László Gyöngyösi, Sándor Imre
Abstract:
In the first decade of the 21st century, many revolutionary properties of quantum channels were discovered. These phenomena are purely quantum mechanical and completely unimaginable in classical systems. Recently, the most important discovery in quantum information theory was the possibility of transmitting quantum information over zero-capacity quantum channels. The phenomenon called superactivation is rooted in the extreme violation of additivity of the channel capacities of quantum channels. Here we show, that using the superactivation effect, it is possible to develop efficient quantum repeaters with the elimination of the very inefficient and expensive purification process.Introduction:
A quantum channel can be used to realize classical information transmission or to transmit quantum information, such as quantum entanglement.Information transmission also can be approached using the question of whether the input consists of un-entangled or entangled quantum states. This leads us to say that for quantum channels, many new capacity definitions exist in comparison to a classical communication channel. In the case of a classical channel, we can send only classical information. Quantum channels extend the possibilities, and besides the classical information we can send entanglement-assisted classical information, classical private information, and of course, quantum information. On the other hand, the elements of classical information theory cannot be applied in general for quantum information—in other words, they can be used only in some special cases.
There is no general formula to describe the capacity of every quantum channel model, but one of the main results of the recent researches was the “very simplified”
picture, in which the various capacities of a quantum channel (i.e., the classical, private, quantum) are all non-additive. There are many phenomenon in quantum systems that cannot be described classically, such as entanglement, which makes it possible to store quantum information in the quantum correlation of quantum states. Entangled quantum states are named EPR states after Einstein, Podolsky, and Rosen, and a subset of them after are called Bell states, after J. Bell. Quantum entanglement was discovered in the 1930s, and it may still yield many surprises in the future. Currently it is accepted that entanglement has many classically indescribable properties and many new communication approaches can be based on it. Quantum entanglement plays a fundamental role in advanced quantum communications, such as teleportation, quantum cryptography, and quantum communication processes.
By the end of the 20th century, many advanced and interesting properties of quantum information theory had been discovered, and—as it seemed—most questions concerning quantum channel capacities had already been answered. At the dawn of this millennium new problems have arisen whose solutions are still unknown, and which have opened the door to many new promising results, such as the superactivation of zero-capacity quantum channels. The theoretical background of the superactivation of quantum capacities is currently unsolved; however, it is based on the extreme violation of
the additivity property—in other words, on the non-additivity of the various quantum channel capacities.
Research Method:
In our work, we apply computational geometry in quantum space.With the help of efficient computational geometric methods, the superactivation of optical quantum channels can be analyzed very efficiently. We would like to analyze the properties of the quantum channels using current classical computer architectures – since, currently we have no quantum computers - with the most efficient currently available algorithms. To this day, the most efficient classical algorithms for this purpose are computational geometric methods. We use these classical computational geometric tools to discover the still unknown “superactive” zero-error capacity optical quantum channels.
At present, computational geometry algorithms are an active, widely used and integrated research field. To study the geometry of superactivation, we define a new abstract geometrical object, the quantum informational superball.
Future Quantum Communications:
The superactivation of quantum channels makes it possible to use two zero-capacity quantum channels with a positive joint capacity for their output. Currently, we have no theoretical background to describe all possible combinations of superactive zero-capacity channels; hence, there may be many other possible combinations. Before our research work, the superactivation of classical capacity seemed to be completely unimaginable. As we have proven, the superactivation of classical capacity is also possible. Moreover, as we have found, it works for the most generalized quantum channel models, which describe the most natural physical processes.The biggest problem in future quantum communications is the long-distance delivery of quantum information. Since the quantum states cannot be copied, the amplification of quantum bits is a more complex compared to classical communications. The success of future long-distance quantum communications and global quantum key distribution systems strongly depends on the development of efficient quantum repeaters.
Standard Quantum Repeaters:
The quantum repeater is based on the transmission of entangled quantum states between the repeater nodes. The entanglement creation uses the quantum communication channel; hence some noise is added to the transmitted states.In the next step, the created entanglement has to be purified. The purification is an error-correcting scheme, and it uses local quantum operations only – hence these operations can be realized in the separated base stations locally.
The working mechanism of the quantum repeater with the entanglement sharing and the swapping is illustrated in Fig. 1.
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Fig. 1. The standard model of the quantum repeater with noisy quantum channels. The structure of the quantum repeater consists of a chain of base stations and the information is transmitted via quantum teleportation.
The main task is the improvement of the fidelities of the shared entangled states. The most important part of quantum communication between quantum repeaters is the entanglement purification. This step purifies the noisy quantum states, however this process requires a lot of resources in the quantum nodes, and it cannot be implemented efficiently in current solutions. The creation of high-fidelity entanglement between the nodes requires a lot of entangled pairs, while the purification process is a computationally very complex. In order to recover fidelity of entanglement from noisy quantum states purification is needed. The efficiency of the quantum repeaters in future would be incremental only if the efficiency of the purification process could be dramatically increased.
