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Abstract
Quantum random access code (QRAC) serves the communication task to encode a long message into a quantum system and allow the receiver to decode the initial information with a higher success probability than classical random access code (RAC). Here, we present an experimental demonstration of sequential 3 → 1 QRAC in the prepare-transform-measure scenario with one sender and three independent receivers. The experimental results show that, in the 3 → 1 QRAC scenario, three receivers can independently decode the initial information with an average success probability higher than the classical RAC.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Random access code (RAC) serves the communication task to encode a long bit-string into a shorter one, which possesses the ability to recover the initial information with certain success probability. In the case of $n\overset {p}{\rightarrow } m$ RAC, Alice, the sender, encodes a long $n$-bit message into a shorter $m$-bit string and sends it to Bob, the receiver, who has the probability $p >\frac {1}{2}$ to successfully decode an arbitrary bit of the initial message on average. If the sender encodes the message into quantum system, rather than a classical string, it is called quantum RAC (QRAC). Quantum advantage can be found for $n=2,3$ and $m=1$ as long as qubits are sent and projective measurements are performed. The QRAC was first introduced in [1] and then are developed for qubit systems [2–4], entanglement [5,6] and higher dimensional quantum systems [7]. It can also be implemented in quantum information processing protocols such as quantum cryptography [8,9], network coding [10], random number generation [11,12], and self-testing [13,14].
Measurement plays an important role in both the fundamental quantum mechanics and quantum information processing. Generally, quantum measurements can be described as positive operator-valued measures (POVMs), i.e., a collection of positive operators $\{M_{k}\}$ satisfying the normalization constraint $\sum _{k}M_{k}=I$. The projective measurements in the standard QRAC are also called sharp measurements, belonging to a class of POVMs, where each operator is orthogonal to the others. Sharp measurements get a large amount of information about the measured system and destroy the state of the system, resulting in other observers failing to extract the initial information. Unsharp measurements provide less information about the system and affect it weakly, which allows other observers to decode the initial state. As a special class of positive operator-valued measurements (POVM), the unsharp measurement is a weak version of the projective measurement, which can bring less damage on system. Unsharp measurements have been applied in quantum state tomography [15,16], entanglement amplification [17], sequential quantum correlations [18–22].
Sequential quantum measurements were first introduced into QRAC by Mohan, Tavakoli, and Brunner (MTB) [18]. The MTB’s protocol is a version of $2\to 1$ QRAC scenario with one sender and two independent receivers: the sender, Alice, encodes two-bit messages into a qubit and sends it to the first receiver, Bob. Then Bob performs a measurement based on his random input $y$, gets a classical output $b$ and a post-processing state, which are then sent to the second receiver Charlie, who performs a measurement based on his random input $z$ independent of Bob’s input and output, and gets a classical output $c$. This means that Alice’s input qubit can be used more than once, which is different from the standard QRAC. This method requires Bob to makes an unsharp measurement rather than a sharp measurement, which allows the post-processing state that sent from Bob to Charlie contains enough information. As a result, Charlie can extract enough useful information from it. The MTB protocol has been experimentally demonstrated in [14,23]. In fact, QRACs are key tools for semi-device-independent quantum random number expansion protocols [27–29], therefore, studies on QRACs can promote the development of SDI randomness expansion. Till today, people have generized QRACs from $2\rightarrow 1$ to $n\rightarrow 1$ scenarios [3,4,30], which can be utilized to improve the security and efficiency of randomness expansion. However, most of them considered only two or three parties, and did not involve sequential QRACs.
In this paper, we extend MTB’s protocol to the scenario of four-party cascaded prepare-transform-measure, which contains one sender and three independent receivers by using two unsharp measurements and a sharp measurement. Moreover, by using single photons and linear optics, an experimental demonstration of sequential $3\to 1$ QRAC is present with an average success probability higher than the corresponding classical RAC, verifying the advantages of quantum resources. Our work may promote the development of multi-party randomness expansion or multi-party randomness certification.
