BMe Research Grant
I developed a channel coding technique to transmit classical and quantum information over extremely noisy, initially completely useless quantum channels. I worked out the mathematical background and proved it's correctness in information theoretic terms. The method can be used to transmit private information over insecure channels and for perfect classical or quantum communication with very noisy quantum channels.
Brief Introduction of Research Place
The Department of Telecommunications of Budapest University of Technology and Economics in Hungary represents a solid ground of teaching, researching and developing of Information and Communication Technologies for more than 60 years. The Department covers primary fields such as Acoustics, Cryptography, Computing (both classical and quantum), Multimedia, Networking and Signal processing.
In the first decade of the 21st century, many revolutionary properties of quantum channels were discovered. At the dawn of this millennium new problems have arisen with many open questions, which have opened the door to many new promising results such as superactivation of quantum channels. The superactivation of zero-capacity quantum channels makes possible to use zero-capacity quantum channels for communication.
Besides the relevance of superactivation, its effectiveness is strongly limited by the preliminary conditions (such as the channel maps of the joint structure, or the initial capacities of the channels). Our goal was to eliminate these drawbacks. We introduced an improved method - called polaractivation.
The difference of superactivation and our new phenomenon – called polaractivation – is summarized in Fig. 1. The set of polaractive quantum channels involves the set of superactive channels. Our polaractivation provides a more general framework without any preliminary conditions on the channels or on the initial capacities of the quantum channels involved in the joint structure.
Fig. 1. The problem of superactivation and polaractivation of zero-error capacity of quantum channels as a sub domain of larger problem sets.
Superactivation (Fig. 2) is limited by several preliminary conditions on the properties of the quantum channels and on the maps of the channels involved in the joint channel structure.
Fig. 2. The quantum channels have zero capacities individually, but jointly they define a structure which can be used to transmit information. [G. Smith, J. Yard, Quantum Communication with Zero-capacity Channels. Science 321, 1812-1815 (2008)]
I introduced a new phenomenon called polaractivation. Polaractivation is based on quantum polar encoding and the results are equal to the superactivation effect – positive capacity can be achieved with zero-capacity quantum channels. However, polaractivation has many advantages over the superactivation: it is limited neither by any preliminary conditions on the quantum channel nor on the maps of other channels involved in the joint channel structure. We also prove that the polaractivation works for arbitrary quantum channels. We also derived a lower bound on the symmetric classical capacity, which is required for the polaractivation of private classical capacity. Furthermore, we demonstrated that it works for the zero-error classical and quantum communication.
Polaractivation is based on quantum polar coding and the result is similar to the superactivation effect – positive capacity can be achieved with zero-capacity quantum channels. It requires only the multiple uses of the same quantum channel and our special channel coding scheme.
Fig. 3. Polaractivation has many advantages over superactivation: it is limited neither by any preliminary conditions on the quantum channel nor on the maps of other channels involved in the joint channel structure. Furthermore, it requires only the multiple uses of the same quantum channel.
Polaractivation of Zero-Capacity Quantum Channels
Polar codes belong to the group of error-correcting codes. They introduce no redundancy, only operate on codewords of n qubit of length. They can be used to achieve the symmetric capacity of classical discrete memoryless channels (DMCs). The basic idea behind the construction of polar codes is channel selection called polarization: assuming n identical DMCs we can create two sets by means of an encoder. “Good” channels are nearly noiseless while “bad” channels have nearly zero capacity (Fig. 4). Furthermore, for large enough n, the fraction of good channels approaches the symmetric capacity of the original DMC.
Fig. 4. (a): The ‘bad’ channel: input is not known by Bob. (b): The ‘good’ channel: input is also known on Bob’s side. (c): The recursive channel construction from two lower-level channels. R is the permutation operator.
The polaractivation of any symmetric channel capacities of arbitrary quantum channels requires only the proposed polar encoding scheme and the multiple uses of the same quantum channel. The positive private capacity will be obtained after the multiple uses of the quantum channel. An important difference between superactivation and our polaractivation scheme: while the superactivation of private classical capacity is not possible, the polaractivation is possible (Fig. 5).
Fig. 5. While the superactivation of private classical capacity is not possible, the polaractivation is possible.
Results on the polaractivation of quantum channels
As summarized in Fig. 6, the polaractivation will result in the non-empty set of input polar codewords, and the channel will be able to transmit classical information privately.
Fig. 6. (a): The brief summarization of the proposed polaractivation scheme. In the initial phase, input channels cannot transmit classical information privately. (b): The polaractivation of symmetric private classical capacity makes it possible to construct codewords capable of transmitting private classical information between Alice and Bob.
