Cosmo Lupo, Stefano Pirandola, Paolo Aniello, Stefano Mancini
The set of quantum Gaussian channels acting on one bosonic mode can be classified according to the action of the group of Gaussian unitaries. We look for bounds on the classical capacity for channels belonging to such a classification. Lower bounds can be efficiently calculated by restricting to Gaussian encodings, for which we provide analytical expressions.
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We establish the fundamental limit of communication capacity within Gaussian schemes under phase-insensitive Gaussian channels, which employ multimode Gaussian states for encoding and collective Gaussian operations and measurements for decoding. We prove that this Gaussian capacity is additive, i.e., its upper bound occurs with separable encoding and separable receivers so that a single-mode communication suffices to achieve the largest capacity under Gaussian schemes. This rigorously characterizes the gap between the ultimate Holevo capacity and the capacity within Gaussian communication, sho…
Recently, there have been considerable progresses on the bounds of various quantum channel capacities for bosonic Gaussian channels. Especially, several upper bounds for the classical capacity and the quantum capacity on the bosonic Gaussian channels, via a technique known as quantum entropy power inequality, have been shed light on understanding the mysterious quantum-channel-capacity problems. However, upper bounds for the private capacity on quantum channels are still missing for the study on certain universal upper bounds. Here, we derive upper bounds on the private capacity for bosonic Ga…
The full solution of the optimization problem giving the Gaussian capacity of the single-mode fiducial Gaussian quantum channel is provided. Since it was shown that the Gaussian capacity of an arbitrary (phase-sensitive or insensitive) single-mode Gaussian quantum channel is equal to the Gaussian capacity of this fiducial channel, the solution presented in this work can be regarded as universal. The analytical study of this solution, below and above the energy threshold, shows that the dependence of the Gaussian capacity on the environment noise squeezing is not monotonic. In particular, the c…
The most natural way to describe an information-carrying system containing a specific noise is an additive white Gaussian-noise (AWGN) channel. In bosonic quantum systems (especially the Gaussian case), although the classical information capacity for a phase-insensitive and thermal-noise channel is additive based on a proof of the minimum output entropy conjecture, several open questions remain. By generalizing the Gaussian noise model from thermal noise to general Gaussian noise, we rigorously revisit and calculate these strong upper bounds on the information capacity for single-mode with gen…
The quantum channel capacity gives the ultimate limit for the rate at which quantum data can be reliably transmitted through a noisy quantum channel. Degradable quantum channels are among the few channels whose quantum capacities are known. Given the quantum capacity of a degradable channel, it remains challenging to find a practical coding scheme which approaches capacity. Here we discuss code designs for the detected-jump channel, a degradable channel with practical relevance describing the physics of spontaneous decay of atoms with detected photon emission. We show that this channel can be …
We present an algorithm for calculation of the Gaussian classical capacity of a quantum bosonic memory channel with additive Gaussian noise. The algorithm, restricted to Gaussian input states, is applicable to all channels with noise correlations obeying certain conditions and works in the full input energy domain, beyond previous treatments of this problem. As an illustration, we study the optimal input states and capacity of a quantum memory channel with Gauss-Markov noise [J. Schäfer, Phys. Rev. A 80, 062313 (2009)]. We evaluate the enhancement of the transmission rate when using these opti…
Quantum channels can be activated by a kind of channels whose quantum capacity is zero. This activation effect might be useful to overcome noise of channels by attaching other channels which can enhance the capacity of a given channel. In this work, we show that such an activation is possible by specific positive-partial-transpose channels for Gaussian lossy channels whose quantum capacities are known. We also test more general case involving Gaussian thermal attenuator whose the exact value of quantum capacity has been unknown so far. For a recently suggested narrow upper bound on quantum cap…
The Holevo-Schumacher-Westmoreland (HSW) classical (entanglement-unassisted) channel capacity for a class of qudit unital channels is shown to be C = log2(d) - Smin, where d is the dimension of the qudit, and Smin is the minimum possible von Neumann entropy at the channel output. The HSW channel capacity for tensor products of this class of unital qudit channels is shown to obey the same formula.
We investigate the classical capacity of two quantum channels with memory: a periodic channel with depolarizing channel branches, and a convex combination of depolarizing channels. We prove that the capacity is additive in both cases. As a result, the channel capacity is achieved without the use of entangled input states. In the case of a convex combination of depolarizing channels the proof provided can be extended to other quantum channels whose classical capacity has been proved to be additive in the memoryless case.
