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COMMON PART

Project Number20-71-10072

Project titleThe effect of interaction with the environment on the information properties of quantum channels

Project LeadKronberg Dmitry

AffiliationSteklov Mathematical Institute of Russian Academy of Sciences,

Implementation period | 07.2020 - 06.2023 | extension for 07.2023 - 06.2025 |

Research area 01 - MATHEMATICS, INFORMATICS, AND SYSTEM SCIENCES, 01-212 - Quantum data processing methods

Keywordsquantum information, quantum channels, completely positive map, quantum cryptography, unambiguous state discrimination, open quantum systems

PROJECT CONTENT

Annotation

From the mathematical point of view, the arbitrary evolution of a quantum system, or quantum channel, can be described as a unitary interaction of the system and the environment (the Stinespring representation). This interaction can be both destructive and constructive. The destructive effect of the environment is called decoherence and is expressed in the loss of some quantum properties, which usually also leads to the loss of information that can be extracted from a set of input states. But a constructive effect from interaction with the environment is also possible, which is expressed in the fact that in some cases, with some probability, the input states become more distinguishable, and postselection allows these cases to be selected, which gives a gain in distinguishability of states. It is also possible to obtain information about whether they have become more distinguishable or not, which makes it possible both to select the desired outcomes and to study such transformations in general, taking into account all the outcomes.
The project involves a comprehensive theoretical study in the field of quantum information theory, the result of which will be the development of a physical and informational description of the interaction of the environment and quantum signals propagating through the optical channel, methods for their study and modeling. Despite quantum channels is a more general term compared to opticals channels, authors would like to note that practical implementation of developed models aims at the first place to the field of optical data transmission, however it can be generalized to the other fields of implementation. Various theoretical models representing a synthesis of physical and informational approaches will be presented as the main results of the work.

Expected results

Quantum information technologies is one of the key fields of modern science being at the overlap of quantum physics and classical information technologies. The field of science is dedicated to storing, transmitting and processing of data with the help of various physical systems’ quantum states. Quantum information theory introduces principally new possibilities and their implementations (quantum computing, quantum communication, quantum sensing, etc.) compare to the classical information theory. With the help of quantum theory new fundamental results in the field of estimation of information capacities of a quantum channel. However, a huge variety of new unsolved problems are still there, in particular, problems related to active and passive influence of the environment on information properties of quantum systems. Thus detailed theoretical research is required; it should combine and complement physical and information theoretical approaches.
1. The study of physical (passive) environmental impact on quantum signals that propagate through optical channels. The most common (due to its simplicity) method of coding classical information into quantum optical systems is polarization coding of single photons. Thus study of dissipative dynamics of polarization properties of quantum signals (and their higher moments) allows to develop more accurate models of signal distortions. The latter in turn allows to develop new methods of distortion compensation and data preserving techniques. Also new physical methods that control information properties of quantum channels could be developed.
2. Moreover, information (active) interaction between quantum channels and the environment is of interest. In other words, it is necessary to study how information of a particular environment’s state could be used in order to increase information capacity of the channel or how dynamic adjustment of the local environment could maximize intercepted information without noticing by legitimate users. In particular, it is possible to develop a new method of security estimation based on the proposed paradigm for quantum key distribution systems that utilize a set of linearly independent states.
3.The study of quantum channels with output information about the change in the degree of distinguishability of input states. It is supposed to investigate the possibility of a probabilistic increase in the information content of input states during postselection, that is, when considering only a part of the output states which corresponds to conclusive results. The study of the effect of a drop in the total information content in channels that allow, with some probability, a postselective increase in the distinguishability of input states. It is also planned to investigate the fundamental possibility of implementing this type of channel using elements of linear optics and the construction of an appropriate physical model.
4. Determining the characteristics of an unknown channel from partial or complete information about the input and output states, in particular, obtaining estimates for the information that went into the complementary channel, which is relevant when evaluating eavesdropper information in quantum cryptography. It is planned to investigate a set of sufficient characteristics for an upper bound of a complementary channel capacity without performing a tomographically complete set of measurements, and the possibility of optical realization of such an estimation scheme.
Combination of physical and information models will allow a united approach of describing information quantum systems that completely takes into account both potential (ideal) properties of the system and full variety of (real) features related to the implementation. Therefore, the set of physical-mathematical models will be a result of the project; completely new technologies in the field of quantum information theory could be developed based on the latter results.

