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Online evolution reconstruction from a single measurement record with random time intervals for quantum communication


Online reconstruction of a time-variant quantum state from the encoding/decoding results of quantum communication is addressed by developing a method of evolution reconstruction from a single measurement record with random time intervals. A time-variant two-dimensional state is reconstructed on the basis of recovering its expectation value functions of three nonorthogonal projectors from a random single measurement record, which is composed from the discarded qubits of the six-state protocol. The simulated results prove that our method is robust to typical metro quantum channels. Our work extends the Fourier-based method of evolution reconstruction from the version for a regular single measurement record with equal time intervals to a unified one, which can be applied to arbitrary single measurement records. The proposed protocol of evolution reconstruction runs concurrently with the one of quantum communication, which can facilitate the online quantum tomography.


Quantum tomography (QT) is a standard tool in quantum information (QI) research [1] to characterize a quantum system or determine a quantum operation by extracting information from a set of measurement data. There are two common types of QT: quantum-state tomography (QST) and quantum-process tomography (QPT) [2, 3]. QST has been developed to estimate the state of a variety of systems [4,5,6], with particular interest in the quantum states of light and single photons [7,8,9,10,11,12] since they are irreplaceable carriers of quantum information. QPT has been used to verify the performance of a quantum device or to characterize an unknown quantum process when controlling quantum systems and their dynamics [13,14,15,16,17,18,19,20,21]. Early QT aimed at feasible methods onto different kinds of quantum systems. Recent works pay close attention to promoting the application of QT, such as optimizing the reconstruction frames [11], saving physical resources [20], and achieving concurrency with quantum computation and communication [18].

QST is the foundation of QPT, especially for determination of unknown dynamics. It is required by standard QPT and ancilla-assisted process tomography, though direct characterization of quantum dynamics bypasses it for simplicity [15, 20]. Based on investigations on steady states, much progress has been made in the tomography of time-variant quantum states, i.e., the evolution reconstruction [22,23,24,25,26,27,28].

Two types of measurements on an evolution have been developed [22]. The traditional one is a strong process, where projective measurements are performed directly on the quantum system of interest and the objective evolution will be interrupted after each measurement. In a weak measurement process, the projective measurement can be performed continuously on the environment, which is coupled with the quantum system of interest. As a result, the number of preparation and evolution of initial states is saved since the observed dynamic remains after each measurement. And there are two ways of determining evolutions: Hamiltonian independent and dependent. The evolution of a well-defined quantum system can be determined by characterizing its Hamiltonian (usually with only several parameters) [24], while the Hamiltonian independent way determines quantum states from the measurement results at each point in time by operator expectation value estimation [26] or quantum Bayesian inference [27] and thus can handle unknown dynamics. There are two kinds of schemes for processing the measurement records as well. Solving a stochastic master equation [22] or quantum Bayesian inference determines the evolution point by point in time iteratively fed by measurement records. But Fourier-based method can calculate the operator expectation value function linearly by operating the measurement data according to a recovery series [26].

In a standard QST context, it is taken for granted that enough copies of the quantum system of interest can be prepared [2]. This is the same for the evolution reconstruction, where a quantum system are initialized, evolved and measured repeatedly for many times. So most of previous works characterize the dynamics of quantum systems from multiple measurement records [24, 26, 27]. Normally, the prerequisite of enough copies of a quantum system and its evolution is reasonable when a controllable quantum system is handled in laboratories and many measurement records are obtained accordingly. However, sometimes only a single measurement record can be obtained when the evolution is caused by an uncontrollable process and thus unrepeatable. One typical example is quantum communication, which usually suffers from time-variant deviations of quantum states [29, 30].

Quantum communication is one of major domains of QI, which can exchange secret information between two spatially separated parties, namely Alice and Bob. It performs projective measurements and utilizes discrete variables of photons or continuous variables of light, such as polarization and coherent states [31, 32], for encoding/decoding. These states, sent identically by Alice, will deviate from their origins differently when arriving at Bob due to a time-variant channel. For each specific state employed in quantum communication, the deviation can be represented by a time-variant quantum state [30] and called an evolution equivalently. Since the channels are out of the control of Alice and Bob, the time-variant deviation cannot be repeated and each encoding state cannot be evolved again and again.

Most recently a novel version of Fourier-based method is proposed for evolution reconstruction from a single measurement record [33]. It recovers time-variant expectation values of projectors from sequential measurement data, which just fits quantum communication. However, this method requires regular measurement records with equal time intervals between neighboring measurement data, while the decoding results of quantum communication are generated at random intervals.