We prove the equivalence of an arbitrary single-mode Gaussian quantum channel and a newly defined fiducial channel preceded by a phase shift and followed by a Gaussian unitary operation. This equivalence implies that the energy-constrained classical capacity of any single-mode Gaussian channel can be calculated based on this fiducial channel, which is furthermore simply realizable with a beam splitter, two identical single-mode squeezers, and a two-mode squeezer. In a large domain of parameters, we also provide an analytical expression for the Gaussian classical capacity, exploiting its additi…
For a quantum channel of additive Gaussian noise with loss, in the general case of $n$ copies input, we show that up to first order perturbation, any non-Gaussian perturbation to the product thermal state input has a less quantum information transmission rate when the input energy tend to infinitive.
Classical capacity of quantum non-Gaussian attenuator and amplifier channels
研究非高斯衰减器和放大器通道的经典容量，通过推导上下界并探讨输入状态对输出熵的影响，揭示非高斯性在提升通信速率中的作用。
There exist memoryless zero-capacity quantum channels that when used jointly result in the channel with positive capacity. This phenomenon is called superactivation. Making use of Parrondo's paradox, we exhibit examples of superactivation-like effect for the capacity of classical communication channels as well as quantum and private capacity of quantum channels with memory. There are several ingredients necessary for superactivation of quantum capacity to occur in memoryless case. The first one is the requirement for the quantum channels which are amenable for superactivation to come from two …
Shannon's channel coding theorem describes the maximum possible rate of reliable information transfer through a classical noisy communication channel. It, together with the source coding theorem, characterizes lossless channel communication in the classical regime. Lossy compression scenarios require the additional description provided by rate-distortion theory, which characterizes the trade-off between compression rate and the distortion of the compressed signal. Even in this context, the capacity characterizes the usefulness of a channel -- a channel with more capacity will always outperform…
Dynamics of quantum battery capacity under Markovian channels
本文研究贝尔对角态量子电池容量在马尔可夫信道下的演化规律，揭示振幅阻尼信道下容量增加、退极化信道下容量突然死亡及大n时冻结容量等现象。
Lower Bounding the Secret Key Capacity of Bosonic Gaussian Channels via Optimal Gaussian Measurements
通过最优高斯测量确定玻色高斯信道秘密密钥容量的下界，优化了热损耗和热放大信道的协议并简化了性能评估。
A direct proof of the relation between the one-shot classical capacity and the minimal output entropy for covariant quantum channels is suggested. The structure of covariant channels is described in some detail. A simple proof of a general inequality for entanglement-assisted classical capacity is given.
We present an upper bound for the quantum channel capacity that is both additive and convex. Our bound can be interpreted as the capacity of a channel for high-fidelity quantum communication when assisted by a family of channels that have no capacity on their own. This family of assistance channels, which we call symmetric side channels, consists of all channels mapping symmetrically to their output and environment. The bound seems to be quite tight, and for degradable quantum channels it coincides with the unassisted channel capacity. Using this symmetric side channel capacity, we find new up…
Capacity of dense coding via correlated noisy channel is greater than that in uncorrelated noisy channel. It is shown that weak measurement and reversal measurement can make further effort to improve quantum dense coding capacity in correlated amplitude damping channel, but this effort is very small in correlated phase damping channel and correlated depolarizing channel.
We define the quantum zero-error capacity, a new kind of classical capacity of a noisy quantum channel. Moreover, the necessary requirement for which a quantum channel has zero-error capacity greater than zero is also given.
For a quantum channel with additive Gaussian quantum noise, at the large input energy side, we prove that the one shot capacity is achieved by the thermal noise state for all Gaussian state inputs, it is also true for non-Gaussian input in the sense of first order perturbation. For a general case of $n$ copies input, we show that up to first order perturbation, any non-Gaussian perturbation to the product thermal state input has a less quantum information transmission rate when the input energy tend to infinitive.
