Experimental Demonstration of Sequential Quantum Random Access Codes

A random access code (RAC) is a strategy to encode a message into a shorter one in a way that any bit of the original can still be recovered with non-trivial probability. Encoding with quantum bits rather than classical ones can improve this probability, but has an important limitation: due to the disturbance caused by standard quantum measurements, qubits cannot be used more than once. However, as recently shown by Mohan, Tavakoli, Brunner (New J. Phys. 21 083034, 2019), weak measurements can alleviate this problem, allowing two sequential decoders to perform better than with the best classical RAC. We use single photons to experimentally show that these weak measurements are feasible and non-classical success probabilities are achievable by two decoders. We prove this for different values of the measurement strength and use our experimental results to put tight bounds on them, certifying the accuracy of our setting. This paves the way for further experimental applications of weak and sequential measurements for quantum information tasks.

Introduction.-Arandom access code (RAC) is a communication protocol that requires a transmitter (Alice) to encode a n−bit long random sequence into a shorter m−bit message, and a receiver (Bob) to be able to decode any of the n bits with non-trivial probability p > 1/2.These parameters are often grouped in expression n p − → m that describes the task.A quantum random access code (QRAC) is the very similar situation in which Alice sends m qubits rather than bits.This concept was introduced by Wiesner [1] but caught the interest of the scientific community only after subsequent research by Ambainis et al. [2] who showed quantum strategies that achieve 2 0.85 −−→ 1 and 3 0.78 −−→ 1, which beat the best classical RACs for these choices of n, m.Further studies found that a 4 → 1 QRAC that reaches p > 1/2 does not exist [3] but a 4 m −1 → m always does [4].Other investigations considered different values of n, m [5], the use of qudits (d−level quantum systems) rather than qubits [6][7][8], or the request of decoding more than 1 bit [9].Applications include communication complexity [10], network coding [11], locally decodable codes [12], dimension witnessing of quantum states [13], self-testing of quantum devices [14,15], semi-device-independent quantum randomness extraction [16][17][18] and key distribution [19,20].
Recently, improvements in the theory and implementation of weak and sequential quantum measurements [21][22][23], prompted the introduction of sequential QRACs by Mohan, Tavakoli, Brunner (MTB in what follows) [24].Their protocol is a variation of the 2 → 1 QRAC: Alice encodes a 2−bit message into 1 qubit and sends it to Bob, who, after measuring it, forwards the resulting quantum state to a third party (Charlie) who shares the same goal of Bob: decoding any of the two bits of Alice with non-trivial probability p > 1/2.The core tenets of quantum physics remind us that Bob's measurement disturbs the initial state, making it more difficult for Charlie to extract information from it.However, if Bob uses weak measurements rather than projective ones, he can tune this disturbance and give back some informa-tion to Charlie at the cost of some of his own.This means that Alice's qubit can be used more than once, overcoming a crucial limit of previously studied QRACs, but there is a trade-off between Bob's and Charlie's attainable information that depends on Bob's measurement strength.The observation of decoding probabilities that saturate this trade-off self-tests the use of an unique set of states and measurements, under the assumption that states are two-dimensional and measurements have binary outcomes.Additionally, even imperfect results can bound Bob's measurement strength.
In this work, we verify MTB's protocol in a quantum optics experiment, for different values of the strength parameter.We show that it is possible to observe nearoptimal decoding probabilities and we put tight bounds on Bob's strength using MTB's expressions.
Model.-We briefly introduce the quantitative relations presented by MTB and add some comments.Let x = (x 0 , x 1 ) ∈ {0, 1} 2 be the two-bit sequence that Alice wants to encode.Let y (z) ∈ {0, 1} label the position of the bit in x that Bob (Charlie) randomly chooses to decode.Finally, let b (c) be the result of Bob's (Charlie's) measurement, associating bit 0 with outcome +1 and bit 1 with −1.We define the two correlation witnesses which quantify the probabilities that Bob and Charlie correctly decode the bit they are interested in, averaged over all possible input sequences and bit choices.
If the parties use classical physics, these probabilities are independent of each other and limited by W AB , W AC ≤ 3  4 .This upper bound is reached, for example, if Alice sends the first of her bits, meaning that when Bob and Charlie want to decode the second, they can only guess.Yet, MTB found that the two decoders can both violate this limit in a quantum scenario.The aforementioned trade-off between the information that each of them can extract translates into an upper bound to W AC that depends on the attained value of W AB .In particular: with W AB itself being limited by previous results at 4 [2].MTB also proposed a strategy to saturate this trade-off and proved that it is unique, up to unitary transformations and under the assumption that Alice's state is two-dimensional and all measurements have binary outcomes.This strategy reads: C1 Alice encodes her two bits x 0 , x 1 into one of four pure states and sends it to Bob.These states form the angles of square in the XZ equatorial line of the Bloch sphere and are equidistant from the eigenstates of σ X and σ Z : where s x = 1 − 2x.
C2 Bob weakly measures σ X if y = 0 or σ Z if y = 1 on the qubit.We label η ∈ [0, 1] the strength parameter.This means that Bob uses Kraus operators He then sends the resulting state to Charlie.