REPORTS

Annotation of the results obtained in 2022

An approach for evaluating attacks on quantum key distribution systems with a set of non-orthogonal linearly independent states is presented, which can be implemented using either an active or passive malicious environment. The cornerstone of this approach is the combination of attack strategies involving both active and passive malicious environments. Such a generalized description of the eavesdropper’s actions based on the formalism of quantum control attack (which is described within the framework of the quantum channel model with the assistance of the environment) allows for the maximum amount of information available to the eavesdropper to be identified among all possible strategies within one approach.
The models developed during the project for evaluating the boundaries of permissible deviations in the first moments of statistical quantities of observables to determine the inclusion of a malicious environment in the channel and to estimate the maximal distance of quantum key distribution in the case of various actions of an active malicious environment, depending on the number and configuration of the set of quantum states collectively, provide the necessary calculations to be made in order to evaluate the security of quantum key distribution systems with a set of non-orthogonal linearly independent states that can be implemented in the presence of either an active or passive malicious environment.
In addition, the scalability of approaches for the security estimation of point-to-point systems has been evaluated for multi-user quantum networks with unidirectional key transport. The developed approach presents a method for managing authentication resources and the security parameter of quantum distribution protocols in quantum key distribution networks with arbitrary configurations, in order to maintain the security of unidirectional key transport.
A quantum transformation taking as an input two pure non-orthogonal quantum states and consisting of two symmetric operators is constructed. It is shown that special cases of this transformation are both optimal measurement of these states and probabilistic increase of their distinguishability, including unambiguous state discrimination. Numerically obtained estimates of information extracted from the classical outcomes of the corresponding channel as well as from quantum states at the output, as well as the total information.
An explicit example of a bipartite ensemble is constructed, for which accessible information is additive but becomes superadditive after a simple transformation corresponding to the classical states processing. We explicitly construct an optimal observable providing maximum information for this ensemble, which is a measurement in the Bell basis. The Helstrom measurement is also constructed and it is shown that it is only a subordinate observable for the optimal one.
Methods for estimating the information extracted from the complementary channel were developed in the presence of partial information about the states at the output from the main channel, and explicit physical models were built to implement these methods. In the constructed models, we took into account side channels of information leakage in the form of a tensor product of the operational and non-operational degrees of freedom of a quantum communication signal. As a test protocol for analyzing the influence of side channels of information leakage, the decoy states BB84 protocol was considered. To calculate the secret key generation rate, we developed and applied the effective error method. By calculating the secret key generation rate, we tested the hypothesis that a possible reason for the large discrepancy between the upper and lower secret key estimates is related to the possibility of using quantum memory on the eavesdropper side. The calculation results showed that this hypothesis turned out to be incorrect.

Publications

**1.** *Gaidash A., Miroshnichenko G., Kozubov A.* **Quantum network security dependent on the connection density between trusted nodes** Journal of Optical Communications and Networking, Journal of Optical Communications and Networking Vol. 14, Issue 11, pp. 934-943 (year - 2022) https://doi.org/10.1364/JOCN.457492

**2.** *Pastushenko V.A., Kronberg D.A.* **On classical data processing which affects additivity of quantum accessible information** Lobachevskii Journal of Mathematics, - (year - 2023)

Annotation of the results obtained in 2020

Dynamical equations for mean values of different order correlators of field operators (creation and annihilation operators) were derived based on Liouville equation (special case of Lindblad equation that describes dynamics of arbitrary open quantum system), as well as dynamical equations for various characteristics of single-mode (quadratures, mean photons number and its dispersion) and two-mode (polarization state and dispersion of Stokes parameters) radiation were constructed. Peculiarities of dynamics, various dynamic regimes, and physical effects of light propagation in medium (optical fiber, however proposed method allows to adopt it to other media) were examined. Studied features of radiation dynamics could be exploited in design of various quantum optical systems. For instance, the influence of physical characteristics of light propagation in optical fiber on information properties (quantum bit error rate) of BB84 quantum key distribution protocol was investigated.
The effect of overcoming the Holevo value for the set of two arbitrary noncommuting states is demonstrated. An explicit transformation is constructed after which the Holevo value of the ensemble of two states becomes higher.
An operation is developed which increases the distinguishability of the set of non-orthogonal pure linearly independent states and has an arbitrary probability of success. This operation generalizes the operation of quantum unambiguous state discrimination. A Krauss representation of the proposed operation is constructed.
Transmission of two pure non-orthogonal states is considered, a mathematical model of information leakage is constructed, and information leakage is estimated for the symmetric two-dimensional case of known states on the receiving side. It is shown how the amount of leaked information depends on the visibility of the two-photon interference of the signal states. Explicit models of signal transformation in the form of a projective measurement and cloning operations in the composite signal space are constructed and analysed. It is shown that in the case of encoding information in four polarisation states according to the BB84 protocol, the critical error rate is less than 11%. A physical model of information leakage based on the preparation of optical quantum states using electro-optical components is constructed.