2. Quantum random access code
In the $n\to 1$ classical RAC scenario, Alice receives a random input $x=x_{0}x_{1}{\ldots }x_{n-1}$ (where $x_{i}\in \left \{0, 1\right \}$) in a uniform distribution and encodes this string into a bit, then sends it to Bob. Bob is also given an input $y \in \left \{0,1,\ldots,n-1\right \}$ to decode $x_{y}$. The average success probability of decoding $x_{y}$ is $\frac {1}{2}\left (1+\frac {1}{n}\right )$. For $n=3$, the average success probability of decoding $x_{y}$ is $p = \frac {1}{2} \left ( 1+\frac {1}{3} \right ) =\frac {2}{3}$.
In the $n\to 1$ QRAC scenario, as shown in Fig. 1, Alice prepares a quantum state $\rho _{x}=|\psi _{x}\rangle \langle \psi _{x}|$ based on her random input $\boldsymbol {x}$, and sends it to Bob. The quantum state $|\psi _{x}\rangle$ is represented by the Bloch vector $\gamma (x) = (\sin \theta _x \cos \phi _x, \sin \theta _x \cos \phi _x, \cos \theta _x)$ and
Bob performs a measurement $\left \{ {M_{b \mid y}} \right \}$ based on his input $y \in \left \{0,1,\ldots,n-1\right \}$ and gets a classical output $b\in \left \{0,1\right \}$. The input $\boldsymbol {x},y$ satisfy uniform distribution. And the conditional probability distributions $p\left (b\mid \boldsymbol {x},y\right )$ can be calculated by Born’s rule, When $b = x_{y}$, Bob recovers the initial bit correctly and the average success probability $P_{succ}$ can be calculated by
Fig. 1. Scenario of $n\to 1$QRAC with Alice prepares the qubit state $\rho _{x}$ based on her classical input $x$ and sends it to Bob. Bob selects his measurement bases on $\rho _{x}$ based on his input $y\in \left \{0,1,\ldots,n-1\right \}$ and gets a classical output $b \in \left \{0,1\right \}.$
In the $3\to 1$ QRAC scenario, Alice prepares the qubit state $\rho _{x}$ based on her classical input $x=x_0x_1x_2$, the encoding quantum state can be described by Eq. (1) with
3. $3\to 1$ QRAC with unsharp measurements
The scenario of our four-party $3\to 1$ QRAC is illustrated in Fig. 2. Alice receives three bits $x_{0}$, $x_{1}$, $x_{2}$ and prepares a quantum state $\rho _{x}$, with Bloch vectors $\gamma \left (x\right )=\frac {1}{\sqrt {3} }\left [\left ( -1\right )^{x_{1} },\left (-1\right )^{x_{2}},\left (-1\right )^{x_{3}}\right ]$. The state is sent to Bob, who performs a quantum measurement $\left \{ {M_{b \mid y}} \right \}$ on $\rho _{x}$ based on his input $y \in \left \{0,1,2\right \}$ and gets a classical output $b \in \left \{0,1\right \}$ together with a post-processing state $\rho _{x,y}^{b}$. We characterise the quantum measurement instrument by Kraus operators $\left \{ {K_{b \mid y}} \right \}$ satisfying $M_{b \mid y} = K_{b\mid y}K_{b\mid y} ^{\dagger }$ and $\sum _{b} K_{b \mid y}^{\dagger } K_{b \mid y}=\mathbb {I}$. The Kraus operators $K_{b \mid y}$ are defined by
Fig. 2. Scenario of our experiment with four participants. Alice prepares the qubit state $\rho _{x}$ based on her classical input x and sends it to Bob. Bob selects his measurement bases on $\rho _{x}$ based on his input $y \in \left \{0,1,2\right \}$ and gets a classical output $b \in \left \{0,1\right \}$ and a post-processing state $\rho _{x,y}^{b}$. Charlie selects his measurement bases on $\rho _{x,y}^{b}$ based on his input $z \in \left \{0,1,2\right \}$ and gets a classical output $c \in \left \{0,1\right \}$ and a post-processing state $\rho _{x,y,z}^{b,c}$. Dave selects his measurement bases on $\rho _{x,y,z}^{b,c}$ based on his input $w \in \left \{0,1,2\right \}$ and gets a classical output $d \in \left \{0,1\right \}$. Bob and Charlie perform unsharp measurement and Dave performs sharp measurement.
