Three-observer classical dimension witness violation with weak measurement

Dimension is an important resource in quantum information theory. Based on weak measurement technology, we propose the three-observer dimension witness protocol in a prepare-and-measure setup. By applying the dimension witness inequality based on the quantum random access code and the nonlinear determinant value, we demonstrate that double classical dimension witness violation is achievable if we choose appropriate weak measurement parameters. Analysis of the results will shed new light on the interplay between the multi-observer quantum dimension witness and the weak measurement technology, which can also be applied in the generation of semi-device-independent quantum random numbers and quantum key distribution protocols. Dimension is an important resource in quantum information theory. By exploiting the weak measurement technique, the paper presents a three-observer dimension witness protocol which can be applied, among other protocols, in semi device-independent quantum information theory.

D imension is an important resource in quantum information theory, for instance, a high dimensional quantum system can enhance the performance of the quantum computation 1, 2 , quantum entanglement 3 , quantum communication complexity 4 and others, and it can also reduce the security of certain practical quantum key distribution systems 5 . To estimate the lower bound of the dimensions of a physical system, quantum dimension witness has been proposed and experimentally realized [6][7][8][9] , which has important applications in the semi-device-independent quantum key distribution and quantum random number generation [10][11][12][13][14][15] . Until now, it has been demonstrated that two-observer classical dimension witness violation can be achieved with the Bell inequality test, quantum random access code test, and determinant value test respectively [16][17][18] , but whether the multi-observer classical dimension witness violation can be obtained or not is still an open question.
Similar to a quantum dimension witness, quantum nonlocality also plays a fundamental role in quantum information theory, which can be used to guarantee the security of deviceindependent quantum information protocols 19,20 . In a twoobserver system, two observers can perform independent measurements on their subsystem to test the Clauser-Horne-Shimony-Holt (CHSH) inequality 21 , the violation of which certifies quantum nonlocality. Recently, it has been demonstrated that nonlocality sharing among three observers can be established by applying weak measurement technology [22][23][24] , which demonstrates that subsequent measurements can not be described by using classical probability distributions.
Inspired by the work of sharing nonlocality with weak measurement technology, three-observer classical dimension witness violation will be analyzed in this paper, in which we analyze the dimension witness inequality based on the quantum random access code and the nonlinear determinant value. The analysis result indicate that double classical dimension witness violations can be realized. More interestingly, we demonstrate local and global randomness generation in the three-observer protocol, and the analysis method can also be applied to future multi-observer quantum network studies.

