x is a d-dimensional state.We restrict the analysis to prime number dimensions because in these cases the conditions in Eq. ( 4) are particularly hard to meet without entanglement.The task at hand can be seen as an (unorthodox) variant of a quantum random access code [27,28].
The proof ideas recently developed for quantum random access codes in Ref. [29] can be immediately modified to obtain a general bound, L d , on Eq. ( 10), namely for prime d.The derivation is detailed in Appendix A and it is based on analysing operator norms for sums of the measurement operators.Regardless of the protocol used, the observation of S d > L d implies the certification of entanglement.The bound L d is typically not tight (except for d = 2), i.e. it does not equal the value defined in (10).
The reason for this becomes apparent in Appendix A, where both operator norm inequalities and concavity inequalities are employed, the saturation of which is not guaranteed in general.Nevertheless, to give an indication of how close to optimal the bound is, we have numerically optimised the argument in (10) over the set of quantum states and measurements.For d = 3, 5, 7 we obtain the lower bounds 0.6616, 0.5121 and 0.4233 respectively, which can be compared to the upper bounds 0.6667, 0.5266 and 0.4459 obtained respectively from (11).We note that numerical techniques likely can be used to improve the bound (11), but only for specific values of d [30].
Even though the bound ( 11) is not generally tight, it is good enough to reveal the qualitative abilities of product measurements in a dimension-scalable manner.To showcase that, we focus on the seminal isotropic state, with visibility v ∈ [0, 1].Thus, when running the strategy from the previous section, we compute the smallest visibility for which the state produces a value of S d that exceeds the limit in Eq. (11).For comparison, the isotropic state is known to be entangled if and only if v > 1 d+1 [31].

Corollary 1. For every prime dimension d, entanglement certification in the EAPM scenario with product measurements is possible for the isotropic state when
This exhibits an inverse-square-root scaling in the dimension parameter, thus showing that product measurements become increasingly good at certifying the entanglement.In particular, for d = 2, it reduces to v > 1/ 3, which significantly improves on previous protocols [23], and happens to equal the exact threshold for steerability of ρ iso v for the same number (three) of measurements [32].Moreover, for prime d, Eq. ( 13) exactly matches the bound for steerability of ρ iso v under d +1 mutually unbiased bases obtained from the steering inequality of Ref. [33].

IV. NO ENTANGLEMENT ADVANTAGE WITHOUT STEERING
It is not a coincidence that our above scheme happens to give critical visibilities that closely parallel results known for steering.As we now show, the above results are examples of saturation (or near saturation) of a more fundamental limitation that applies to any protocol in the EAPM scenario using adaptive product measurements.Proposition 1.Let ρ AC be any entangled state that is not steerable from C to A. Then, any probability distribution in the EAPM scenario obtained from adaptive product measurements can be simulated in a quantum model with shared classical randomness.
Proof.Consider first product measurements that are adaptive from system C to system R.The probability distribution can then be written where ρ AC are the unnormalised states remotely prepared on A by measuring C. If ρ AC is unsteerable from C to A, there exists a local hidden state decomposition ρ c 2 |z = λ p(λ)p(c 2 |z, λ)τ λ for some arbitrary-dimensional quantum states {τ λ }.Inserting this, we obtain This can be simulated without entanglement as follows.Let Alice and Charlie share λ, with distribution p(λ).Alice prepares τ λ and applies Λ x to it, sending the d-dimensional state Λ x [τ λ ] to Charlie.He draws c 2 from the distribution p(c 2 |z, λ), then applies the measurement {M R c 1 |z,c 2 }, and lastly draws c from p(c|c 1 , c 2 ).This reproduces the distribution in Eq. (15).
The case of product measurements adaptive from system R to system C is similarly treated.The probability distribution becomes where ρ AC are the unnormalised states remotely prepared on A. The existence of a local hidden state model implies (17) To simulate this distribution without entanglement, one distributes λ, let's Alice prepare τ λ and run it through the map Λ x .Charlie first measures the message, then uses the outcome c 1 to draw c 2 from p(c 2 |z, c 1 , λ) and finally draws c from p(c|c 1 , c 2 ).
