Digital Signal Processing Quantum Pdf

[Carri Seargent]

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Continuous-variable quantum key distribution (CVQKD) offers the specific advantage of sharing keys remotely by the use of standard telecom components, thereby promoting cost-effective and high-performance metropolitan applications. Nevertheless, the introduction of high-rate spectrum broadening has pushed CVQKD from a single-mode to a continuous-mode region, resulting in the adoption of modern digital signal processing (DSP) technologies to recover quadrature information from continuous-mode quantum states. However, the security proof of DSP involving multi-point processing is a missing step. Here, we propose a generalized method of analyzing continuous-mode state processing by linear DSP via temporal modes theory. The construction of temporal modes is key in reducing the security proof to single-mode scenarios. The proposed practicality oriented security analysis method paves the way for building classical compatible digital CVQKD.

Two barriers exist in the security analysis of a digital CVQKD system. One is the discrete modulation format, and the other is DSP. The former results from the destruction of estimating the covariance matrix directly from the measurement results and was recently solved with the semidefinite programming30,31,32 or other novel methods33,34. The other barrier is the difficulty of constructing an appropriate measurement operator to describe the output of DSP.

Here, to narrow the gap between practical systems and theoretical models, we develop a generalized security proof framework for continuous-mode systems processed by linear DSP algorithms. The key step is the temporal-mode (TM) construction using DSP results, which is suitable for the analysis of high-speed, multi-point sampling systems. Specifically, in continuous-mode formalism, time-domain field operators can be introduced by Fourier transformation, based on which the generalized receiver can be well modeled. By properly calibrating the shot-noise unit (SNU), we model the linear processing of sampled data by recombination of time-domain field operators, which defines a specific TM field operator35,36,37. Consequently, the security of DSP is reduced to the security of a specific single TM measurement. Then, the rest of the analysis is compatible with traditional methods.

Our work provides a feasible way of analyzing the security and performance of a continuous-mode system processed by linear DSP algorithms, so it could provide important guidance for the DSP design of digital CVQKD. Linear DSP toolboxes are expected to be directly employed in CVQKD, reinforcing the importance of our work.

We start by introducing the basics of continuous-mode formalism of quantum optics and then describing the state preparation phase. Recall that in traditional CVQKD analysis, the coherent state is represented by the creation and annihilation operators in terms of a single-mode field, given by \(\hata_i^\dagger ,\hat a_i\). By contrast, in a practical system, high-speed modulation inevitably introduces a nonuniform temporal waveform, so continuous-mode formalism of field operators35,37,38 should be introduced, which is widely used in studying continuous-mode quantum optics. By transforming the annihilation and creation operators from their discrete-mode counterparts, the continuous-mode field operators are defined as \(\hat a_i\to \left(\Delta \omega \right)^\frac12\hata\left(\omega \right)\) and \(\hat a_i^\dagger \to \left(\Delta \omega \right)^\frac12\hata^\dagger \left(\omega \right)\), where Δω denotes the mode spacing, which satisfies the commutation relation \([\hata(\omega ),\hata^\dagger (\omega ^\prime )]=\delta (\omega -\omega ^\prime )\). In the time domain, it is useful to define the Fourier transforms of \(\hata\left(\omega \right)\), namely \(\hata\left(t\right)\), given by \(\hata\left(t\right)=\frac1\sqrt2\pi \int\,d\omega \hata\left(\omega \right)\exp \left(-i\omega t\right)\). The creation operator \(\hata^\dagger \left(t\right)\) follows a similar definition.

Multiple sampling points may exist within one period Tr. Generally, types of linear data processing exist: (i) directly choose one sampled data of each period as final data of this period, for instance, the sampling point near the peak of the envelope of measured state; (ii) using the sampled data within one period to define the final data of this period, for instance, calculating the weighted averaging of all sampled data within the same period; and (iii) generally, a DSP algorithm may use the sampled data from multiple periods, for instance, the root raised cosine (RRC) filter42 introduces a convolution over multiple periods.