In this paper, we address the reconstruction of a time-variant state in quantum communication by developing a method for single measurement records with random time intervals. We will show how to compose measurement records from the encoding/decoding results of quantum communication firstly. Then, we introduce the method of recovering the expectation value function of a projector from the acquired single measurement record with random time intervals. Next, we apply our method to reconstruct a time-variant two-dimensional quantum state in quantum communication. The recovery and reconstruction are both evaluated by simulations. The results show that a time-variant polarization state can be completely reconstructed from the encoding/decoding results of the six-state protocol with small errors even over a long channel with large loss.

Our method is extended from and covers the Fourier-based evolution reconstruction from a regular single measurement record that is introduced in Ref. [33], and thus can be applied to arbitrary single measurement records. Our work will help to advance discrete-variable quantum communication by providing information of qubits over a time-variant realistic channel for system stabilization such as polarization control [29] or for security analysis as Ref. [34] doing for the case of B92 protocol. The proposed protocol of evolution reconstruction requires no physical resource and brings no constraint and change to the one of quantum communication. Hence, we also illustrate a novel way of throughout online quantum tomography.

Composing single measurement records in quantum communication

Quantum communication can be described with the language of prepare-and-measure (P&M) schemes [35], which coincides with the quantum tomography naturally. To reconstruct the evolution of a quantum system, one usually repeats the procedure that is made up of initial state preparation, state evolution, and state measurement. A quantum communication process is made up of the three experimental steps of the evolution reconstruction in the view of tomography, where a qubit is coded, transported and decoded in turn, and the deviation of the qubits caused by the time-variant channel could be regarded as the evolution of the initial state.

Although nonorthogonal state bases are randomly chosen to code a random bit string in quantum communication, for a specific basis, same qubits are encoded as the same state. Therefore, the qubit encoding in the quantum communication could be regarded as preparing a set of initial states, each of which can result in a unique evolution reconstruction.

Take the six-state protocol [36] as an example and suppose Alice and Bob have a communication record as depicted in Fig. 1a after communicating for a while. They agree on one of the six states, e.g., \(\left| H \right\rangle \), to perform tomography. They pick up all of the qubits that Alice encoded as \(\left| H \right\rangle \) and Bob detected successfully from this record. Then, they have a measurement record corresponding to an ensemble of identically prepared states, i.e., \(\left| H \right\rangle \), which is shown in Fig. 1b. And this record consists of three sub-records, one for each of the three decoding bases. As the qubits are encoded as one of the six states randomly, there are other five measurement records corresponding to the left five states, i.e., \(\left| V \right\rangle , \left| + \right\rangle , \left| - \right\rangle , \left| R\right\rangle \), and \(\left| L \right\rangle \), separately. And each of these measurement records contains three sub-records as well.

Fig. 1

The mapping from a record of quantum communication (a) to measurement records for quantum tomography (b). The letters “H”, “V”, “\(+\)”, “−”, “R” and “L” denote the states \(\left| H\right\rangle , \left| V \right\rangle , \left| + \right\rangle , \left| - \right\rangle , \left| R \right\rangle \), and \(\left| L \right\rangle \), respectively

As the qubits sent by Alice are affected by a time-variant channel, each of the above measurement records results from a specific time-variant state at the end of the channel to be encoded by Bob. Let the propagation delay from Alice to Bob be \(\tau \), an initial state, e.g., \(\left| H \right\rangle \), sent at \(t-\tau \) will be received as an unknown state \(\left| {X\left( t \right) }\right\rangle \) because of the rotation by the time-variant channel. Then, two identical quantum states sent at \(t_0 -\tau \) and \(t_0+\triangle {t}-\tau \) will be received as \(\left| {X\left( {t_0 }\right) } \right\rangle \) and \(\left| {X\left( {t_0 +\triangle {t}}\right) } \right\rangle \), respectively. It is apparent a time-variant state represented by \(\left| {X\left( t \right) } \right\rangle \) will be sampled and measured many times as shown in Fig. 2 when the communication process goes on. As a result, a single measurement record of \(\left| {X\left( t \right) } \right\rangle \) will be acquired as above and the reconstruction of its evolution can be expected.