A noteworthy corollary of this argument is that, product measurements adaptive from C to R, in an EAPM scenario with N inputs for Charlie, one-way steerability under just N measurements is necessary for an entanglement-advantage.This makes a significant difference since steerability under a limited number of measurements is known to be considerably more constrained than steerability under unboundedly many measurements [32].In view of this, the scheme from the previous section, which led to Corollary 1 via independent product measurements, is optimal for d = 2 since it coincides with the steering bound of ρ iso v under three measurements.For larger d, it is unlikely that our result from the previous section can be much improved, because of the steering results for N = d + 1 bases in [33,34].However, by employing potentially unboundedly many measurements (instead of d +1 as in our case), it may be possible to approach the ultimate steering limit [35] on the parameter v.
Proposition 1 provides a fundamental limitation on the abilities of product measurements in the EAPM scenario.Although we already found that significantly noise-tolerant entanglement certification is possible, it is impossible to certify any state which is entangled but not steerable.Therefore, in what follows, we go beyond the EAPM scenario to show that this obstacle can be overcome, allowing for even stronger entanglement certification.

V. THE SYMMETRIC EAPM SCENARIO
In order to circumvent the limitation on product measurement schemes imposed by Proposition 1, we consider an extended version [24,36] of the original EAPM scenario which we refer to as the symmetric EAPM scenario.The extension is modest in terms of an implementation perspective and is conceptually natural.In the original EAPM scenario, classical information is only encoded into half the entangled state, namely by Alice, into system A. In the symmetric EAPM scenario, we want to encode classical information also in the other half of the entangled state.To make this possible, we add a third party, Bob, who selects an input y and encodes it into the second source particle before relaying it to Charlie.See illustration in Fig 2 .Let us now write the state as ρ AB , distributed to Alice and Bob.They each select x and y and perform CPTP maps We remark that all parties can also share classical randomness, which can be included in the state ρ AB .Again, we are interested in how protocols using shared entanglement and product measurements can outperform protocols using no shared entanglement.The correlations from the latters are described as where α x and β y are d-dimensional states sent from Alice and Bob, respectively, to Charlie.Note that in contrast to the original EAPM scenario, the measurement {M c|z } can now be entangled.
A direct inspection shows that the argument used to arrive at Proposition 1 cannot be repeated for the symmetric EAPM scenario.Indeed, the argument relies on the fact that Alice's operations do not influence the other particle, making one-way steering relevant.The counterparts to these states arriving to Charlie are now influenced by Bob.We shall see that this is not a superficial observation; the symmetric EAPM scenario can indeed certify unsteerable entanglement.To show this, let us focus on qubit systems and the following scheme.
Alice and Bob each select two bits, x ∈ (x 0 , x 1 ) ∈ {0, 1} 2 and y ∈ ( y 0 , y 1 ) ∈ {0, 1} 2 .Charlie selects one of three inputs z ∈ {0, 1, 2}, each with binary outputs c ∈ {0, 1}.The aim is for Charlie to compute a binary function for each z, specifically the functions z = 0 : computed modulo 2. The average success probability becomes In analogy with the discussion in section III, this task can be performed deterministically with shared entanglement and product measurements.In complete analogy with the protocol in section III, we let ρ AB = φ + 2 and we let Charlie perform the separate Pauli observables in Eq. ( 7).Alice and Bob both select among the same four Pauli unitaries, namely U x and U y , as given in Eq. ( 6).Evaluating Eq. ( 21) then gives R 2 = 1.
The key question is to determine the largest value of R 2 achievable in a quantum model without shared entanglement.Consider first a classical protocol, in which α x and β y in Eq. ( 19) are all diagonal in the same basis.An optimal strategy is for Alice and Bob to simply send x 0 and y 0 respectively, leading to a deterministic output for z = 0 but random outputs for z ∈ {1, 2}, and thus a value of R 2 = 2 3 .We prove in Appendix B that this cannot be improved in a generic quantum protocol without shared entanglement, i.e. any model of the form (19) obeys R 2 ≤ 2 3 .Consequently, any quantum-over-classical advantage in the scheme must be due to entanglement.Notably, the same is not true for the scheme presented in section III; there the classical limit can be overcome using quantum communication without entanglement, and then be further enhanced by adding entanglement.