The second receiver, Charlie, carries out the same operation as Bob with $\left \{ K_{c\mid z} \right \} \left ( z \in \left \{0,1,2\right \}\right )$, gets a classical output $c\in \left \{0,1\right \}$ and a post-processing state $\tilde {\rho }_{x}^{C}=\frac {1}{3} \sum _{z,c}{K_{c\mid z} \tilde {\rho }_{x}^{B} K_{c\mid z} ^{\dagger }}$. As the last receiver, Dave performs a sharp measurements $\left \{ D_{d\mid w} \right \}$ on $\tilde {\rho }_{x}^{C}$ according to his input $w \in \left \{0,1,2\right \}$ and gets a classical result $d \in \left \{0,1\right \}$. All the inputs $x,y,z,w$ satisfy a uniform distribution. The conditional probability distributions $p\left (b\mid \boldsymbol {x},y\right )$, $p\left (c\mid \boldsymbol {x},z\right )$, $p\left (d\mid \boldsymbol {x},w\right )$ are used to evaluate the success probability of three independent observers in our QRAC. When $b = x_{y}$, $c = x_{z}$, $d = x_{w}$, the success probability $P_{succ}^{B}$, $P_{succ}^{C}$, $P_{succ}^{D}$ can be calculated by
In the scenario of four-party cascaded measurements, we assume that Bob and Charlie perform unsharp measurements with sharpness parameters $\eta _{1}$ and $\eta _{2}$ respectively, whereas Dave performs a sharp measurement. Though the inputs for receivers are independent to each other, the average successful probabilities are not. In the MTB protocol, it is found that a trade-off between the two receivers for $2\to 1$ QRAC depending on the sharpness of the measurement of the first receiver, and the extremal point for equal probability is $\frac {5+2\sqrt {2}}{10} > \frac {3}{4}$. In order to allow three independent observers to successfully extract the initial information, the relationship between the two unsharp measurements should be found. In other words, the three success probabilities should meet the relationship of $P_{succ}^{B}$, $P_{succ}^{C}$, $P_{succ}^{D}>\frac {2}{3}$ simultaneously. Through carrying out numerical simulation, we have tried several relationships between $\eta _{1}$ and $\eta _{2}$ and found that an approximate extremal point for equal probability when the relationship $\eta _{2}=\eta _{1}^{0.615}$ is satisfied.
First, we assume that Bob and Charlie perform the measurements with the same sharpness $\eta _{1}=\eta _{2}$. The result of numerical simulation is shown in Fig. 3. The values for $P_{succ}^{C} \approx P_{succ}^{D}$ are closed to the classical RAC bound $\frac {2}{3}$. Considering experimental imperfections, it is difficult to verify this case in the experiment. Then we fixed the ratio between the two parameters as $\eta _{2}=0.8\eta _{1}$, see Fig. 4. It is found that Charlie shows no quantum advantage against the classical RAC bound. When we used a nonlinear relationship between the two parameters, $\eta _{2}=\eta _{1}^{0.615}$, an approximate extremal point for equal probability is found, as shown in Fig. 5. The values of $P_{succ}^{B}, P_{succ}^{C}, P_{succ}^{D}$ are closed to 0.686, when $\eta _{1}=0.645$.
Fig. 3. Correlations between the sharpness of measurement and the three success probabilities $P_{succ}^{B}$, $P_{succ}^{C}$, $P_{succ}^{D}$ when $\eta _{1}=\eta _{2}$.
Fig. 4. Correlations between the sharpness of measurement and the three success probabilities $P_{succ}^{B}$, $P_{succ}^{C}$, $P_{succ}^{D}$ when $\eta _{2}=0.8\eta _{1}$.
Fig. 5. Correlations between the sharpness of measurement and the three success probabilities $P_{succ}^{B}$, $P_{succ}^{C}$, $P_{succ}^{D}$ when $\eta _{2}=\eta _{1}^{0.615}$. The red solid line, the blue dotted and dashed line, and the black dotted line each corresponds to the theoretical probability of Bob, Charlie and Dave. The bars represent experimental probability with different measurement sharpness parameter when random errors are taken into account.
4. Experimental details and results
The optical setup is shown in Fig. 6, which includes the single-photon source, the state preparation and the measurements. A heralded single-photon source at 808 nm is generated via spontaneous parametric down-conversion (SPDC) process using a type-II PPKTP crystal pumped by a 404nm laser (MOGLabs LDL/404-nm semiconductor laser). The down-converted photon-pair is divided into two paths by a polarized beam splitter. The photon in path 2 is detected by single-photon detector as the trigger and the one in path 1 is coupled into a single-mode fiber with an aspheric lens as the signal photon, where the coincident count rate is about 12kHz. The signal photon is collimated and aligned in the horizontal polarization state $\left |H\right \rangle$.
