Quantum
communication
channels

Classical communication
systems based on optics and optoelectronics changed our society.
On the other hand, when the intensity of few-photon signals is
reduced, we unavoidably enter the realm of quantum physics.
Quantum communication, namely the art of transferring quantum
states from one place to another, has arrived at a crucial stage
of developing a commercial and technological impact on our
society. This is witnessed by the first quantum cryptosystems
available on the market. At the same time, quantum communication
is a fascinating field: it combines concepts and techniques from
basic quantum physics and information science to optical
engineering.

Quantum communication channels use quantum systems to transfer classical or quantum information. In the first case, classical bits are encoded by means of quantum states. In the latter, one may want to distribute entanglement between two or more communicating parties or to transfer an unknown quantum state between different units of a quantum computer. The fundamental quantities characterizing a quantum channel are the classical and the quantum channel capacities, that are defined as the maximum number of bits/qubits that can be reliably transmitted per channel use .

Quantum channels are the natural theoretical framework to investigate both quantum communication and computation in a noisy environment. In the first case, information is transmitted in space, in the latter in time. In both cases, noise can have relevant low frequency components, which traduce themselves in memory effects. That is, consecutive uses of a channel can be correlated. Memory effects may be important, for instance, in quantum communication protocols realized by means of photons travelling across fibers with birefringence fluctuating with characteristic time scales longer than the separation between successive light pulses. Moreover, solid state implementations of quantum hardware show a characteristic low-frequency noise.

In spite of their physical relevance, quantum channels with memory are very hard to analyze and their classical and quantum capacities rarely known. Indeed, due to the peculiar features of quantum entanglement, one has to solve a sequence of optimization problems of increasing size (Hilbert space dimension) as the number of channel uses grows, and then take the limit of infinite number of uses. We have computed [1] the quantum capacity of a Markov chain dephasing channel with memory and shown that it is greater than in the memoryless case. Furthermore, based on theoretical arguments and numerical simulations, we have conjectured that memory-induced quantum capacity enhancement takes place also when the dephasing environment is modelled by a bosonic bath. We plan to further explore the connection between quantum channels with memory and the statistical mechanics of interacting many-body systems. This may open fruitful new possibilities for the study of information transmission across quantum channels with memory.

Quantum communication channels use quantum systems to transfer classical or quantum information. In the first case, classical bits are encoded by means of quantum states. In the latter, one may want to distribute entanglement between two or more communicating parties or to transfer an unknown quantum state between different units of a quantum computer. The fundamental quantities characterizing a quantum channel are the classical and the quantum channel capacities, that are defined as the maximum number of bits/qubits that can be reliably transmitted per channel use .

Quantum channels are the natural theoretical framework to investigate both quantum communication and computation in a noisy environment. In the first case, information is transmitted in space, in the latter in time. In both cases, noise can have relevant low frequency components, which traduce themselves in memory effects. That is, consecutive uses of a channel can be correlated. Memory effects may be important, for instance, in quantum communication protocols realized by means of photons travelling across fibers with birefringence fluctuating with characteristic time scales longer than the separation between successive light pulses. Moreover, solid state implementations of quantum hardware show a characteristic low-frequency noise.

Dephasing channel with memory

In spite of their physical relevance, quantum channels with memory are very hard to analyze and their classical and quantum capacities rarely known. Indeed, due to the peculiar features of quantum entanglement, one has to solve a sequence of optimization problems of increasing size (Hilbert space dimension) as the number of channel uses grows, and then take the limit of infinite number of uses. We have computed [1] the quantum capacity of a Markov chain dephasing channel with memory and shown that it is greater than in the memoryless case. Furthermore, based on theoretical arguments and numerical simulations, we have conjectured that memory-induced quantum capacity enhancement takes place also when the dephasing environment is modelled by a bosonic bath. We plan to further explore the connection between quantum channels with memory and the statistical mechanics of interacting many-body systems. This may open fruitful new possibilities for the study of information transmission across quantum channels with memory.

Quantum capacity of a Markov
chain dephasing channel (in the asymptotic limit when number n of channel uses tends to
infinite)

Enhancement of transmission rates in quantum
memory channels with damping

We have considered the
transfer of quantum information down a single-mode quantum
transmission line. Such quantum channel is modeled as a damped harmonic
oscillator, the interaction between the information carriers -a
train of n qubits- and the oscillator being of the
Jaynes-Cummings kind [2,3]. The oscillator acts as a local environment,
coupled to a memoryless reservoir damping both its phases and
populations, which mimics any cooling process resetting the
oscillator to its ground state. Memory effects appear if the
state of the oscillator is not reset after each channel use. The
model is visualized by a qubit-micromaser system, the qubit
train being a stream of two-level Rydberg atoms injected at low
rate into the cavity; it also describes the dynamics of a
quantum memory, which may be implemented by coupling n
superconducting qubits to a microstrip cavity, in a circuit-QED
architecture.

We have shown [2] that the quantum information transmission worsens with increasing transmission frequency due to the increase of memory effects. However, the decrease is found to be only moderate, so that the quantum transmission rate increases with increasing transmission frequency. Therefore, operating the memory channel at high transmission frequency, thus accepting prima facie deleterious memory effects, will be more beneficial than using low frequency. These results are relevant also for the secure transmission of classical information, that is, for cryptographic purposes.

Finally, we have analytically computed the single-shot classical capacity and the quantum capacity for a fully correlated (i.e., with full memory) channel with damping [4], and also investigated the performance of a partially correlated amplitude dampig channel [5].

We have shown [2] that the quantum information transmission worsens with increasing transmission frequency due to the increase of memory effects. However, the decrease is found to be only moderate, so that the quantum transmission rate increases with increasing transmission frequency. Therefore, operating the memory channel at high transmission frequency, thus accepting prima facie deleterious memory effects, will be more beneficial than using low frequency. These results are relevant also for the secure transmission of classical information, that is, for cryptographic purposes.

Finally, we have analytically computed the single-shot classical capacity and the quantum capacity for a fully correlated (i.e., with full memory) channel with damping [4], and also investigated the performance of a partially correlated amplitude dampig channel [5].

Enhancement of transmission
rate when increasing the memory factor

References

[1] A. D'Arrigo, G. Benenti and
G. Falci, Quantum capacity of
dephasing channels with memory, New J. Phys. 9, 310 (2007).

[2] G. Benenti, A. D'Arrigo and
G. Falci, Enhancement of
transmission rates in quantum memory channels with damping,
Phys. Rev. Lett. 103,
020502 (2009).

[3] A. D'Arrigo, G. Benenti and
G. Falci, Transmission of
classical and quantum information through a quantum memory
channel with damping, Eur. Phys. J. D66, 147 (2012).

[4] A. D'Arrigo, G. Benenti, G. Falci and C. Macchiavello, Classical and quantum capacities of a fully correlated amplitude damping channel, Phys. Rev. A 88, 042337 (2013).

[5] A. D'Arrigo, G. Benenti, G. Falci and C. Macchiavello,*Information transmission over an amplitude damping channel
with an
arbitrary degree of memory*,
Phys. Rev. A **92**, 062342 (2015).

[4] A. D'Arrigo, G. Benenti, G. Falci and C. Macchiavello, Classical and quantum capacities of a fully correlated amplitude damping channel, Phys. Rev. A 88, 042337 (2013).

[5] A. D'Arrigo, G. Benenti, G. Falci and C. Macchiavello,