Results
Quantum dimension witness based on quantum random access code. The first quantum dimension witness inequality based on the quantum random access code in the two-observer system is given by 10,11,18 Àpð1j100Þ þ pð1j101Þ À pð1j110Þ À pð1j111Þ: In the two-dimensional Hilbert space, it has been proved that the upper bound of the classical dimension witness value W 1 is 2, while the upper bound of the quantum dimension witness value W 1 is 2 ffiffi ffi 2 p . We will apply this dimension witness equation to analyze the dimension witness values between Alice and Bob W 1AB and between Alice and Charlie W 1AC .
Based on the state preparation and measurement model given in the Method section, the dimension witness value between Alice and Bob is given by The dimension witness value between Alice and Charlie is given by Note that ε = 0 indicates W 1AB ¼ 2 ffiffi ffi 2 p and W 1AC = 0, which demonstrates that Charlie's quantum state has no interaction with Bob's system. In this case, only W 1AB violates the classical upper bound, which is reduced to a two-observer system (Alice-Bob). Similarly, ϵ ¼ π 2 indicates W 1AB = 0 and W 1AC ¼ 2 ffiffi ffi 2 p , which is reduced to a two-observer system (Alice-Charlie). Thus, our protocol is more general compared with the previous 2observer protocols, which sheds new light on the dimension witness in the network environment.
The detailed quantum dimension witness values W 1AB and W 1AC with different weak measurement parameter ε values are given in Fig. 1 Quantum dimension witness based on the determinant value.
The second quantum dimension witness inequality based on the nonlinear determinant value test in the two-observer system is given by 13,14 W 2 ¼ pð1j000Þ À pð1j010Þ pð1j100Þ À pð1j110Þ pð1j001Þ À pð1j011Þ pð1j101Þ À pð1j111Þ : Assuming that the state preparation and measurement devices are independent, it has been proved that this nonlinear dimension witness can tolerate an arbitrarily low detection efficiency. In the two-dimensional Hilbert space, the upper bound of the quantum dimension witness value is 1, while the classical dimension witness value is 0. Similar to the previous subsection, the dimension witness values between Alice and Bob W 2AB and between Alice and Charlie W 2AC can be analyzed.
Based on the state preparation and measurement model given in the Method section, the dimension witness value between Alice and Bob is given by The dimension witness value between Alice and Charlie is given by Note that ε = 0 indicates W 2AB = 1 and W 2AC = 0, which demonstrates that Charlie's quantum state has no interaction with Bob's system. In this case, only W 2AB violates the classical upper bound, which is reduced to a two-observer system (Alice-Bob).
The detailed quantum dimension witness values W 2AB and W 2AC with different weak measurement parameters ε are given in Fig. 2.
The analysis result demonstrates that double classical dimension witness violations (W 2AB > 0, W 2AC > 0) can be obtained if 0 < ε < π. However, since Bob's system may be influenced by Alice and Charlie's system, the security of the measurement outcome b should be guaranteed by considering the double classical dimension witness violation W 2AB > 0 and W 2AC > 0.
Semi-device-independent random number generator. The classical dimension witness violation can generate semi-deviceindependent quantum random numbers, for which we can only assume knowledge of the dimension of the underlying physical system, but otherwise nothing about the quantum devices. The generated random numbers in our protocol are the measurement outcomes b and c, and the eavesdropper can not guess the measurement outcomes even if the state preparation and measurement devices are imperfect.
Randomness generation can be divided into global randomness and the local randomness, where global randomness must analyze the global conditional probability distribution p(b,c|x,y,z), while local randomness must analyze the local conditional probability distribution p(b|x,y). With the given conditional probability distributions, the random number generation efficiency can be estimated by following min-entropy functions 25 To analyze the global randomness generation efficiency H min1 , the maximal guessing probability 1 16 P x;y;z max b;c ðpðb; cjx; y; zÞÞ can be estimated by 1 where the second line is based on the condition for which Charlie's measurement outcome c can not be effected by Bob, and the corresponding randomness generation efficiency H min1 is given by where the first part is the randomness generation in Charlie's side, and the second part is the randomness generation in Bob's side.
In the two-observer system, with a given random access code based dimension witness value W 1 , the relationship between W 1 and the randomness generation efficiency H ′ min2 ðW 1 Þ 26 is given by However, the eavesdropper can apply Charlie's input parameter z to guess Bob's measurement outcome b, thus the previous method can not be directly applied in our protocol. To estimate the local randomness generation in Bob's side, we will estimate the dimension witness value between Alice and Bob W 1AB(z) with different input random number z ∈ {0,1} as follows with the detailed calculation is given in the Method section, and the corresponding local randomness generation efficiency in Bob's side is H ′ min2 ðW 1 ¼ ffiffi ffi 2 p cosðϵÞ þ ffiffi ffi 2 p Þ. With the given determinate value based on dimension witness W 2 in the two-observer system, the relationship between the quantum dimension witness value W 2 and the randomness Similar to the previous calculation, the dimension witness value between Alice and Bob W 2AB(z) with different input random number z can be given by with the detailed calculation is given in the Method section, and the corresponding local randomness generation efficiency in Bob's side is H ′′ min2 ðW 2 ¼ cosðϵÞÞ. Based on the previous analysis result, the detailed local randomness generation efficiency with different weak measurement parameter ε values are given in Fig. 3. Note that the local random number generation in Charlie's side can be directly estimated by the two-observer protocol. Thus double local randomness generation can be realized if we choose an appropriate weak measurement parameter.
Comparing with the random access code based protocol, the determinant value based protocol can provide the double classical dimension witness violation except the special case (ε = kπ, k = 0, 1, 2,...), thus it has the advantage in the experimental realization. However, the disadvantage is that this protocol has a strong assumption that the state preparation and measurement devices are independent.

Discussion
We proposed a three-observer dimension witness protocol, where the weak measurement technology was applied to analyze the double classical dimension witness violations. The results of our analysis shed new light on understanding the quantum dimension witness in the network environment. The three-observer dimension witness protocol can be assumed to be a sequential measurement protocol, and it will be interesting to analyze a higher dimensional multi-observer quantum system with sequential measurement 27 technology. We demonstrated the randomness generation in the three-observer protocol, and the analysis results demonstrate that weak measurement may have significant applications in multi-observer semi-device-independent quantum information theory. This study also provides tremendous motivation for further experimental research.