Fig. 2

The sampling and measurement of a time-variant state implied in the process of quantum communication. Dashed bold circles indicate many other samples. \(\left| H \right\rangle \left\langle H \right| \), \(\left| + \right\rangle \left\langle + \right| \), and \(\left| R\right\rangle \left\langle R \right| \) are the projectors corresponding to the three nonorthogonal bases of the six-state protocol separately. Each green arrow denotes a measurement on the sample (Color figure online)

Recovering the expectation value function of a projector from a random single measurement record

As shown in Figs. 1b and 2, the data of the acquired single measurement record in quantum communication are generated at random time intervals. The random intervals are contributed to partly by the loss of photons in the channel. But it is mainly caused by the random choice of encoding bases and the random bit string, which leads to identical qubits not being sent to sample the evolution at equal intervals any more, even if there is no attenuation in the channel. Additionally, the qubit decoding aggravates the random intervals of each sub-record of these measurement records, since the decoding basis is also randomly chosen. One cannot select a subset of data from the encoding/decoding results to compose a regular measurement record with equal intervals because the loss of photons in the channel and the choice of encoding/decoding bases are random. To recover the expectation value function of a projector from a single measurement record with random intervals, we extend the Fourier-based method of evolution reconstruction from the version for the case of a regular single measurement record as follows.

Assume that the projection of \(\left| {X\left( t \right) } \right\rangle \) on \(\left| H \right\rangle \left\langle H \right| \), denoted by \(\rho _H(t)\), is band-limited with a Nyquist interval of \(T_S\) and the encoding/decoding period is \(T_S/W\), where W is a natural number. When there are Q measurement results in the time interval \((n-1/2)T_S \sim (n+1/2)T_S\), let the \(i\hbox {th}\) result be denoted by \(R\left( {\left( {n+{k_i }/W-1/2} \right) T_S} \right) \), where Q and \(k_{i}\) are natural numbers, 1 \(\le { k}_{1}\cdots<{ k}_{i}<{\cdots } <{ k}_{Q}\le \hbox {W}\), and n is an integer. To find a clue from the measurement results to the expectation value function, we define a random variable as

$$\begin{aligned} c\left( n \right)= & {} \frac{k_{1} }{W}R\left( {\left( {n+{k_{1} }/W-1/2} \right) T_S } \right) +\,\sum _{i=\hbox {2}}^Q {\frac{k_i -k_{i-1} }{W}R} \left( {\left( {n+{k_i }/W-1/2} \right) T_S } \right) \nonumber \\&\quad +\,\frac{W-k_Q }{W}R\left( {\left( {n+{k_Q }/W-1/2} \right) T_S } \right) . \end{aligned}$$

The expected value, i.e., the first moment, of c(n) is

$$\begin{aligned} E\left( {c\left( n \right) } \right)= & {} \frac{1}{T_S }\rho _H \left( {\left( {n+{k_{1} }/W-1/2} \right) T_S } \right) \frac{k_{1} T_S }{W} \nonumber \\&\quad +\,\frac{1}{T_S }\sum _{i={2}}^Q {\rho _H } \left( {\left( {n+{k_i }/W-1/2} \right) T_S } \right) \frac{\left( {k_i -k_{i-1} } \right) T_S }{W}, \nonumber \\&\quad +\,\frac{1}{T_S }\rho _H \left( {\left( {n+{k_Q }/W-1/2} \right) T_S } \right) \frac{\left( {W-k_Q } \right) T_S }{W} \end{aligned}$$

which is an approximation of the definite integral of \(\rho _H(t)\) in the interval \((n-1/2)T_S \sim (n+1/2)T_S\) divided by \(T_S\) as shown in Fig. 3.

Fig. 3

Approximation of the definite integral of \(\rho _H(t)\) by summing rectangles with random widths. Dashed bold circles and rectangles indicate many other samples and the corresponding items of approximation, respectively

As the average width of the rectangles in the approximation, i.e., \(T_S/Q\), becomes zero in the limit Q \(\rightarrow \) \(\infty \), one can find that c(n) converges to the mean of the expectation value function \(\rho _H(t)\) in the interval of \(((n-1/2)T_S \sim (n+1/2)T_S)\) in probability when Q increases as

$$\begin{aligned} \mathop {\lim }\limits _{Q\rightarrow \infty } c\left( n\right) \buildrel \textstyle .\over = \frac{1}{T_{S} }\int _{(n-1/2)T_{S}} ^{(n+1/2)T_{S} } {\rho _H \left( {t{\prime }} \right) dt{\prime }}, \end{aligned}$$

where \(\buildrel \textstyle .\over = \) means being equal in probability. Thus, according to the Fourier-based method introduced in Ref. [33], the expectation value function \(\rho _H(t)\) can be recovered as

$$\begin{aligned} \rho _H(t)\buildrel \textstyle .\over = T_{S} \sum _{n=-\infty }^\infty {\mathop {\lim }\limits _{Q\rightarrow \infty } c\left( n\right) F^{-1}\left[ {\frac{G\left( \omega \right) e^{-jnT_{S} \omega }}{Sa\left( {{\omega T_{S} }/2} \right) }} \right] }, \end{aligned}$$

where \(F^{-1}\) stands for the inverse Fourier transform operator, \((T_S={\pi \omega }_{m})\) and

$$\begin{aligned} G(\omega )=\left\{ \begin{array}{ll} 1&{}\quad \left| \omega \right| \le \omega _m \\ 0&{}\quad \left| \omega \right| >\omega _m \\ \end{array}\right. \end{aligned}$$