The polaractivation of symmetric private classical capacity depends on the amount of available symmetric classical capacity, which can be achieved by the proposed polar coding technique. The Bhattacharya parameters represent the noise of the channel: 0 for an idealistic channel, 1 for a completely noisy channel.
Fig. 7 (a): The ratio of the Bhattacharya parameters of “good” and “bad” channels as the function of the achievable symmetric classical capacity expressed by the codewords of Alice and Bob. As the set converges to the critical lower bound, the symmetric private classical capacity will becomes positive, otherwise positive private capacity is not possible. (b): The ratio of the Bhattacharya parameters of “good” and “bad” channels as the function of the achievable symmetric private classical capacity expressed by the private codewords of Alice and Bob. For high enough number n of channel uses, the critical lower bound on the private classical capacity can be exceeded, as depicted by the black dashed line.
Fig. 8. (a): The Bhattacharya parameters of the individual amplitude and phase-errors. The private classical capacity will be greater than zero only in this domain. (b): The achievable symmetric classical capacity and symmetric private classical capacity as the function of Bhattacharya parameters. Individually, the channels are so noisy that they are in the red area; thus they cannot transmit any private classical information. With the help of quantum polar coding, the initial error probabilities are re-transformed, and the symmetric classical capacity will be above the critical lower bound, which makes possible to transmit private classical information.
Fig. 9. (a): The achievable symmetric classical capacity (blue) and symmetric private classical capacity (yellow) as the functions of the Bhattacharya parameters of the amplitude and phase transmissions of the noisy quantum channel. (b): The private communication is possible if and only if the polar codes can ensure the symmetric classical capacity above the critical lower bound. In the red area, the channels are so noisy that private communication is not possible. Initially, the input channels are in this area, however, with the help of polar coding, error-probabilities are re-transformed, and the critical lower bound on the symmetric classical capacity can be exceeded, which makes private communication possible.
Fig. 10. The achievable private classical communication rate expressed as Bob’s valuable codewords in the function of Eve’s valuable codewords for a non-degraded eavesdropper (a) and for a degraded eavesdropper (b).
Efficiency of polaractivation
Polaractivation is very efficient. Assuming n uses of the noisy quantum channel, the complexity of the proposed scheme is
After the channels are being polarized, the polaractivation of arbitrary quantum channels results in a non-empty set of polar codewords which set achieves the symmetric capacity of the quantum channel (Fig. 11).
Fig. 11. While the set of polar codewords which can transmit information is empty in the initial phase (a), by the proposed polaractivator scheme this set can be transformed into a non-empty set (b).
Fig. 12. The valuable information is being transmitted by a quantum polar codeword over the polarized channel structure.
Quasi-superactivation of classical capacity of zero-capacity quantum channels
One of the most surprising recent results in quantum Shannon theory is the superactivation of the quantum capacity of a quantum channel. This phenomenon has its roots in the extreme violation of additivity of channel capacity and enables to reliably transmit quantum information over zero-capacity quantum channels. In this work, we demonstrate a similar effect for the classical capacity of a quantum channel which previously was thought to be impossible. We show that a nonzero classical capacity can be achieved for all zero-capacity quantum channels and it only requires the assistance of an elementary photon-atom interaction process – the stimulated emission.
Importance of our discovery
Sending classical information over a channel combination in which each channel has zero classical capacity seemed to be impossible. The transmission of classical information over zero-capacity quantum channels seemed to be the biggest problem of all, and it also had its roots in Hastings’ counterexample. As was found in 2009, the Holevo information is non-additive in general; on the other hand, this result did not give an answer to the general case of the classical capacity, and also left open many new questions. It was found that in some very special cases quantum information can be transmitted in a similar scenario, however, the most general issue – the transmission of classical information over such a structure – could not be solved.
The phenomenon we propose in this work is called quasi-superactivation. The result is similar to the superactivation effect – positive capacity can be achieved with noisy quantum channels that were initially completely useless for communication. An important difference is that quasi-superactivation is limited neither by any preliminary conditions on the initial private capacity of the channel nor on the maps of other channels involved in the joint channel structure.
Quasi-superactivation only requires the addition of quantum entanglement and the use of stimulated emission; then arbitrary zero-capacity quantum channels can be used for classical communication (Fig. 13). We show that classical information can also be transmitted over the combination of zero-capacity quantum channels using quasi-superactivation and it only requires the most natural process that occurs during stimulated emission. Another important difference is that contrary to the superactivation of classical capacity of quantum channels – which is theoretically impossible, here we prove that – quasi-superactivation is possible.