Publications

Annotation of the results obtained in 2021

An approach to describe interaction with an active environment is presented; it can describe the quantum control attack based on a combination of protocol and hardware vulnerabilities in quantum key distribution systems. It combines an intercept-resend attack and control of the detection node (detector blinding attack). In the basic version of protocols with a few number of states used (for example, B92 and BB84), detection control is not so important, but when scaling the number of states, state overlap plays a significant role. Of interest are protocols that operate with an arbitrary even number of symmetric linearly independent non-orthogonal (for example, coherent) states. The cornerstone of the approach under consideration is that it combines both the distinction of quantum states by eavesdropping in the channel and various methods of state imposing. In principle, detection control eliminates any bit correlations between legitimate users that are unknown to the attacker, and can be considered a necessary part of most intercept-resend attacks, including a faked-state attack, which is impossible without a hardware vulnerability. In addition, the problem associated with a unified quantum description of intercept-resend attacks was solved by combining the concepts of the von Neumann measurement scheme and the ambiguity of square root extraction for operators.
The effect of overcoming the Holevo bound for a class of quantum ensembles when using postselective measurements is demonstrated. A criterion for possibility of unambiguous states discrimination for an arbitrary ensemble is obtained with quantum max-relative entropy. Also the maximum value of confidence functional when using maximum confidence quantum measurement is expressed through the mentioned entropy.
Specific examples of system-environment interactions that are used by an eavesdropper to attack symmetric coherent state quantum cryptography protocols are presented. A generalization of the photon number splitting attack is constructed, as well as an attack by probabilistic state amplification, an attack by unambiguous discrimination of key bits, and an attack with the use of heterodyne measurement over a part of the signal. The explicit values of QBER or critical channel length, on which the attacks are possible, are obtained for the constructed attacks.
According to the work plan, we considered the case of information leakage from a quantum channel for mixed quantum states. Mixed states lead to incomplete interference of signals (non-unit interference visibility), so we considered explicit types of attacks on a quantum channel, depending on the visibility of interference of quantum signals. This approach enables, without loss of generality and without specifying the explicit form and mechanism of information leakage, to estimate the reduction in the secrecy of the quantum communication protocol due to information leakage in side channels. As a basic protocol, we considered the single-photon version of the BB84 protocol, but the results are applicable to its decoy state modification. A wide class of attacks both with and without postselection were considered as explicit actions of the eavesdropper: intercept-resend attack, individual attack and collective attack. For these attacks, explicit dependences of the critical level of errors in the channel were obtained, at which the secrecy of information transmission is lost. It is shown which optimal types of attacks correspond to a given value of interference visibility.

Publications

**1.** *Avanesov A.S., Kronberg D.A.* **On eavesdropping strategy for symmetric coherent states quantum cryptography using heterodyne measurement** Lobachevskii Journal of Mathematics, Vol. 42, No. 10, pp. 2285–2294 (year - 2021) https://doi.org/10.1134/S1995080221100048

**2.** *Gaidash A.A., Kozubov A.V., Miroshnichenko G.P., Kiselev A.D.* **Quantum dynamics of mixed polarization states: effects of environment-mediated intermode coupling** Journal of the Optical Society of America B, Vol. 38, Issue 9, pp. 2603-2611 (year - 2021) https://doi.org/10.1364/JOSAB.425226

**3.** *Kenbaev N.R., Kronberg D.A.* **Quantum postselective measurements: Sufficient condition for overcoming the Holevo bound and the role of max-relative entropy** Physical Review A, 105(1), 012609 (year - 2022) https://doi.org/10.1103/PhysRevA.105.012609

**4.** *Kenbaev N.R., Kronberg D.A.* **Quantum measurement with post-selection for two mixed states** AIP Conference Proceedings, 2362, 050001 (year - 2021) https://doi.org/10.1063/5.0054964

**5.** *Kozubov A.V., Gaidash A.A., Kiselev A.D., Miroshnichenko G.P.* **Filtration Mapping as Complete Bell State Analyzer for Bosonic Particles** Scientific Reports, 11, 14236 (year - 2021) https://doi.org/10.1038/s41598-021-93679-7

**6.** *Kozubov A.V., Gaidash A.A., Miroshnichenko G.P.* **Quantum control attack: Towards joint estimation of protocol and hardware loopholes** Physical Review A, 104, 022603 (year - 2021) https://doi.org/10.1103/PhysRevA.104.022603

**7.** *Kronberg D.A.* **Об уязвимостях квантовой криптографии на геометрически однородных когерентных состояниях** Квантовая электроника, том 51, номер 10, страницы 928–937 (year - 2021) https://doi.org/10.1070/QEL17625

**8.** *Kronberg D.A.* **Об увеличении различимости квантовых состояний с произвольной вероятностью успеха** Труды МИАН, том 313, страницы 124–130 (year - 2021) https://doi.org/10.4213/tm4178