Now we have a series for linear recovery of the expectation value function of a projector from a single measurement record with random time intervals by estimating the mean of the expectation value in each Nyquist interval. It can be seen that the random single measurement record will change to a regular one when \(Q=W\) and \( k_{i}=i\) accordingly, and the estimation variable c(n) defined as Eq. (1) can accommodate this change. Therefore, the Fourier-based method for evolution reconstruction from single measurement records can be unified by the series shown as Eq. (4).

To examine the recovery method for a random single measurement record, we conduct a simulation with \(Q =\) 80 and \(W=72 \times 10^{6}\). The parameters are set to fit the situation of realistic quantum communication, which will be detailed in the next section. The original expectation value function is set the same as the one in Ref. [33] for comparison, which is

$$\begin{aligned} \rho _H \left( t \right) =e^{-0.1\pi t}\sin ^{2}\left( {\pi t} \right) \hbox { }\left( {t\ge 0} \right) . \end{aligned}$$

In the simulation, the Nyquist interval \(T_{S}\) is 0.2, and the reconstruction time span is \(\hbox {0 s}{\le t\le }\hbox {20 s}\). The recovered expectation values are sketched in Fig. 4. It is apparent that the recovery from a random single measurement record is almost as accurate as its counterpart from a regular record, in spite of stochastic errors caused by the projection noise, approximation of limit, and realistic truncation in frequency and time domains.

Fig. 4

Original and recovered time functions of expectation values. The recovered expectation values less than zero, which are improper obviously, have been modified to zero

Reconstruction of a time-variant state from the discarded qubits of quantum communication

The six-state protocol provides three sub-records for three nonorthogonal projectors, denoted by \(\left| H \right\rangle \left\langle H \right| , \left| + \right\rangle \left\langle + \right| \), and \(\left| R \right\rangle \left\langle R \right| \), which can be used to recover three expectation value functions and reconstruct a time-variant two-dimensional state [37]. To perform tomography, it is indispensable that Alice tells Bob which qubits are encoded as a specific state or Bob reports to Alice what a reception record he gets. Since these messages are transmitted through ordinary insecure classical channels [31], the qubits revealed in the tomography should not generate quantum keys any longer. Thus, evolution reconstruction tends to decrease the performance of quantum communication.

For the sake of mediating between quantum tomography and communication, we recycle the qubits discarded by quantum communication to compose a measurement record. These discarded qubits come from two steps of the communication process. One is that Alice and Bob determine which photons are successfully received with the correct basis, which supplies with the qubits decoded with wrong bases and skipped afterward. The other is that Alice and Bob test for eavesdropping, which provides the qubits compared with their origins and thereupon sacrificed. The two parts of the whole measurement record are completely independent of the final quantum keys and compose three sub-records for the three nonorthogonal bases separately. Hence, we can use them to recover the expectation value functions of three nonorthogonal projectors and reconstruct a two-dimensional state without any damage to the security of quantum communication.

It is worthwhile to note that the first part of discarded qubits constitutes two sub-records, both of which have as many measurement results as the raw keys of a specific state, while the second part comprises another sub-record, which declines to a proportion of that resting with the eavesdropping test. For a specific quantum state of the six-state protocol, let the encoding/decoding rate, the total equivalent channel attenuation (including the parts caused by a variety of defective efficiencies) and the proportion of the raw keys used for eavesdropping test be denoted by f, \(\xi \), and \(\varepsilon \), respectively. Then, the mean generation rate of measurement results of its minimal sub-record will be

$$\begin{aligned} \left\langle {r_Q } \right\rangle ={\varepsilon f10^{-\xi /{10}}}/{18}. \end{aligned}$$

As the more data in the record the more accuracy of quantum tomography, the performance of evolution reconstruction is positive correlated with the generation rate of raw keys. On the other hand, the generation rate of raw keys depends not only on the repetition rate of the quantum communication system, but also on the channel attenuation. Thus, the higher the repetition rate or the less the channel attenuation is, the better the evolution reconstruction will be.