Our Solution:
The zero-error capacity of the quantum channel describes the amount of information which can be transmitted perfectly through a noisy quantum channel. The superactivation of the zero-error capacity of quantum channels makes it possible to use noisy quantum channels with perfect information transmission. The zero-error capacity of the quantum channel can be a very important measure where perfect communication is required or the resources for communication are very limited. The superactivation of quantum channels may be the starting-point of a large-scale revolution in quantum information theory and in the communication of future quantum networks where quantum channels are extremely noisy.In Fig. 2, we show possible ways of applying superactivation in the quantum repeater structure. The superactivation makes it possible to transmit information perfectly between the repeater stations, using very noisy quantum channels. Superactivation can be applied to an optical-fiber based optical quantum communication network, or in a free-space environment, both in a dense metropolitan area or over very long distances to improve the quality of information transmission.
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1Fig. 2. The superactivation of zero-error capacity can help to transmit information through a very noisy quantum network, and the information can be transmitted perfectly through a temporarily unavailable quantum channel.
The proposed new method can be very valuable tool to realize noiseless quantum communication between the repeater nodes. Using the superactivation of quantum channels, the efficiency of the quantum repeater can be increased, since the purification steps can be completely removed, and the long-distance quantum communication techniques can be revolutionized.
Superactivated Quantum Repeaters:
The newly developed quantum repeater uses noisy quantum channels such as in the case of standard quantum repeaters. On the other hand, these quantum channels can be used for perfect information transmission, and the entangled quantum states can be sent through the channels with maximal fidelity. As a conclusion, no further purifications needed during the communication.The basic model of the superactivated quantum repeater is depicted in Fig. 3.
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Fig. 3. The newly developed quantum repeater with noisy quantum channels and perfect information transmission.
The proposed joint channel structure between the repeater nodes contains two different classes of Gaussian quantum channels with zero zero-error capacity individually.
The general view of the zero-error information transmission over Gaussian quantum channel is shown in Fig. 4. The input message is denoted by Xi. The encoder
E and the decoder D were implemented by optical devices, the quantum channel is a Gaussian quantum link. The i-th entangled photon pair Ψi consists of entangled particles
{
00 , 01}
i β β
Ψ ∈ , and ρi( )1 = Ψ( )i1 Ψ( )i1 ,ρi( )2 = Ψ( )i2 Ψ( )i2 denote the density matrices of first and the second qubits of the EPR state. To realize the superactivation of the Gaussian quantum channel, we construct a joint channel denoted by G12 =G1⊗G , 2 using two Gaussian quantum channels denoted by G and 1 G . 2
Optical
Fig. 4. Information transmission with zero-error over a Gaussian quantum channel.
Numerical Results:
We have discovered that the possibility of the superactivation of the zero-error capacity depends on the length of the input codewords( )
N encoded by the EPR photons, and on the number of input codewords( )
M that can be distinguished by the POVM (Positive Operator Valued Measure) operators (called the non-adjacent codewords) performed by the optical decoder. We analyzed the classical zero-error capacity with using input blockcode length up to N=21 EPR pairs, with M N( )
=332non-adjacent input codewords, these results are shown in Fig. 5(a). Axis x represents the length of the input codewords encoded by EPR photon pairs, the y-axis illustrates the number of input state subspace length (M) with non-adjacent input codewords N. In Fig.
5(b) the results for the quantum capacity are shown. In that case we used blockcode length up to N=45 EPR pairs, with M N
( )
=2148 non-adjacent input codewords.Fig. 5. (a): Superactivated classical zero-error capacity. The non-adjacent input codewords as a function of the length of the input quantum blockcode. We have found only one possible constellation for the superactivation in the analyzed domain. (b) Results on the
quantum zero-error capacity. The superactivation of quantum zero-error capacity requires different input settings.
The results demonstrate that it is possible to transmit classical and quantum information perfectly over very noisy Gaussian quantum channels. For the zero-error quantum capacity, As we have found, within this large parameter domain, there is only one combination of input length of EPR photon pairs (N) and input state subspace length (M, the number of POVM elements of the joint measurement) for which the zero-error capacity of two Gaussian quantum channels can be superactivated. On the other hand, for larger values of N, and M, or different channel models, other solutions could be possible.
Our goal is to discover these still unrevealed combinations in the near future.
Performance Analysis:
The results of on the performance analysis of the superactivated quantum repeater are shown in Fig. 6. The red line depicts the rate ofsuperactived quantum zero-error capacity. In Fig. 6(a) the dashed line depicts the entanglement generation rate between the repeater stations without entanglement purification using general noisy quantum channel, the solid line depicts the superactivated channel. In 6(b), the lines represent one round of purification using standard quantum channels (dashed line) and superactivated channels (solid line). The F represents the final fidelities in the repeater nodes.
Fig. 6 (a). The rate of a standard quantum repeater (dashed line) and our newly designed quantum repeater (solid line) over a total distance of 1500 km, without purification using standard noisy quantum channels. (b). Comparison of superactivated zero-error quantum capacity with one-round of entanglement purification with standard quantum channels (dashed line) and superactivated quantum channel (solid line). The superactivated channels preserved entanglement nearly perfectly and the rate of entanglement
generation increased significantly.
From the results follows that using standard noisy quantum channels, the entanglement generation rate without purification is very low. On the other hand, if the quantum states are not purified, but the noisy quantum channels between the repeater nodes are
“superactivated”, then the rate of entanglement generation for the hybrid quantum repeater becomes about eight-times higher in average in comparison to the case if the quantum states are not purified and “standard” noisy quantum channels were used.