Fig. 13. Alice’s classical register is denoted by X, the input system is A while P is the purification state. The environment of the channel is denoted by E, the output of the channel is B. The quantum channel has positive classical capacity if and only if the channel output system B will be correlated with Alice’s classical register X.
In Fig. 14, we show our channel construction. The first channel can be any zero-capacity quantum channel that produces a maximum possible mixed output state (The channel output is completely uncorrelated with the input, i.e., the classical capacity of the channel is zero, since the channel destroys every classical correlation.). The second channel in the channel construction is the so-called cloning quantum channel, which channel model describes a natural process that occurs during stimulated emission.
The cloning channel describes the effect of optical amplification as a result of the fundamental interaction of an atom with an impinging photon. The effect is known as stimulated emission and occurs, for instance, in erbium-doped optical fibers. Furthermore, it was also found that the qudit Unruh channel has deep connection with the cloning channels.
As follows, the stimulated emission has deep relevance in the quasi-superactivation of the classical capacity of quantum channels. The process that occurs during the stimulated emission is described and modeled by the cloning quantum channel.
Fig. 14. (a): The first quantum channel can be any quantum channel that produces a maximally mixed output state. The second channel is the cloning channel. Alice’s classical register is denoted by X, the channel input is A, while P is the purification state (it describes the connection with the environment). In the sending process Alice correlates her quantum system A with her classical register X (orange-shaded rectangle). The first channel destroys every classical correlation between register X and channel output B. The input of the second channel is the output B of the first channel. The purification of system B is denoted by the blue-shaded rectangle. The environment of the first channel is depicted by E. The output of the channel is O, while D is the cloned output and F is the environment. (The environments of the channels are initialized in the pure input system.)(b): The quantum channel M cannot transmit any classical information. It consists of two channels where the first channel can be any quantum channel that generates a maximally mixed output state and the second is the cloning quantum channel, the classical capacity is zero.
As we have found, while individually quantum channel M cannot used to transmit any classical information (Fig. 15(a)), but something strange occurs if we use two of these zero-capacity channels jointly where each channel is constructed from a zero-capacity quantum channel and a cloning quantum channel (Fig. 15(b)).
As we have found, two zero-capacity quantum channels in a joint structure can activate each other, and the joint classical capacity will be positive, while the individual classical capacities are zero.
Fig. 15. (a): Individually quantum channel M cannot transmit any classical information. The channel destroys every classical correlation between Alice’s classical register X and channel output O.
(b): For the combination of the two zero-capacity quantum channels with zero classical capacities, the joint classical capacity will be positive. Any correlation between classical register X and output systems will occur that result in positive classical capacity.
Fig. 16 helps explain what is happening in the background and how the quasi-superactivation of classical capacity works.
Fig. 16. The detailed view of joint channel construction helps reveal the effect. Individually, neither M1, nor M2 can transmit any classical information. On the other hand, if we use entangled auxiliary input and the amount of entanglement in the input qubits is chosen from a very limited domain, then the two channels can activate each other and classical information can be transmitted. Using input states with this special amount of entanglement, the outputs of the joint channel construction will be correlated with each other and some correlation will also occur with the classical register X. However, individually every classical correlation will vanish, jointly some correlation can be produced at the channel output which leads to positive classical capacity.
The requirements for the amount of entanglement in the input system for the quasi-superactivation are also summarized in Fig. 17.
Fig. 17. (a): If the amount of entanglement in the auxiliary input system of the joint channel is chosen inappropriately, then the channel construction cannot be used to transmit any classical information, and the quasi-superactivation effect will not occur. In this case, any classical correlation between register X and channel outputs will completely vanish.
(b): The classical capacity of the joint channel structure will be positive if and only if the amount of entanglement in the auxiliary input system is chosen from a very tight domain by Alice. In this case, classical correlation between register X and outputs will occur on the channel output.
As we have found, the level of quasi-superactivation depends on the amount of entanglement in the inputs of the joint structure and the classical capacity can be greater than zero only in a well-specified strict domain (Fig. 18).
Fig. 18. The level of quasi-superactivated classical capacity of the joint structure depends on the amount of entanglement in the EPR input that was fed by Alice to the inputs of the joint channel structure.
In space-earth quantum communications, the relative motion of the ground station and the satellite causes a rotation in the polarization of the quantum states. Current approaches to the compensation for these types of polarization errors require high computational costs and extra physical apparatuses. Our newly developed lightweight quantum error-correction scheme fixes the polarization errors without redundant encoding, which is critical in space-earth quantum communication systems. The proposed solution can be implemented in practice without any extra hardware or software costs, providing an easily implemented on-the-fly polarization compensation scheme for future space-earth quantum communications.