Prior entanglement between sender and receiver, which exactly doubles the classical capacity of a noiseless quantum channel, can increase the classical capacity of some noisy quantum channels by an arbitrarily large constant factor depending on the channel, relative to the best known classical capacity achievable without entanglement. The enhancement factor is greatest for very noisy channels, with positive classical capacity but zero quantum capacity. Although such quantum channels can be simulated classically, no violation of causality is implied, because the simulation requires at least as …
It is easy to show coincidence of the entanglement-assisted classical capacity and the Holevo capacity for any c-q channel between finite dimensional quantum systems. In this paper we prove the converse assertion: coincidence of the above-mentioned capacities of a quantum channel implies that the $χ$-essential part of this channel is a c-q channel (the $χ$-essential part is a restriction of a channel obtained by discarding all input states useless for transmission of classical information). The above observations are generalized to infinite dimensional quantum channels with linear constraints …
A formula for the capacity of a quantum channel for transmitting private classical information is derived. This is shown to be equal to the capacity of the channel for generating a secret key, and neither capacity is enhanced by forward public classical communication. Motivated by the work of Schumacher and Westmoreland on quantum privacy and quantum coherence, parallels between private classical information and quantum information are exploited to obtain an expression for the capacity of a quantum channel for generating pure bipartite entanglement. The latter implies a new proof of the quantu…
We study the classical capacity of a forgetful quantum memory channel that switches between two qubit depolarizing channels according to an ergodic Markov chain. The capacity of this quantum memory channel depends on the parameters of the two depolarizing channels and the memory of the ergodic Markov chain. When the number of input qubit's is two, we show that depending on channel parameters either the maximally entangled input states or product input states achieve the classical capacity. Our conjecture based on numerics is that as the number of input qubits are increased the classical capaci…
Super-additivity of quantum capacity in simple channels
本文研究了“generalized platypus channel”的信道容量，证明其私有和经典容量均为1，并证实其与qudit erasure channels及multilevel amplitude damping channels结合时仍具量子容量超加性。
The classical product state capacity of a noisy quantum channel with memory is investigated. A forgetful noise-memory channel is constructed by Markov switching between two depolarizing channels which introduces non-Markovian noise correlations between successive channel uses. The computation of the capacity is reduced to an entropy computation for a function of a Markov process. A reformulation in terms of algebraic measures then enables its calculation. The effects of the hidden-Markovian memory on the capacity are explored. An increase in noise-correlations is found to increase the capacity…
Conjugate degradable channels are channels whose quantum capacity is calculable. They were defined and studied in [1] where, however, only channels that are both degradable and conjugate degradable were found. In this paper we bring the very first example of conjugate degradable channels that are not degradable. We also identify the physical origin of these channels and show that they belong to the class of optimal universal asymmetric cloners. We thus not only positively answer the question whether conjugate degradable channels form a new class of channels with a single-letter capacity formul…
We study the capacity of a quantum channel where channel acts like controlled phase gate with the control being provided by a one-dimensional quantum spin chain environment. Due to the correlations in the spin chain, we get a quantum channel with memory. We derive formulas for the quantum capacity of this channel when the spin state is a matrix product state. Particularly, we derive exact formulas for the capacity of the quantum memory channel when the environment state is the ground state of the AKLT model and the Majumdar-Ghosh model. We find that the behavior of the capacity for the range o…
We study optimal rates for quantum communication over a single use of a channel, which itself can correspond to a finite number of uses of a channel with arbitrarily correlated noise. The corresponding capacity is often referred to as the one-shot quantum capacity. In this paper, we prove bounds on the one-shot quantum capacity of an arbitrary channel. This allows us to compute the quantum capacity of a channel with arbitrarily correlated noise, in the limit of asymptotically many uses of the channel. In the memoryless case, we explicitly show that our results reduce to known expressions for t…
In this thesis we analyse the type of states and ensembles which achieve the capacity for certain quantum channels carrying classical information. We first concentrate on the product-state capacity of a particular quantum channel, that is, the capacity which is achieved by encoding the output states from a source into codewords comprised of states taken from ensembles of non-entangled states and sending them over copies of the quantum channel. Using the "single-letter" formula proved independently by Holevo and by Schumacher and Westmoreland we obtain the product-state capacity of the qubit qu…
Communication over a noisy quantum channel introduces errors in the transmission that must be corrected. A fundamental bound on quantum error correction is the quantum capacity, which quantifies the amount of quantum data that can be protected. We show theoretically that two quantum channels, each with a transmission capacity of zero, can have a nonzero capacity when used together. This unveils a rich structure in the theory of quantum communications, implying that the quantum capacity does not uniquely specify a channel's ability for transmitting quantum information.
We give examples of qubit channels that require three input states in order to achieve the Holevo capacity.