We remark that the proof presented in Appendix B applies more generally.It can be used to bound the average success probability in any input-output scenario in which Charlie has binary outputs and the winning conditions are XOR between balanced functions of Alice's input and Bob's input.Nevertheless, we focus on the specific case in Eq. ( 20) because of its relevance for certifying the entanglement of isotropic states 1 .Indeed, a simple calculation now shows that every entangled isotropic state is certified, i.e.R 2 > 2 3 when v > 1  3 .In contrast, the state is steerable under generic projective measurements only when v > 1  2 [35].Notably, this result completely solves the main open problem raised in Ref. [23].
More generally, consider the so-called maximally entangled fraction of ρ AC , where Λ 1 and Λ 2 are CPTP maps with d-dimensional output spaces.A non-trivial maximally entangled fraction corresponds to EF d (ρ) > 1 d , and it is the key parameter for quantifying fidelity-based quantum teleportation [37].We find that it gives a sufficient condition for whether a state can be certified via product measurements in our scheme.
Proposition 2. Every state ρ AB with a non-trivial qubit maximally entangled fraction can be certified in the symmetric EAPM scenario using product measurements.In particular, it can achieve the value Moreover, this value is optimal when ρ AB is a pure two-qubit state.
Proof.Here, we show only that Eq. ( 23) is attainable, with remaining details given in Appendix C. Upon receiving the shares of ρ AC , let Alice and Bob first apply some arbitrary CPTP maps Λ 1 and Λ 2 respectively, whose output systems are d-dimensional.Subsequently, they each implement the previously given protocol, i.e. they perform unitaries U x and U y respectively and Charlie measures the observables in Eq. ( 7).We can express the average success probability as where The right-hand-sides are obtained after some simplifications.Due to the protocol's symmetry, we have B (1)  z = B (2) z .One then observes that, z B (1)  z ⊗ B (2) z = 16(4φ Inserted into Eq.( 24) and allowing for an optimisation over the channels Λ 1 and Λ 2 , we obtain Eq. ( 23).
Thus, a state's usefulness in teleportation is a sufficient condition for certification in the symmetric EAPM scenario.Notably, many states with a non-trivial maximally entangled fraction do not admit any steering.The most immediate example is the isotropic state in the interval 1  3 < v < 1 2 [35].We remark that we have numerically explored the trade-off between R 2 and the set of (mixed) two-qubit states with a bounded maximally entangled fraction, and we again find that Eq. ( 23) is the optimal value of R 2 for every such state.

VI. TOWARDS HIGH-DIMENSIONAL SCHEMES
Having found that the symmetric EAPM scenario can for some classes of states enable even optimal entanglement advantages under product measurements, we proceed with investigating whether the same is possible also for highdimensional systems.To this end, we draw inspiration from the scheme in section III and extend it to the symmetric EAPM scenario.
Let d be an odd prime number.Let Alice and Bob each select one of d 2 inputs, x = (x 0 , x 1 ) ∈ {0, . . ., d − 1} 2 and y = ( y 0 , y 1 ) ∈ {0, . . ., d − 1} 2 .Charlie selects z ∈ {0, . . ., d} and outputs c ∈ {0, . . ., d − 1}.The winning conditions correspond to computing the following functions z ̸ = d : The average success probability of computing these functions is Notice that for d = 2, this reduces to the qubit scheme from section V.A protocol based on product measurements, analogous to that used in section III, can deterministically compute each of the winning functions.That is, choose ρ AB = φ + d , choose Alice's and Bob's unitaries as in Eq. ( 6) and choose Charlie's measurements as in Eq. ( 8), with the d + 1 mutually unbiased bases {|e m,z 〉}.One then calculates that R d = 1.