The proposed error correction scheme is based on the usage of pilot states. The pilot states are ordinary known quantum states, sent by Alice to the quantum channel. However, Alice does not place any data in these states - these states will merely be used for error correction. The pilot states will store the unknown map of the quantum channel. These pilot states – using a simple Hadamard and Controlled-NOT (CNOT) gates – can be used to correct an arbitrarily high number of data quantum states sent through the quantum channel. However, all of these states are unknown, Bob is able to construct these states using our simple quantum circuit. The angle of the polarization rotation is stored in the pilot quantum state without making process tomography on the channel. The simplicity of the proposed error-correction quantum circuits allows easy implementation in practice.
Fig. 19. (a): The relative motion of the ground and satellite stations causes a rotation in the polarization of the qubits. (b): The correction of an arbitrary unknown state (d) can be achieved by the unknown pilot states (theta) and elementary quantum gates (CNOT, Hadamard).
My proposed pilot channel coding scheme was presented at the First NASA Quantum Future Technologies Conference (NASA Ames Research Center, Moffett Field, California, USA.), and at the Second International Conference on Quantum Error Correction (QEC11), Dec. 2011, University of Southern California, Los Angeles, USA).
In this work, I introduced the term polaractivation. The result of polaractivation is similar to the superactivation effect, but without the necessary preconditions on the quantum channels or on the joint structure. The polar coding is a revolutionary channel coding technique, which makes it possible to achieve the symmetric capacity of a noisy communication channel by the restructuring of the initial error probabilities. In the case of a quantum system, the problem is more complicated, since the error characteristic of a quantum communication channel significantly differs from the characteristic of quantum communication channels.
I introduced the term quasi-superactivation. We proved that by adding quantum entanglement to zero-capacity quantum channels, classical information transmission is possible. The quasi-superactivation is similar to the superactivation effect, thus positive capacity can be achieved with noisy quantum channels that were initially completely useless for classical communication. However, an important difference is that quasi-superactivation is limited neither by any preliminary conditions of the originally introduced superactivation effect nor on the maps of other channels involved in the joint channel structure. As we have proved, additionally to the existence of zero-capacity quantum channels with positive quantum capacity, finding zero-capacity quantum channels with individually zero classical capacities, which if employed in a joint channel construction can transmit classical information is also possible.
We hope that our results help reveal the strange and mysterious world of quantum information, and also characterize and exploit the hidden possibilities in information transmission over quantum channels in future communication systems and networks. The proposed scheme uses only very simple elements, which allows for a very effective implementation and verification in practice.
The results discussed above are supported by the grant TAMOP-4.2.1/B-09/1/KMR-2010-0002, 4.2.2.B-10/1--2010-0009 and COST Action MP1006.
E. Arikan. Channel polarization: A method for constructing capacity achieving codes for symmetric binary-input memoryless channels. IEEE Transactions on Information Theory, 55(7):3051–3073, July 2009. arXiv:0807.3917.
H. Mahdavifar and A. Vardy. Achieving the Secrecy Capacity of Wiretap Channels Using Polar Codes. arXiv:1001.0210v2 [cs.IT], April 2010.
M. Wilde and S. Guha. Polar codes for classical-quantum channels.
arXiv:1109.2591v1 [quant-ph], September 2011.
K. Bradler, An infinite sequence of additive channels: the classical capacity of cloning channels. IEEE Trans. Info. Theory, vol. 57, no. 8, arXiv:0903.1638, (2011)
K. Bradler, P. Hayden, D. Touchette, and M. M. Wilde, Trade-off capacities of the quantum Hadamard channels, Journal of Mathematical Physics 51, 072201, arXiv:1001.1732v2, (2010).
A. Holevo, “The capacity of the quantum channel with general signal states”, IEEE Trans. Info. Theory 44, 269 - 273 (1998).
B. Schumacher and M. Westmoreland, “Sending classical information via noisy quantum channels,” Phys. Rev. A, vol. 56, no. 1, pp. 131–138, (1997).
S. Lloyd, “Capacity of the noisy quantum channel,” Phys. Rev. A, vol. 55, pp. 1613–1622, (1997)
P. Shor, “The quantum channel capacity and coherent information.” lecture notes, MSRI Workshop on Quantum Computation, Available online at http://www.msri.org/publications/ln/msri/2002/quantumcrypto/shor/1/. (2002).