Methods
Weak measurement protocol. Weak measurement is a powerful method to extract less information about a system with smaller disturbance 28 , which has proven to be useful for signal amplification, state tomography, solving quantum paradoxes and others [29][30][31][32][33] . In this work, we use the weak measurement definition given in refs. [22][23][24] , and the corresponding analysis model is given in Fig. 4.
There are three observers Alice, Bob and Charlie in the analysis model, and the purpose of our protocol is to establish double classical dimension witness violations under the two-dimensional Hilbert space restriction. More precisely, Alice prepares the two-dimensional quantum state ρ x ∈ C 2 and sends it through the quantum channel with a different classical input random number x ∈ {00,01,10,11}. Then, Charlie receives the quantum state and applies the following operation with the input random number z ∈ {0,1} where the above operation illustrates that the rotation maps the initial Hilbert space basis of {|0〉,|1〉} to a new basis of fjω z i; jω ? z ig depending on the given z value. Charlie also has an two-dimensional ancillary quantum state jþi ¼ 1 ffiffi 2 p ðj0i þ j1iÞ in the quantum channel, and then, the following control operation is applied when Charlie receives the quantum state |1〉 where ε is the weak measurement parameter. If Charlie receives the quantum state | 0〉, he will apply the identity operation I ¼ 1 0 0 1 , thus the total unitary operation in Charlie's side can be given by where we define W þz B ¼ jω z ihω z j, W Àz B ¼ jω ? z ihω ? z j. Note that Charlie will apply the identity operation I if the weak measurement parameter ε is 0, which can be simply proved since W þz B I þ W Àz B I ¼ I. In this case, Charlie will only obtain the initial ancillary quantum state þ j i after the control operation, which demonstrates that Charlie can not gain any information about Alice's state ρ x , and there is no interaction introduced by Charlie correspondingly.
To prove the dimension witness in the three-observer system, we should analyze the density matrix to illustrate Bob and Charlie's system. The initial quantum state can be given by  Fig. 4 Analysis model of three-observer classical dimension witness violation. Alice prepares the two-dimensional quantum state ρ x , and Bob and Charlie apply the two-dimensional strong measurement and weak measurement respectively By applying the previous unitary transformation U, the quantum state ρ BCx will be transformed into The quantum state ρ ′ Bx ¼ Tr C ρ ′ BCx will be transmitted to Bob, as follows Similarly, the quantum state ρ ′ Cx ¼ Tr B ρ ′ BCx will be transmitted to Charlie, as follows After receiving the quantum state ρ ′ Bx , Bob will apply the two-dimensional projective measurement depending on the input random number y ∈ {0,1}. If Bob's input random number is 0, the corresponding measurement basis in Bob's side is given by If Bob's input is 1, Bob's measurement basis is where the measurement outcomes |ν 0 〉〈ν 0 | and |ν 1 〉〈ν 1 | indicate classical bit 1, and the measurement outcomes jν ? 0 ihν ? 0 j and jν ? 1 ihν ? 1 j indicate classical bit −1. After receiving the quantum state ρ ′ Cx , Charlie will apply the two-dimensional projective measurement fjtihtj; jt ? iht ? jg; ð23Þ where the measurement outcomes |t〉〈t| and |t ⊥ 〉〈t ⊥ | respectively indicate the classical bit 1 and −1. Before the state measurement, Charlie randomly chooses rotation fR z ; R þ z g with respect to the input random number z ∈ {0,1}. In the case where the weak measurement parameter ε is 0, there is no interaction on Charlie's side, and then, Charlie's density matrix will be transformed into ρ ′ Cx ¼ Tr B ðW þz B ρ x þ W Àz B ρ x Þjþihþj ¼ jþihþj, and Bob's density matrix will be transformed into ρ ′ Bx ¼ ρ x . In the following section, we will focus on the situation 0 ≤ ε ≤ π, and analyze the interaction introduced by the weak measurement, which can be used to obtain the information gain in Charlie's side.
The general representation of a qubit can be illustrated by using the density matrix formalism IþrÁσ 2 , whereσ is the Pauli matrix vector three-dimensional vector such that r k k 1, thus the conditional probability distribution to illustrate Alice and Bob's system can be given by where B b y is the measurement operator acting on the two-dimensional Hilbert space with input parameter y and output parameter b by considering the prepared quantum state ρ ′ Bx ðzÞ. The conditional probability distribution to illustrate Alice and Charlie's system is given by where C c is the measurement operator acting on the two-dimensional Hilbert space to obtain the measurement outcome c with the state ρ ′ Cx ðzÞ. Based on the previous observed statistics p(b|xy) and p(c|xz), we can calculate the dimension witness value between Alice and Bob and between Alice and Charlie respectively.
To prove the double classical dimension witness violation under Eq. (1), we propose the following density matrix prepared in Alice's side ρ 00 ¼ 1 ffiffi The measurement operator in Bob's side is given by ν 0 ¼ ð0; 0; 1Þ; The measurement operator in Charlie's side is given by The rotation operator in Charlie's side is given by