Random access code (RAC) is a type of collaborative communication task which is suitable for a wide variety of applications. Usually, RAC is implemented in a prepare-and-measure (PM) scenario^[1], where the sender encodes the long message (string of more than one bit) into a short code (one bit usually) and sends it to the receiver, who tries to decode an arbitrarily chosen bit from the initial message. If the sender is allowed to encode the messages into a qubit state, the receiver can successfully retrieve the random bit with an average probability higher than that of the classical version, via corresponding measurements, which is the so called quantum RAC (QRAC). QRAC was introduced in Refs. [2–4], which is useful to certify quantum systems, e.g.,  dimension witnesses^[5], self-testing^[6–8], and comparison of different quantum resources^[9,10]. It can also be implemented in quantum information processing protocols such as quantum key distribution^[11,12], network coding^[13], and random number generation^[14].

The PM scenario of RAC/QRAC above involves one sender and one receiver. For two receivers, Bob and Charlie, two classical RACs can be implemented parallelly and independently, as the classical code can be copied or broadcast.

However, in a quantum version, as the number of all possible messages is larger than two, there must be at least a pair of non-orthogonal states in the set of encoded states, which cannot be cloned perfectly. The quantum system can be accessible by the receivers sequentially. Assuming that Bob, the first receiver, performs quantum operations on the quantum message from the sender, Alice, he gets a classical result, which reveals some information of the original message, and a quantum output, which will be delivered to Charlie, the second receiver, who also tries to retrieve the original message. There is no doubt that the average successful probabilities for both receivers are affected by Bob’s operation. As the quantum system will collapse in one of the eigenstates of the sharp (projective) measurement operator, a weaker measurement performed by Bob is helpful for Charlie’s further retrieving. As a special kind of positive operator-valued measurement (POVM)^[15–20], unsharp measurement is a weak version of projection measurement, introducing less damage to a system^[21–26], as the trade-off between information gain and disturbance. It plays an important role in certain quantum information processing tasks, such as quantum tomography^[27,28], state discrimination^[29,30], and randomness certification^[31,32]. In the sequential QRACs, the total probability of successful retrieval with optimal trade-off using weak measurement has been characterized in Refs. [33,34] and demonstrated^[35] in a photonic experiment.

All of the scenarios above^[33–35] adopt $2\to 1$ QRAC (encode 2 bit into a qubit), and it is shown that unsharp measurement can be better than the sharp projective one. Here, in this Letter, we extend it to the $3\to 1$ QRAC. The total successful probability of sequential QRAC for two receivers with Bob’s unsharp measurement is derived. For comparison, we also investigate the sequential QRAC, where Bob uses sharp measurement. However, numerical results show that unsharp measurement does not always show merits in the case of $3\to 1$.

In the $n\to 1$ RAC, Alice, the sender, encodes an $n$ bit-string, $\mathit{x}\in {\{0,1\}}^n$, into 1 bit $a$, via a classical function $a=f(\mathit{x})$. Given the random input $y\in \{0,1,\dots ,n-1\}$ corresponding to the bit to be retrieved, the receiver, Bob, gets the estimate $b\in \{0,1\}$ using a decode function, $b=g(a,y)$. As the simplest case, Alice sends one of the origial bits, and Bob guesses all of the others randomly. So, the average success probability is $\frac{1+1/n}{2}$.

In the $n\to 1$ QRAC, as shown in Fig. 1, the sender encodes her classical $n$ bit message $\mathit{x}$ into one qubit ${\rho }_x$, and Bob extracts the required bit as $b\in \{0,1\}$ according to the random variable $y\in \{0,1,\dots ,n-1\}$ he receives by performing some measurement ${\{M_{b|y}\}}_{b\in \{0,1\}}$, where $M_{b|y}>0$ and $\sum _b{M}_{b|y}=\mathbb{I}$. Generally, the statistical results of this PM scheme can be expressed by conditional probability by the Born rule, $P(b|\mathit{x},y)=\mathrm{tr}({\rho }_x{M}_{b|y})$. The average probability of a successful guess $P_{\text{succ}}$ is expressed as $$P_{\text{succ}}=\frac{1}{n{2}^n}\sum _{\mathit{x},y}P(b=x_y|\mathit{x},y).$$(1)

For $n=2$, the optimal probability is $\frac{1}{2}+\frac{1}{2\sqrt{2}}$^[4]. It exceeds the limit of the classic scheme 3/4.

For the sequential QRAC, as depicted in Fig. 2, the second receiver, Charlie, receives a random classical variable $z\in \{0,1,\dots ,n-1\}$ and performs corresponding measurements ${\{N_{c|z}\}}_{c\in \{0,1\}}$ on the post-processing state ${\rho }_{x,b}^y$ delivered by Bob. The classical output for Charlie is $c\in \{0,1\}$. The total probability of a successful guess for the senario is written as $$P_{\text{succ}}=\frac{1}{n{2}^n}\sum _{x,y}P(b=x_y|\mathit{x},y)+\frac{1-\alpha }{n{2}^n}\sum _{\mathit{x},z}P(c=x_z|\mathit{x},z),$$(2)with $\alpha \in [0,1]$ indicating the contribution of Bob to a successful guess.