I. Devetak, “The private classical capacity and quantum capacity of a quantum channel,” IEEE Trans. Inf. Theory, vol. 51, pp. 44–55, quant-ph/0304127, (2005).
G. Smith, J. Yard, Quantum Communication with Zero-capacity Channels. Science 321, 1812-1815 (2008)
Smith, J. A. Smolin and J. Yard, Gaussian bosonic synergy: quantum communication
via realistic channels of zero quantum capacity, arXiv:1102.4580v1, (2011).
R. Duan, Superactivation of zero-error capacity of noisy quantum channels.arXiv:0906.2527, (2009)
T. S. Cubitt, G. Smith, Super-Duper-Activation of Quantum Zero-Error Capacities, arXiv:0912.2737v1. (2010).
Cubitt, D. Leung, W. Matthews and A. Winter, Improving
Zero-Error Classical Communication with Entanglement, Phys. Rev. Lett. 104,
230503 (2010), arXiv:0911.5300 [quant-ph]
T. Cubitt, J. X. Chen, and A. Harrow, Superactivation of the Asymptotic Zero-Error Classical Capacity of a Quantum Channel, arXiv: 0906.2547. (2009)
F.G.S.L. Brandao, J. Oppenheim and S. Strelchuk, "When does noise increase the quantum capacity?", arXiv:1107.4385v1 [quant-ph] (2011)
M. Christandl, A. Winter, Uncertainty, Monogamy, and Locking of Quantum Correlations, IEEE Trans Inf Theory, vol 51, no 9, pp 3159-3165 (2005)., arXiv:quant-ph/0501090.
The results have been published at conferences on quantum information:
The results on additivity and superactivation analysis of quantum channels have been published in many journals (published by IEEE, Elsevier, Springer, Wiley, etc.)
Proceedings of the IEEE (Special Centennial Celebration Issue)
Two articles have been published in the Special Centennial Celebration Issue of the Proceedings of the IEEE.
100th Year Anniversary Celebration Volume of the Proceedings of the IEEE (Special Centennial Celebration Issue: Reviewing the Past, the Present, and the Future of Electrical Engineering Technology and the Profession)
Sandor Imre, Laszlo Gyongyosi: Introduction to Quantum-assisted and Quantum-based Solutions, with Lajos Hanzo, Harald Haas, Dominic O’Brien and Markus Rupp, in "Prolog to the Section on Wireless Communications Technology", Proceedings of the IEEE, Volume: 100, Issue: Special Centennial Issue, ISSN: 0018-9219. (Impact Factor: 5.151, IEEE Highest), 2012.
Sandor Imre, Laszlo Gyongyosi: Quantum-assisted and Quantum-based Solutions in Wireless Systems, with Lajos Hanzo, Harald Haas, Dominic O’Brien and Markus Rupp, in: "Wireless Myths, Realities and Futures: From 3G/4G to Optical and Quantum Wireless", Proceedings of the IEEE, Volume: 100, Issue: Special Centennial Issue, ISSN: 0018-9219. (Impact Factor: 5.151, IEEE Highest), 2012.
Sandor Imre and Laszlo Gyongyosi: Advanced Quantum Communications - An Engineering Approach
Publisher: Wiley-IEEE Press (New Jersey, USA), John Wiley & Sons, Inc., The Institute of Electrical and Electronics Engineers.
Book Details: Hardcover: 524 pages, ISBN-10: 1118002369, ISBN-13: 978-11180023, Sept. 2012.
Teaching activity and Conference Organizing
My papers on the arXiv
Laszlo Gyongyosi received the M.Sc. degree in Computer Science with Honors from the Budapest University of Technology and Economics (BUTE) in 2008. He is a Ph.D. Candidate at the Department of Telecommunications, BUTE. His research interests are in Quantum Channel Capacities, Quantum Computation and Communication, Quantum Cryptography and Quantum Information Theory. Currently, he is completing a book on advanced quantum communications, and he teaches courses in Quantum Computation. In 2009, he received Future Computing Best Paper Award on quantum information, in 2010, he was awarded the Best Paper Prize of University of Harvard, USA. In 2010, he obtained a Ph.D. Grant Award from University of Arizona, USA. In 2011, he received the PhD Candidate Scholarship at the Budapest University of Technology and Economics, the Ph.D. Grant Award of Stanford University, USA, the award of University of Southern California, USA, and the Ph.D. Grant Award of Quantum Information Processing 2012 (QIP2012), University of Montreal, Canada. In 2012 he received the PhD Grant Award of APS DAMOP 2012 from the Division of Atomic, Molecular, and Optical Physics of the American Physical Society (APS), California, USA, 2012.