Consider now a fully classical model.A simple protocol is, just like for R 2 , to send e.g.x 0 and y 0 to Charlie and thus let him output correctly (c = w z ) when z = d but output at random when z ̸ = d.This leads to R d = 2/(d + 1).One may wonder weather there exist a quantum strategy without entanglement that improves this bound.Nonetheless, in analogy with what was proven for the qubit case in section V, we are unable to find any such protocol.Particularly, when employing the strategy mentioned above, that was optimal for shared entanglement, but now to the case without shared entanglement, we get the classical score R d = 2/(d +1)-see Appendix D for more details.While for d = 2 we proved analytically that the bound cannot be improved, in Appendix B, for the cases of d = 3 and d = 5, we have used a numerical search based on the see-saw method [38] to optimise R d over the operations of Alice, Bob and Charlie without shared entanglement.Specifically, we optimise R d in Eq. ( 28) for all possible correlations according to Eq. ( 19) for any set of local quantum states α x , β y in Alice and Bob's laboratories respectively, and measurements M c|z in Charlie's laboratory.The optimization is rendered as a semidefinite program with variables iterating in a see-saw manner.That is, we begin sampling random quantum states α x and β y with dimension d and optimise R d for any measurements of Charlies, M c|z .The optimal M c|z are stored and R d is again optimised but now over all possible states of Alice, α x .Again, the optimal α x are stored and now the optimisation runs over all possible states of Bob, β y .This routine of three separate optimisations is then repeated until the estimated value of R d converges (within a precision factor of 10 −4 ).In over 300 separate trials for each d, we have without exception found the obtained value of R d , coincides with the classical bound.On this basis, we make the following conjecture.

Conjecture 1. For every odd prime d, the largest average success probability achievable in a quantum model without shared entanglement is
Interestingly, if the conjecture is true, it implies that the strong entanglement advantages previously proven for qubit systems can be extended to high-dimensional systems.In Appendix D, we prove that the connection between the maximally entangled fraction and the average success probability, seen in Proposition 2, generalises to larger d.

Proposition 3. For every odd prime d and state ρ AB , there exists a quantum model achieving the average success probability
Moreover, numerics for d = 3 suggests that for pure states the value ( 30) is optimal, but reveals that the same is not always true for mixed states.If Conjecture 1 is true, Proposition 3 implies that every state with a non-trivial maximally entangled fraction exceeds the limitation (29) and is therefore certified as entangled.In particular, every entangled isotropic state ρ iso v has a non-trivial maximally entangled fraction and therefore this family of states is optimally certified.

VII. DISCUSSION
We have shown that product measurements are sufficient for revealing the advantages of noisy forms of entanglement in prepare-and-measure scenarios, and that this can be achieved via simple communication tasks.In the standard EAPM scenario, we showed that visibility requirements for white noise can decrease as the inverse squareroot of the dimension.However, we also showed that this scalability is fundamentally limited by a need for steerability for entanglement advantages.By proposing the symmetric EAPM scenario, we showed how this limitation can be overcome, sometimes even in an optimal way.This is exemplified by every entangled qubit Werner state being certified, as well as every state useful for fidelity-based teleportation.Beyond qubits in the symmetric EAPM scenario, we also showed how these results can be generalised to primedimensional systems, but this ultimately requires a proof of Conjecture 1. Extending our methods to arbitrary, nonprime, dimensions is a natural next step.
Our results pave the way for theoretical exploration and experimental implementation of strong forms of semidevice-independent entanglement certification, which may apply also to finer entanglement concepts such as Schmidt numbers or fidelity estimation with a target state.It is appropriate to label our scenarios as semi-deviceindependent, because they require none of the quantum devices to be perfectly characterised, but only that the number of degrees of freedom in the channel is bounded.Therefore, this form of entanglement certification is far stronger than standard entanglement witnessing, where the devices are assumed to be flawless.For instance, the latter is known to be vulnerable to false positives when devices do not precisely correspond to the desired measurement [42][43][44].
The main results of this work are summarised in the first two rows of Table I, and the rest of the table compares our results with other relevant types of protocols.The table focuses on the well-known isotropic state for the sakes of simplicity and providing a concrete benchmark for the protocols.However, in general our results apply to arbitrary states, as no assumption on the entanglement source is required.Using Table I, we now proceed to discuss our results in this broader context of entanglement certification.