In general, we can use a quantum instrument^[36] to describe Bob’s two-outcome measurement, which is characterized by Kraus operators ${\{K_{b|y}\}}_{b\in \{0,1\}}$ satisfying $\sum _b{K}_{b|y}^{†}{K}_{b|y}=\mathbb{I}$. It is an ordered set of the completely positive trace non-increasing map ${\mathrm{\Gamma }}_{b|y}({\rho }_x)=K_{b|y}{\rho }_x{K}_{b|y}^{†}$ acting on the input state ${\rho }_x$, with the probability $p(b|\mathit{x},y)=\mathrm{tr}({\rho }_x{K}_{b|y}^{†}{K}_{b|y})$. The normalized post-measurement state will be described by ${\rho }_{x,b}^y=\frac{K_{b|y}{\rho }_x{K}_{b|y}^{†}}{p(b|\mathit{x},y)}$ on the outcome $b$. The sharp measurement, projection-valued measure (PVM), can be considered as a special case, where $K_{b|y}{K}_{b\prime |y}=K_{b|y}{\delta }_{bb\prime }$.

For $n=2$, the difference of strategies for Bob, the unsharp POVM and the PVM, is discussed in detail^[34]. As shown in Fig. 3, the PVM strategy is divided into three schemes: (i) unitary, where Bob does not extract information for any $y$; (ii) measure and prepare, where Bob performs the corresponding PVM according to $y$; and (iii) mixed, which is the synthesis of the first two strategies. It is found that the POVM scheme is always better than PVM schemes for all $\alpha$, In other words, unsharp measurements can provide advantages.

Let us consider the $3\to 1$ QRAC task in this section. For the one receiver case, the successful probability can be written as $$P_{\text{succ}}^B=\frac{1}{24}\sum _{x,y}P(b=x_y|\mathit{x},y),$$(3)where $x_y$ is the $y$th bit of the input string $\mathit{x}$. $P_{\text{succ}}^B\le 2/3$, for a classical verion, while $P_{\text{succ}}^B=\frac{1}{2}+\frac{1}{2\sqrt{3}}\approx 0.7887$ for a standard QRAC.

In the sequential QRAC process, Bob receives the code qubit from Alice, and Charlie receives the post-measurement state from Bob. Here, the state set {${\rho }_x$} sent by Alice corresponds to the eight vertices of the inscribed cube of the Bloch sphere, i.e., $${\rho }_x=\frac{1}{2}\mathbb{I}+\frac{1}{2\sqrt{3}}\sum _i{(-1)}^{x_i}{\sigma }_i,$$(4)where ${\{{\sigma }_i\}}_{i=0,1,2}$ denotes the three Pauli matrices $\{{\sigma }_x,{\sigma }_y,{\sigma }_z\}$, respectively. Bob’s unsharp measurements can be written in Kraus operators, $$K_{b|y}=\frac{\sqrt{\lambda }+\sqrt{1-\lambda }}{2}\mathbb{I}+\frac{\sqrt{\lambda }-\sqrt{1-\lambda }}{2}{(-1)}^b{\sigma }_y,$$(5)where $\lambda \in [0.5,1]$ is the maximal eigenvalue of Bob’s operations, as the sharpness of the corresponding measurement. When Bob performs a projective measurement, where $\lambda =1$, we will get the optimal value $P_{\text{succ}}^B\approx 0.7887$, which is higher than the classical bound 2/3.

As a sequential communication task, Charlie receives a post-measurement state. Without any information of Bob’s measurement choice $y$ or outcome $b$, the post-measurement state is written as $${\rho }_x^{\prime }=\frac{1}{3}\sum _{b,y}{K}_{b|y}{\rho }_x{K}_{b|y}^{†}.$$(6)

Subsequently, he performs the two-outcome measurement. Here, the best strategy of Charlie is the sharp measurement, $\{N_{c|z}=\frac{1}{2}[\mathbb{I}+{(-1)}^c{\sigma }_z]\}$. The average successful probability $P_{\text{succ}}^C$ for Charlie is $$P_{\text{succ}}^C=\frac{1}{24}\sum _{\mathit{x},z}P(c=x_z|\mathit{x},z).$$(7)

Although Charlie’s choice of measurements is independent of Bob’s measurement choice and outcome, $P_{\text{succ}}^C$ will be potentially affected by Bob’s operations. Given the sharpness $\lambda$ of Bob’s measurment, we can obtain $$P_{\text{succ}}^B=\frac{1}{2}+\frac{2\lambda -1}{2\sqrt{3}},$$(8)$$P_{\text{succ}}^C=\frac{1}{2}+\frac{1+4\sqrt{\lambda (1-\lambda )}}{6\sqrt{3}}.$$(9)