To normalize the output data from DSP, one key step is to define and calibrate the SNU, which is the most distinguishable phase from classical optical communication. Considering \(\hatD_t_j^N\) as the final data for the period in which tj lies, we can easily verify that for the vacuum input, the mean is \(\left\langle 0\right\vert \hatD_t_j^N\left\vert 0\right\rangle =0\), and the variance \(\sigma _\rmSNU^2=\left\langle 0\right\vert \hatD_t_j^N\hatD_t_j^N\left\vert 0\right\rangle\) is

Therefore, the final data (output from DSP and being normalized) can be treated as a quadrature measurement of \(\Xi _\rmDSP^t_j\)-TM. As long as the data represent a quadrature measurement result, they can be used to construct the covariance matrix and are thus compatible with traditional security analysis methods. The abovementioned analysis also highlights one key point of SNU calibration, which is that the sampled data of vacuum input should be processed by the same DSP as the usual signal input case before the data are used to calculate the variance of shot noise.

This completes our security analysis framework for linear DSP, which can be summarized in two key points. One is properly calibrating SNU, which naturally leads the final data to be a quadrature measurement result with respect to the TM defined jointly defined by LO, detector, sampling, and DSP. The second is to avoid the complex measurement model introduced by intersymbol crosstalk, where TMs corresponding to different periods should be orthogonal. If these two conditions apply, the final data can be directly used to construct the covariance matrix, and then we can calculate the secret key rate through the current security analysis method.

In this study, we have developed a generalized practical system model with continuous-mode formalism of quantum optics, based on which the IQ modulation at the transmitter side and band-limited homodyne detection with the sampling process at the receiver side can be well described. Then, with proper calibration of SNU, the output data of a linear DSP can be modeled by the quadrature measurement result with respect to a specific TM, jointly defined by the LO, filter, sampling, and DSP algorithms. This immediately results in good compatibility with traditional security analysis methods, which completes the security proof of linear DSP algorithms. Linear DSP toolboxes are expected to be directly employed in CVQKD, which highlights the importance of our work.

Since we restrict the discussed DSP algorithms to linear algorithms, the processing of the output signal is equivalent to the independent processing of the incident signal and the electronic noise. Then the output of DSP shares the same form as the output of the current trusted model, given by

Therefore, the trusted model of a practical detector with DSP can be simplified as Fig. 4b, as long as we re-calibrate the variance of the equivalent electronic noise \(v_\rmel^\rmDSP\).

From Eq. (26), we can see that the measurement results are equivalent to a mode-matching loss added before the receiver side, which is modeled by a beam splitter (BS). After the first-order and second-order moments of measured data are given, it is more intuitive to see that the transmittance of equivalent BS is η. In the above discussion, we assume that \(\left\vert \gamma \right\rangle\) is a TM coherent state to simplify the calculation of Eqs. (27) and (28). While one can further exam that, for an arbitrary input state, Eqs. (27) and (28) still hold. The above derivations also hold considering heterodyne detection.

All authors contributed to the scientific discussions and the theoretical developments of the study. Z.C. and Z.L. carried out the theoretical calculations, X.W. performed the simulation, Z.C. wrote the manuscript, and X.W., S.Y., Z.L., and H.G. provided revisions.

In recent years, substantial progress has been made both in quantum computing and quantum communications. There are numerous operational quantum communications networks in Canada, China, Europe, and the United States, just to mention a few of these advanced high-security systems. China has successfully completed the Micious experiment and demonstrated secure quantum key distribution (QKD) over a distance of 1200 km using a satellite and there is also a fiber-based China-Austria QKD link. However, the ambitious research objective of the community is to make the vision of unbreakable global quantum communications portrayed at a glance in the following illustration a reality.

The QKD protocol is a secret key negotiation technique, which is potentially compatible with the existing security solutions, once the secret key has been agreed. Hence, it could find its way into existing secure networks. However, as all technical solutions, it has to obey certain tradeoffs and requires both a quantum as well as a classical channel. Furthermore, in classical secure networks the secret key has to be at least as long as the message, which imposes undesired limitations both on the classical and on the family of quantum networks.

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