For a specific state of encoding, e.g., \(\left| H \right\rangle \), without loss of generality, let it be rotated before decoding by the channel to a time-variant state same as the one in Ref. [33], which is

$$\begin{aligned} \left| {X(t)} \right\rangle =e^{-0.05\pi t+j\pi t}\sin \left( {\pi t} \right) \left| H \right\rangle +\sqrt{1-e^{-0.1\pi t}\sin ^{2}\left( {\pi t} \right) }\left| V \right\rangle . \end{aligned}$$

We simulated the reconstruction with the parameters set as \(\varepsilon =0.1\), and \(f=60, 120, 180, 240, 300, 360, 420\) MHz, \(\xi =20, 30, 40\) dB in turn separately in order to fit the conditions of realistic quantum communication. The two key parameters Q and W of last section will be 40 and \(72\times 10^{6}\), respectively, when \(f=360\) MHz, \(\xi =40\hbox { dB}\), and \(T_{S}=\hbox {0.2 s}\).

The encoding, transmission and decoding of qubits over a lossy channel are simulated according to the six-state protocol. The photons prepared as \(\left| H \right\rangle \) by Alice are set to be \(\left| {X(t)} \right\rangle \) as Eq. (8) before measured by Bob, as the qubits are rotated by a time-variant channel. After opening the encoding/decoding bases and eavesdropping test, the discarded decoding results of \(\left| H \right\rangle \) are selected to compose three single sub-records as depicted in Fig. 1. The expectation value functions of \(\left| H \right\rangle \left\langle H \right| , \left| + \right\rangle \left\langle + \right| \), and \(\left| R \right\rangle \left\langle R \right| \) are recovered according to Eq. (4), and thereupon the time-variant state \(\left| {X(t)} \right\rangle \) is reconstructed.

The average reconstruction errors [33] are shown in Fig. 5. It is shown that the reconstruction error deceases when the encoding/decoding rate increases or the channel attenuation reduces. The results demonstrate that the projection noise will be suppressed better by more measurement results in a Nyquist interval as the mean generation rate increases according to Eq. (7). It is also suggested that larger channel attenuation can be balanced by a higher encoding/decoding rate in order to achieve a good reconstruction.

Fig. 5

The average reconstruction errors. The state is reconstructed by recovering its expectation values of three projectors. The recovered expectation values less than zero or more than one have been modified to zero or one, respectively

For the cases considered here, the errors will be less than 2% when the encoding/decoding rate is more than 100 MHz even if the channel attenuation is as high as 40 dB, e.g., at the point A in Fig. 5 where the reconstruction error is 1.87%. It implies that our approach is applicable to typical fiber or free-space channels of metro quantum communication, where the largest total attenuation reported is 37 dB [38,39,40].

In order to provide an applicable tomography protocol, the process of reconstructing a time-variant state from the discarded qubits of quantum communication is outlined as follows. Firstly, Alice and Bob wait a period of time to perform quantum communication, which is the span where the evolution will be reconstructed. Afterward, they share the discarded qubits through a classical channel. Then, the single measurement records are composed. The last step is to reconstruct the evolution by recovering the expectation value functions of projectors. It is apparent that this protocol requires no custom measurements or extra physical resources and all of these steps are consistent with those of the six-state protocol, including composing measurement records and recycling discarded qubits.


We have reconstructed a time-variant quantum state from the encoding/decoding results of quantum communication by extending the Fourier-based method of evolution reconstruction from the version for regular single measurement records to a unified one for arbitrary single measurement records. Random single measurement records can be composed from the record of quantum communication for reconstruction of specific encoding/decoding states. The band-limited expectation value function of a projector can be recovered from a random single measurement record by forming a time-interval weighted sum of its measurement results. A time-variant two-dimensional state can be reconstructed from the discarded qubits of the six-state protocol. The simulated results prove that our method is robust to typical metro quantum channels, especially that here the reconstruction error of an evolution with a Nyquist interval of about 0.2 is as low as 1.87% when the channel attenuation is 40 dB and the encoding/decoding rate is 120 MHz. A protocol of evolution reconstruction is proposed, which can run concurrent with the one of quantum communication with few requirements. Our work will facilitate the online quantum tomography in discrete-variable quantum communication.


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Our work was supported by the National Natural Science Foundation of China (Nos. 11404407 and 61371121) and the Natural Science Foundation of Jiangsu Province (Nos. BK20140072, BK20161471).

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Correspondence to Zhiyong Xu.

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Zhou, H., Su, Y., Wang, R. et al. Online evolution reconstruction from a single measurement record with random time intervals for quantum communication. Quantum Inf Process 16, 247 (2017).

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  • Online evolution reconstruction
  • Single measurement record
  • Random time interval
  • Quantum communication