A complementary relationship between $P_{\text{succ}}^B$ and $P_{\text{succ}}^C$ is found, as shown in Fig. 4. The superiority of quantum RAC over the classical one is obvious, as either $P_{\text{succ}}^B$ or $P_{\text{succ}}^C$ is greater than the classical bound. Further, by bringing Eqs. (8) and (9) into Eq. (2), the optimal $\lambda$ is found as a function of $\alpha$: $$\lambda =\frac{1}{2}+\frac{3\alpha }{2\sqrt{4-8\alpha +13{\alpha }^2}}.$$(10)

Eventually, we could obtain the optimal average success probability: $$P_{\text{succ}}^{\mathrm{POVM}}=\frac{1}{2}+\frac{1-\alpha }{6\sqrt{3}}+\frac{1}{6\sqrt{3}}\sqrt{4-8\alpha +13{\alpha }^2}.$$(11)

Now let us consider the strategies without weak measurement. For the unitary strategy, where Bob does nothing but guessing, and Charlie receives the identical state that Alice sends, we have $$P_{\text{succ}}=\frac{\alpha }{2}+(1-\alpha )(\frac{1}{2}+\frac{1}{2\sqrt{3}}).$$(12)

For the PVM scheme, we also use the methods proposed in Ref. [32]. The total successful probability is written as $$P_{\text{succ}}^{\mathrm{PVM}}=\frac{1}{72}\sum _{x,y,z,b,c}\mathrm{tr}[{\sigma }_{x,b}^y{N}_c^z][\alpha {\delta }_{b,x_y}+(1-\alpha ){\delta }_{c,x_z}],$$(13)where ${\sigma }_{x,b}^y$ is the unnormalized post-measurement state. For pure state ${\rho }_x=|{\varphi }_x\rangle \langle {\varphi }_x|$ and Bob’s projective measurement on the base $\{|{\psi }_0^y\rangle ,|{\psi }_1^y\rangle \}$, $${\sigma }_{x,b}^y={|\langle {\psi }_b^y|{\varphi }_x\rangle |}^2|{\psi }_b^y\rangle \langle {\psi }_b^y|.$$(14)

To optimize the parameters in the measure-and-prepare strategy, we use the unit vectors on the Bloch sphere to denote an arbitrary pure state of Eq. (4) and get $${|\langle \psi |\varphi \rangle |}^2=\mathrm{tr}({\rho }_1{\rho }_2)=\frac{1}{2}(1+{\overrightarrow{r}}_1·{\overrightarrow{r}}_2).$$(15)

Note that the above formula is a function about angles ($\theta$, $\phi$) while using spherical coordinate vector $\overrightarrow{r}=(\sin \theta \cos \phi ,\sin \theta \sin \phi ,\cos \theta )$. The result of numerical simulation is shown in Fig. 5.

It is found that the optimal strategy for $3\to 1$ QRAC is quite different from that of $2\to 1$ QRAC. In this guessing game, the unsharp scheme cannot always have advantages. The average success probability of the “measure and prepare” strategy is always 0.75 when $\alpha$ is approximately in the range of $[0,0.71)$. It is always higher than that of the POVM strategy for $\alpha \ge 0.15$. The average successful probability 0.75 can be achieved by another kind of classical RAC^[4]. In the classical scheme, the 8 bit-strings $\{\mathit{x}\}$ are divided into two categories according to the number of ones in it, i.e., ${\{\mathit{x}\}}_0=\{000,001,010,100\}$ and ${\{\mathit{x}\}}_1=\{111,110,101,011\}$, which are encoded into classical bits to be sent, 0 and 1, respectively. Therefore, when the receiver gets bit 0, he can guess that the original bit-string is 000 in set ${\{\mathit{x}\}}_0$, and, when it gets bit 1, the guess is 111 in set ${\{\mathit{x}\}}_1$. In both cases, the average success probability is $(1+\frac{2}{3}·3)/4=0.75$.

In this Letter, we have discussed the sequential version of the $3\to 1$ QRAC task with two receivers through unsharp measurements. As a trade-off between information gain and state disturbance, the successful probability for the first receiver and the second one will increase and decrease, respectively, with the sharpness of the first one’s measurement increasing. The optimal average probability of successful retrieval with unsharp measurements is derived. Furthermore, two strategies for the first receiver are discussed, i.e., the unitary one and the measure-and-prepare one with PVM. It is found that for most of the weight for averaging, the measure-and-prepare strategy with PVM achieves a higher total success probability of 0.75 than the POVM one, which is different from the $2\to 1$ case. Moreover, this success probability can be reproduced by a classical scenario. Therefore, our present work can provide useful references for further implementation of QRAC and RAC. We believe that our theory can be expanded to the scheme with more receivers or higher-dimensional QRACs and can be applicable to the implementation of a quantum random number generator (QRNG) based on QRACs^[37].