The evolution of quantum battery capacity of GHZ-like states under Markovian channels
研究GHZ-like态在马尔可夫通道下的量子电池容量演化，发现其容量在特定通道下呈现冻结现象。
I show that classical capacity per unit cost of noisy bosonic Gaussian channels can be attained by employing generalized on-off keying modulation format and a projective measurement of individual output states. This means that neither complicated collective measurements nor phase-sensitive detection is required to communicate over optical channels at the ultimate limit imposed by laws of quantum mechanics in the limit of low average cost.
Recently there has been considerable activity on the subject of additivity of various quantum channel capacities. Here, we construct a family of channels with sharply bounded classical, hence private capacity. On the other hand, their quantum capacity when combined with a zero private (and zero quantum) capacity erasure channel, becomes larger than the previous classical capacity. As a consequence, we can conclude for the first time that the classical private capacity is non-additive. In fact, in our construction even the quantum capacity of the tensor product of two channels can be greater th…
Channel capacities of quantum channels can be nonadditive even if one of two quantum channels has no channel capacity. We call this phenomenon \emph{activation} of the channel capacity. In this paper, we show that when we use a quantum channel on a qubit system, only a noiseless qubit channel can generate the activation of the zero-error classical capacity. In particular, we show that the zero-error classical capacity of two quantum channels on qubit systems cannot be activated. Furthermore, we present a class of examples showing the activation of the zero-error classical capacity in low-dimen…
The qubit depolarizing channel with noise parameter $η$ transmits an input qubit perfectly with probability $1-η$, and outputs the completely mixed state with probability $η$. We show that its complementary channel has positive quantum capacity for all $η>0$. Thus, we find that there exists a single parameter family of channels having the peculiar property of having positive quantum capacity even when the outputs of these channels approach a fixed state independent of the input. Comparisons with other related channels, and implications on the difficulty of studying the quantum capacity of the …
We investigate multiparty communication scenarios where information is sent from several sender to several receivers. We establish a relation between the quantum capacity of multiparty communication channels and their distillability properties which enables us to show that the quantum capacity of such channels is not additive.
We determine the ultimate classical information capacity of a linear time-invariant bosonic channel with additive phase-insensitive Gaussian noise. This channel can model fiber-optic communication at power levels below the threshold for significant nonlinear effects. We provide a general continuous-time result that gives the ultimate capacity for such a channel operating in the quasimonochromatic regime under an average power constraint. This ultimate capacity is compared with corresponding results for heterodyne and homodyne detection over the same channel.
In this paper we fill the gap in previous works by proving the formula for entanglement-assisted capacity of quantum channel with additive constraint (such as bosonic Gaussian channel). The main tools are the coding theorem for classical-quantum constrained channels and a finite dimensional approximation of the input density operators for entanglement-assisted capacity. The new version contains improved formulation of sufficient conditions under which suprema in the capacity formulas are attained.
We consider the transmission of classical information over a quantum channel by two senders. The channel capacity region is shown to be a convex hull bound by the Von Neumann entropy and the conditional Von Neumann entropy. We discuss some possible applications of our result. We also show that our scheme allows a reasonable distribution of channel capacity over two senders.
The quantum capacity of thermal noise channel is studied. The extremal input state is obtained at the postulation that the coherent information is convex or concave at its vicinity. When the input energy tends to infinitive, it is verified by perturbation theory that the coherent information reaches its maximum at the product of identical thermal state input. The quantum capacity is obtained for lower noise channel and it is equal the one shot capacity.
The property of the optimal signal ensembles of entanglement assisted channel capacity is studied. A relationship between entanglement assisted channel capacity and one-shot capacity of unassisted channel is obtained. The data processing inequalities, convexity and additivity of the entanglement assisted channel capacity are reformulated by simple methods.
The classical capacity of a quantum channel with arbitrary Markovian correlated noise is evaluated. For the general case of a channel with long-term memory, which corresponds to a Markov chain which does not converge to equilibrium, the capacity is expressed in terms of the communicating classes of the Markov chain. For an irreducible and aperiodic Markov chain, the channel is forgetful, and one retrieves the known expression for the capacity.
We prove an upper bound on the private capacity of the single-mode noiseless bosonic wiretap channel. Combined with a previous lower bound, we obtain the low photon-number asymptotic expression for the private capacity. We then show that the multiple-mode noiseless bosonic wiretap channel is equivalent to parallel single-mode channels, hence the single-mode bounds can be applied. Finally, we consider multiple-spatial-mode propagation through atmospheric turbulence, and derive a private-capacity lower bound that only requires second moments of the channel matrix.