C3 Charlie performs projective measurements of σ
In this situation, the following relations hold: which, when combined, make expr.(3) an equality.Notably, at least one of these witnesses is always above the classical limit of 3 4 and if For η = 4 5 , they take the same value 1 2 + √ 2 5 .However, we add that these results cannot be extended to a third decoder.Even if Charlie also uses weak measurements with strength η and relays the resulting qubit to David, there are no values of (η, η ) that provide correlation witnesses greater than 3  4 for all three decoders.It would be natural to wonder whether MTB's protocol can improve the decoding probability of the entire input sequence.If Bob and Charlie cooperate and agree to always decode different bits, the joint probability of both being correct follows the law: It holds that W ABC ≤ 1 2 , with the bound being reached only for η = 1 √ 2 .This agrees with the limits present in the literature: a m-qubit system cannot make the decoding probability of a n-bit message better than 2 m /2 n [5, Th. 2.4.2].
The uniqueness of strategy C1-C3 allows MTB to conclude that finding W AB and W AC correlated to saturate expr.(3) self-tests that the state preparation was that of C1 and the measurements were those of C2 and C3.Moreover, even if the values of the witnesses are suboptimal, they can still provide some information on the measurements, in particular they give a lower and upper bound on parameter η: These bounds are tight when conditions C1-C3 are fulfilled.
Method.-Our experiment aims at verifying all these relations and showing that it is feasible to meet conditions C1-C3 and find the optimal trade-off.We also use eqs.(7) and (8) to bound the value of η.We choose single photons as our experimental platform and their polarization as the degree of freedom that encodes the information.We exploit a bulk source of polarizationentangled photon pairs at 808 nm based on a PPKTP crystal at the center of a Sagnac interferometer [25] to produce our states: since entanglement is not necessary, we set it to produce a separable bipartite state (that we label |ψ = |HV ) by shining the pump laser only in one direction of the Sagnac loop.One photon of each pair is selected in the |V polarization to filter out imperfections in state preparation and background light and is sent to a SPAD detector.Its presence heralds the other photon of the pair which reaches the core of the setup.This is divided into three stages that play the role of Alice, Bob and Charlie.First, a half-wave plate (HWP, Alice) changes the state from |H = tr herald (|ψ ψ| |V herald V herald |) to one of the four optimal states of condition C1.A Mach-Zehnder interferometer (MZI, Bob) based on polarizing beam displacers (PBDs) performs the weak measurement.To do so, a PBD entangles the polarization degree of freedom with the path qubit, then, a pair of HWPs on the two paths followed by a subsequent one that spans across both set the strength of the measurement.Rotation of the latter HWP allows us to scan the values of parameter η and evaluate its effects on the correlation witnesses.A second PBD finally closes the interferometer: its two exits correspond to the outcomes of the polarization measurement, but one is blocked.Rotation of waveplates before and after the MZI allows us to choose which measurement Bob is performing (σ X or σ Z ) and which outcome is selected by the exit path that carries on to the rest of setup.Charlie must measure projectively polarization in the σ X or σ Z basis, therefore its setup consists of a fixed linear polarizer (LP) preceded by a HWP that selects basis and outcome.Finally, light is coupled into a single mode fiber and sent to a SPAD detector.Its electrical signals are correlated with those of the herald and coincidences (within a ±1 ns window) are counted for a fixed exposure time of 2 seconds.The total number of coincidences in this time and for each measurement choice is usually ∼ 8 • 10 3 .Fig. 1 depicts a scheme of the setup.
We notice that our implementation represents a proof of principle demonstration of a QRAC without active random choice of preparation and measurements.We iterate sequentially over all the possible configurations of preparation (x), measurement choice (y, z) and outcome (b, c) and record all the corresponding coincident counts.These are proportional to the joint probability of the outcomes selected by Bob and Charlie, and we use them to compute the conditional probabilities required by eqs.( 1) and ( 2) to find the correlation witnesses.
Results.-We measure W AB and W AC for 11 different values of the strength parameter, equally spaced in [0, 1].We use the HWP inside Bob's MZI to set its value η set .
Fig. 2 plots W AC as a function of W AB and compares it with the optimal trade-off that saturates expr.(3).The quantum features of the experiment are most evident from the fact that not only all points are outside of the classical region, but they also lie on the boundary of the set of quantum correlations between the witnesses, which certifies that we were able to match the optimal conditions C1-C3.
Fig. 3 compares the individual witnesses with the expected values of eqs.( 4) and (5).We clearly see that we could sample the very interesting region in which both W AB and W AC are non-classical.
Fig. 4 confirms the validity of eq. ( 6) and shows that if Bob and Charlie cooperate to decode the entire input sequence, they cannot succeed with probability better than 1 2 .However, this scheme does allow them to saturate the upper bound for a specific measurement strength.
Finally, we evaluated the self-testing capabilities of the protocol, computing upper and lower bounds on η from the experimental W AB and W AC using eqs.(7) and (8).Fig. 5 plots them as function of η set .The tightness of the bounds is another proof that our setup achieved the optimal conditions C1-C3.