Firstly, Table I shows that our protocol for the EAPM scenario, which is based on measuring products of complete sets of MUBs, has the same certification performance as steering protocols based on complete sets of MUBs [33,39]; at least when one uses the best known closed expression for the performance of the two protocols.However, the exact performance of both protocols is underestimated since a precise analytical solution is not known in both cases.Notably, if one considers general steering protocols, with infinitely many measurements, the critical visibility can be further reduced [35].It is an interesting conceptual question whether there exists product measurement protocols in the EAPM scenario that, in the limit of many measurements, can reach the critical visibility for steering, which is v = (H d − 1)/(d − 1), where H d = d k=1 1/k.However, our protocols in the symmetric EAPM scenario, again using products of complete sets of MUBs, outperform significantly the general steering bound.Due to our use of product measurements, we achieve this while using similar experimental resources as employed in steering experiments.Interestingly, from the point of view of the assumptions made on the system, we require only a dimension bound on the channel, which is often less severe than the assumption that one measurement device is flawlessly characterised, which is employed in steering.Notably, a strict dimension assumption can also be relaxed so that undesired small highdimensional components associated with the implementation can be taken into account [45].
Secondly, it is well-known that d-dimensional dense coding protocols can detect every isotropic entangled state, namely v > 1 d+1 [24].As we proved for qubits and conjectured for higher dimensions, the same holds for our protocol in the symmetric EAPM scenario.In this sense, we preserve the certification power for the isotropic state while greatly reducing experimental requirements; from deterministic and complete Bell state measurements to product measurements of separate systems.It is relevant to note that we are not the first to realise that product measurements can give rise to strong quantum correlations in the [35] Local hidden states Bell state measurement Bell nonlocality v > 0.6961 [40] v > 0.6734 [41] for d → ∞ Local hidden variables EAPM scenario, as this was reported in Ref. [23].However, the protocol proposed there works only for qubits, and while it is optimally implemented with product measurements, it has a very small noise tolerance.Specifically, it achieves v = 1/ 2 ≈ 0.7071, compared to our v = 1/ 3 ≈ 0.5774 in the EAPM scenario and v = 1/3 in the symmetric EAPM scenario.Note that the assumptions in these protocols are always the same, namely a dimension bound on the channel.Thirdly, we can compare our certification results to those obtained in Bell inequality tests.Certification via nonlocality is conceptually the strongest, since it requires no assumptions beyond the validity of quantum theory, but the certification performance is more limited.Little is known about the possibility of violating Bell inequalities with isotropic states beyond dimension two.To our knowledge, the best bound on v is that reported in [41]; it decreases monotonically with d but converges only to v = 0.6734 in the limit of large d.A more certain comparison is possible in the qubit case; here the optimal known visibility is v ≈ 0.6961 and is known that no Bell inequality can reduce it below v ≈ 0.6875 [40].In contrast, our protocols in both the symmetric and standard EAPM scenarios achieve certification at signficantly smaller visibilities.
Moreover, as noted in Table I, it is possible that product measurement protocols in the symmetric EAPM scenario are fundamentally limited by some operational notion of nonclassicality that is weaker than one-way steering but stronger than entanglement.Given our results, one may be inclined to suggest that the relevant concept is usefulness in fidelity-based teleportation.However, this is not accurate because we can numerically find entanglement advantages from states with a trivial maximally entangled fraction.
Furthermore, in our protocols, Charlie always performs product measurements.However, sometimes it can be practically costly to communicate the quantum messages from Alice and Bob to Charlie.We note that this can be circumvented by "splitting" Charlie into two separate parties, one neighbouring Alice and one neighbouring Bob, with independent inputs z and z ′ .By associating these inputs to the respective single-particle measurements entering Charlie's product measurement, we can recover the same statistics as in our protocols by imposing the post-selection condition z = z ′ .