Fig. 6. Experimental setup of sequential $3\to 1$ QRACs. Part (a): a heralded single-photon source at 808 nm via spontaneous parametric down-conversion (SPDC) using a PPKTP crystal pumped by a 404-nm laser. Part (b): state preparation and measurement. BPF: Band pass filter; PPKTP: periodically poled KTiOPO4 (PPKTP) crystal; PBS: polarzing beam splitter; HWP: half-wave plate; QWP: quarter-wave plate; BD: calcite beam displacer; CL: collimation lens; APD: single-photon detector.
In the state preparation part, Alice prepares quantum state in Eq. (1), $\rho _{x}$ with $x=x_{0}x_{1}x_{2}$ ($x_{i}\in \left \{0, 1\right \}$) by a combination of polarized beam splitter (PBS), half-wave plate (HWP), and quarter-wave plate (QWP) in sequence, as shown in Fig. 6. The corresponding wave plate settings for preparing all Alice’s states are shown in Table 1.
Table 1. Degrees of the HWP and QWP for the state preparation of Alice.
In the state measurement part, both Bob and Charlie perform their unsharp measurements by using a balanced Mach-Zehnder interferometer (MZI), which includes two calcite beam displacers (BDs) and three HWPs. A combination of HWP and QWP is shown as a box in Fig. 6. The first BD in MZI makes the vertically polarized component $\left | V \right \rangle$ pass into the upper path and the horizontally polarized component $\left | H \right \rangle$ enter the lower path. Two slices of half-wave plates are placed in the upper and lower paths respectively. The upper slice is fixed at an angle of $45^{\circ }$ while the lower one at $0^{\circ }$. The third half-wave plate is used to tune the measurement sharpness parameter by changing its angle $\theta _{1}$. After the second BD, only the photon in the combined path is processed with a post-measurement for the Kraus operator $K_{H,\eta (\theta )} =\sqrt {\frac {1+\eta }{2} } \left | H \right \rangle \left \langle H \right | + \sqrt {\frac {1-\eta }{2} }\left |V\right \rangle \left \langle V \right |$, and then is delivered for further measurement. Thus, the MZI can perform the unsharp measurement in $K_{H,\eta (\theta )}$. Two sets of assembles consisting of a HWP and a QWP can be used to select other Kraus operators in the orthogonal complete bases ($\sigma _{Y}$, $\sigma _{X}$, $\sigma _{Z}$) for Bob’s measurement according to his input $y$ and the output $b$, where one performs an unitary rotation to transform $|\phi _{b|y}\rangle \rightarrow |H\rangle$ (box 2) and the other for the inverse transform (box 3). Bob rotates both HWP and QWP in box 2 and box 3 to select one of the three measurement bases $\left \{ \sigma _{Y} ,\sigma _{X},\sigma _{Z} \right \}$. The outcome of these measurement $b \in \left \{0,1\right \}$ corresponds to a photon which is related to $K_{b \mid y}$ coming from the output path of the MZI and then is delivered for further measurement. And Charlie utilizes the same elements for his unsharp measurement. A combination of QWP, HWP, PBS allows Dave to perform a sharp measurement.
Two silicon avalanche photodiode detectors (SPCM-AQRH-13-FC) are employed to detect the trigger photon and the signal photon after Dave’s measurement. The coincidence counts in 3$s$ are collected for each input $(x,y,z,w)$ and output $(b,c,d)$. According to the coincidence counts, we can get the joint probability $p\left ( b,c,d,x,y,z,w \right )$. Assuming the input $x,y,z,w$ are independently and identically distributed, we can employ $p\left ( b,c,d\mid x,y,z,w \right )$ to calculate the success probabilities. In principle, both the message to be encoded by the sender, and the measurement choices of each receiver should be random, which can be realized by electrical-optical modulators (EOMs) and quantum random number generators (QRNGs), which are unavailable in our laboratory. Here, in this demonstration, we use wave plates in piezo rotation mounts, controlled by a computer.