We show some chain-rules for the capacity of classical-quantum and quantum channels. We use the concept of Nash-Equilibrium in game-theory, and its existence in suitably defined games, to arrive at the chain-rules.
A long-standing problem on the classical capacity of bosonic Gaussian channels has recently been resolved by proving the minimum output entropy conjecture. It is also known that the ultimate capacity quantified by the Holevo bound can be achieved asymptotically by using an infinite number of channels. However, it is less understood to what extent the communication capacity can be reached if one uses a finite number of channels, which is a topic of practical importance. In this paper, we study the capacity of Gaussian communication, i.e., employing Gaussian states and Gaussian measurements to e…
The applications of the general formulae of channel capacity developed in the quantum information theory to evaluation of information transmission capacity of optical channel are interesting subjects. In this review paper, we will point out that the formulation based on only classical-quantum channel mapping model may be inadequate when one takes into account a power constraint for noisy channel. To define the power constraint well, we should explicitly consider how quantum states are conveyed through a transmission channel. Basing on such consideration, we calculate a capacity formula for an …
We study the symmetric-side-channel-assisted private capacity of a quantum channel, for which we provide a single-letter formula. This capacity is additive, convex, and, for degradable channels, equal to the unassisted private capacity. While a channel's (unassisted) capacity for for private classical communication may be strictly larger than its quantum capacity, we will show that these capacities are equal for degradable channels, thus demonstrating the equivalence of privacy and quantum coherence in this context. We use these ideas to find new bounds on the key rate of quantum key distribut…
The corrected capacity of a quantum channel is defined as the best one-shot capacity that can be obtained by measuring the environment and using the result to correct the output of the channel. It is shown that (i) all qubit channels have corrected capacity log 2, (ii) a product of N qubit channels has corrected capacity N log 2, and (iii) all channels have corrected capacity at least log 2. The question is posed of finding the channel with smallest corrected capacity in any dimension d.
Establishing the strong converse theorem for a communication channel confirms that the capacity of that channel, that is, the maximum achievable rate of reliable information communication, is the ultimate limit of communication over that channel. Indeed, the strong converse theorem for a channel states that coding at a rate above the capacity of the channel results in the convergence of the error to its maximum value 1 and that there is no trade-off between communication rate and decoding error. Here we prove that the strong converse theorem holds for the product-state capacity of quantum chan…
We show that for the tensor product of an entanglement-breaking quantum channel with an arbitrary quantum channel, both the minimum entropy of an output of the channel and the Holevo-Schumacher-Westmoreland capacity are additive. In addition, for the tensor product of two arbitrary quantum channels, we give a bound involving entanglement of formation for the amount of subadditivity (for minimum entropy output) or superadditivity (for channel capacity) that can occur.
The capacity of classical channels is convex. This is not the case for the quantum capacity of a channel: the capacity of a mixture of different quantum channels exceeds the mixture of the individual capacities and thus is non-convex. Here we show that this effect goes beyond the quantum capacity and holds for the private and classical environment-assisted capacities of quantum channels.
The superactivation of zero-capacity quantum channels makes it possible to use two zero-capacity quantum channels with a positive joint capacity at the output. Currently, we have no theoretical background for describing all possible combinations of superactive zero-capacity channels; hence, there may be many other possible combinations. In this PhD Thesis I provide an algorithmic solution to the problem of superactivation and prove that superactivation effect is rooted in information geometric issues.
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 the 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 e…
A condition for the zero-error capacity of quantum channels
本文提出量子信道零误差容量为正的一个条件：Kraus算子共有至少两个特征态，该条件作为零误差容量的下界，且易于验证。
The superactivation of zero-capacity 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. In practice, to discover such superactive zero-capacity channel-pairs, we must analyze an extremely large set of possible quantum states, channel models, and channel probabilities. There is still no extremely efficient algorithmic tool for this purpose. This pa…
Quantum Entanglement Assistance Improves the Capacity and Activates the Zero-Error Capacity of Classical Channels with Causal CSIT
研究显示，在存在因果信道状态信息的情况下，量子纠缠可提升经典信道容量并激活零错误容量。
For a partially degradable (PD) channel, the channel output state can be used to simulate the degraded environment state. The quantum capacity of a PD channel has been proven to be additive. Here, we show that the private classical capacity of arbitrary dimensional PD channels is equal to the quantum capacity of the channel and also single-letterizes. We prove that higher rates of private classical communication can be achieved over a PD channel in comparison to standard degradable channels.
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