Correlation Witnesses
Eq. ( 4) Eq. ( 5) Classical limit Exper.W AB Exper.W AC FIG. 3. Experimental correlation witnesses (dots) as a function of the strength parameter that we set using Bob's HWP.
We also show the behavior predicted by eqs.( 4) and ( 5) (solid lines).Error bars are 1 standard deviation, obtained by propagation from the poissonian error on the detected counts.We can see that there is region in which both witnesses are above the classical limit ( Discussion.-Our experiment confirms the relations presented by MTB and proves that it is possible for two decoders in a QRAC to share higher success probabilities than admitted by classical physics.The quantum weak measurement is the key to this, as it allows reducing the disturbance on the state observed by the first decoder so that it can be used again by the second.This is a new situation in which weak measurements prove to be useful and to be able to overcome the limitations of axiomatic projective measurements. This protocol offers a new way to self-test quantum devices with limited assumptions.In this sequential configuration, the correlation between Alice and Charlie can  7) η up Eq. ( 8) FIG. 5. Lower and upper bounds on the strength parameter, obtained by applying relations ( 7) and ( 8) to the experimental correlation witnesses.Error bars are 1 standard deviation, obtained by propagation from the poissonian error on the detected counts.bound Bob's measurement strength.This is naturally important to verify the accuracy of the strength setting in experimental setups that implement similar qubit measurements, but is also meaningful in adversarial scenarios, in which Alice and Charlie try to detect a man-inthe-middle (Bob), or infer the properties of his actions.In particular, if 5 , then W AC > W AB (see fig. 3), meaning that Alice and Charlie can extract a cryptographic key secure from Bob's eavesdropping using a semi-device-independent protocol like that of ref. [19] (although here we have the additional assumption that Bob's measurements have binary outcomes).
It would also be interesting to study robust self-testing relations for MTB's scheme that can bound not only η but also other properties of the quantum devices in suboptimal conditions.If needed, other assumptions could be added, e.g.perfect knowledge of Alice's preparations could help characterize Bob and Charlie's operations in a measurement-device-independent scenario.
We conclude with a comment on our setup: Bob's MZI is a practical scheme to perform weak polarization measurements and tune their strength via precise waveplate rotations.This kind of device can be used for many experiments that investigate the features of weak measurements in different strength regimes.
Part of this work was supported by MIUR (Italian Ministry of Education, University and Research) under the initiative "Departments of Excellence" (Law 232/2016).
In addition, the authors would like to thank M. Avesani for the useful conversations about information theory and F. Picciariello for his contribution to setting up the experiment.

FIG. 1 .
FIG.1.Scheme of the experimental setup.The Sagnac loop of the source is omitted for clarity.

FIG. 2 .
FIG. 2. Experimental correlation witnesses (dots) plottedagainst each other and compared with the optimal trade-off of eq.(3) (solid line).Error bars are 1 standard deviation, obtained by propagation from the poissonian error on the detected counts.