In our experiment, the measurement sharpness parameter meets the relationship of $\eta _{2}=\eta _{1}^{0.615}$. For each set of measurement sharpness $(\eta _1, \eta _2)$, 1728 coincident counts are collected. We implement Monte Carlo method to calculate the probabilities and errors, where Poissonian statistics are taken into account. When $\eta _{1}=0.645\left ( \theta _{1}=32.54^{\circ } \right )$, $\eta _{2}=0.7636\left ( \theta _{2}=34.94^{\circ }\right )$, the success probabilities of $P_{succ}^{B}$, $P_{succ}^{C}$, $P_{succ}^{D}$ are presented in Table 2 and Fig. 5. The final success probabilities in our experiment are $0.6863(16)$, $0.6794(17)$, $0.6802(13)$, which agrees well with theoretical probabilities $0.6862$, $0.6858$, $0.6858$. And two other sets are presented in Table 3 for $\eta _{1}=0.5\left (\theta _{1}=30^{\circ }\right )$, $\eta _{2}= 0.6516\left ( \theta _{2}=32.69^{\circ } \right )$, and in Table 4 for $\eta _{1}=0.8\left ( \theta _{1}=35.78^{\circ } \right )$, $\eta _{2}= 0.8712\left ( \theta _{2}=37.67^{\circ } \right )$.
Table 2. Success probabilities of Bob, Charlie and Dave with $\eta _{1}=0.645$, $\eta _{2}= 0.7636$. The averaged probabilities of $P_{succ}^{B}$, $P_{succ}^{C}$, $P_{succ}^{D}$ are $0.6863(16)$, $0.6794(17)$, $0.6802(13)$ , and corresponding theoretical probabilities are $0.6862$, $0.6858$, $0.6858$, respectively.
Table 3. Success probabilities of Bob, Charlie and Dave with $\eta _{1}=0.5$, $\eta _{2}= 0.6516$. The averaged probabilities of $P_{succ}^{B}$, $P_{succ}^{C}$, $P_{succ}^{D}$ are $0.6389(12)$, $0.6689(12)$, $0.7254(11)$ , and corresponding theoretical probabilities are $0.6443$, $0.6716$, $0.7204$, respectively.
Table 4. Success probabilities of Bob, Charlie and Dave with $\eta _{1}=0.8$, $\eta _{2}= 0.8712$. The averaged probabilities of $P_{succ}^{B}$, $P_{succ}^{C}$, $P_{succ}^{D}$ are $0.7263(14)$, $0.6830(16)$, $0.6417(13)$ , and corresponding theoretical probabilities are $0.7309$, $0.6845$, $0.6397$, respectively.
5. Conclusion
In this work, we extend MTB’s protocol to a four-party cascaded prepare-transform-measure scenario and experimentally realize a sequential $3\to 1$ QRAC. The experimental results show that three receivers in a QRAC scenario can recover the initial message independently with success probabilities higher than the corresponding classical RAC when the first and second observers perform unsharp measurements and the measurement sharpness satisfies certain relationship. It is worth noting that the piezo rotation mounts of wave plates implemented in our experiment provide the stability and precision much better than manual alignments, avoiding some artificial experimental errors.
In addition, there are some imperfections in our system. For example, the interference visibility of the BD interferometer is $98.70 \pm 0.11 \%$ [24], which might introduce surplus phase difference. Moreover, considering the statistical fluctuation of single-photon sources and the accuracy of the wave plate, might result in slight deviations between experimental data and theoretical predictions.
It would be interesting to extend our protocol to entanglement-assisted random access code (EARAC) scenarios or to higher dimensions [25,26]. Considering the QRAC is a key tool of semi-device-independent quantum random number expansion protocols [27–29], our present work might be employed to improve the security and efficiency of randomness expansion, especially for multi-party scenarios. Predictably, the present scheme can also be used to generate quantum random numbers for multi-user network.
Funding
Nanjing University of Posts and Telecommunications Special Funds (NUPTSF) (NY220122, NY220123); Natural Science Foundation of Jiangsu Province (BK20210582); National Key Research and Development Program of China (2017YFA0304100, 2018YFA0306400); Leading edge technology Program of Jiangsu Natural Science Foundation (BK20192001); National Natural Science Foundation of China (12074194, 12104240, 62101285, U19A2075).
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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