Another relevant discussion is that of closing the detection loophole.Our protocols were not designed with the aim of minimising detection requirements, but they nevertheless perform well in this regard.Deterministic and complete entangled measurements on separate photons are well-known to be complicated and require additional resources such as nonlinear optics or auxiliary qubits, see e.g.[13,14,46].This is particularly well-known for the seminal Bell state measurement [2], and it typically means that it is significantly harder to reach high total detection efficiencies with such measurements.Moreover, even implementing such measurements in dimensions larger than two is a formidable challenge.Using protocols based on product measurements offers a clear advantage.For instance, in the symmetric EAPM scenario implemented with qubits, we require a detection efficiency per photon of roughly 57.7%.Recent Bell inequality experiments have shown single photon detection efficiencies far above this value [47,48].In contrast, we are not aware of any relevant two-photon Bell state measurement implemented with an efficiency close to this value.Notably, the theoretical efficiency per photon needed in entanglement certification via dense coding is the same as in our protocol in the symmetric EAPM scenario.Furthermore, thanks to the dimensional scalability of product measurements, both Proposition 1 and Conjecture 1 suggests that the advantages in detection efficiency are even more significant for larger dimensions, as the efficiency threshold per photon will decrease monoton-ically with d.For instance, recent experiments on entangled four-dimensional photons show detection efficiencies around 71.7% [49], well above the regime needed for protocols of our type.
by Alice (Bob), and the measurement M c|z by Charlie-not necessarily separable, the winning score reads where we also absorbed the local unitaries V γ into the local channels such that Λγ ]-since Alice and Bob have the chance to optimise their local operations anyways.The average score rates can be further simplified to (C4) Next, note that the shared state Φ AB (θ ) can be expressed in terms of Pauli matrices Furthermore, since Λα is a quantum channel, for any arbitrary Bloch vector ⃗ To see this, firstly applying the channel to the maximally mixed state yields Λα [ , and apply the channel to them.The linearity of the channel dictates that Applying the local channels to Φ AB (θ ) gives -note that we have absorbed a minus sign in the vector ⃗ m 4 x .By plugging into Eq.(C4), and noting that f z (x, y) = g z (x) + h z ( y) mod 2, with balanced functions g z (x) and h z ( y), we have where τ = 1 4 (cos 2 (θ ), sin 2 (θ ), sin 2θ , sin 2θ ) ≥ 0 within the domain of θ .We have also made use of the fact that for any balanced function f (x), x (−1) f (x) = 0.
We can bring R 2 to a similar form as Eq.(B5), where  This can be thought of as an unnormalised correlated-coin state in the z'th MUB.Indeed, a direct calculation for the computational basis, namely z = d, analogously leads to R Hence, we have Since we can allow any channels for Alice and Bob, we can choose those that correspond to the maximally entangled fraction of ρ.Hence, we have obtained c is taken modulo d.Notably, in odd prime dimensions, the local bases {|e m,z 〉} are mutually unbiased.These are known to admit the form e m,d = |m〉 and |e m,z 〉 = 1 d d−1 l=0 ω l(m+zl) |l〉 for z ̸ = d, where ω = e 2πi d FIG.2.Symmetric entanglement-assisted prepare-and-measure scenario between the senders (Alice, Bob) and the receiver (Charlie).The information, (x, y), is encoded into the shares of the entangled state.
at least one of the two vectors ⃗ r 0 α ± ⃗ s α has a norm above one, which contradicts the physicality of the channel.Thus we should always have |⃗ s α | ≤ 1.

TABLE I .
[40]ary of results and comparison with other approaches.The Qubit and Qudit columns indicate bounds on the critical visibility for certifying the isotropic state.In the case of the symmetric EAPM scenario (also dense coding and general steering), the results are optimal.In the case of Qubit Bell nonlocality the result is known to be nearly optimal[40]but for Qudit Bell nonlocality the optimal bound is largely an open problem.We have included a comparison with the best known steering bound for protocols based on complete sets of mutually unbiased bases since our construction in the EAPM scenario also is based on these bases.The coloured box assumes Conjecture 1. Proposition 1 shows a fundamental limitation in the EAPM scenario.Whether a corresponding limitation exists in the symmetric EAPM scenario is an open problem but it must be an entanglement concept that is weaker than usefulness in fidelity-based quantum teleportation. )