The highest fidelity of quantum error-correcting codes of length n and rate R is proven to be lower bounded by 1 - exp [-n E(R)+ o(n)] for some function E(R) on noisy quantum channels that are subject to not necessarily independent errors. The E(R) is positive below some threshold R', which implies R' is a lower bound on the quantum capacity. This work is an extension of the author's previous works [M. Hamada, Phys. Rev. A, 65, 052305, 2002 (e-Print quant-ph/0109114, LANL, 2001), and M. Hamada, submitted to IEEE Trans. Inf. Theory, 2002 (e-Print quant-ph/0112103, LANL, 2001)], which presented the bound for channels subject to independent errors, or channels modeled as tensor products of copies of a completely positive linear map. The relation of the channel class treated in this paper to those in the previous works are similar to that of Markov chains to sequences of independent identically distributed random variables.
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.
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 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…
We calculate the entanglement-assisted classical capacity of symmetric and asymmetric Pauli channels where two consecutive uses of the channels are correlated. It is evident from our study that in the presence of memory, a higher amount of classical information is transmitted over quantum channels if there exists prior entanglement as compared to product and entangled state coding.
We study the capacity of d-dimensional quantum channels with memory modeled by correlated noise. We show that, in agreement with previous results on Pauli qubit channels, there are situations where maximally entangled input states achieve higher values of mutual information than product states. Moreover, a strong dependence of this effect on the nature of the noise correlations as well as on the parity of the space dimension is found. We conjecture that when entanglement gives an advantage in terms of mutual information, maximally entangled states saturate the channel capacity.
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 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…
We present an experimental method to detect lower bounds to the quantum capacity of two-qubit communication channels. We consider an implementation with polarisation degrees of freedom of two photons and report on the efficiency of such a method in the presence of correlated noise for varying values of the correlation strength. The procedure is based on the generation of separable states of two qubits and local measurements at the output. We also compare the performance of the correlated two-qubit channel with the single-qubit channels corresponding to the partial trace on each of the subsyste…
We study information transmission over a fully correlated amplitude damping channel acting on two qubits. We derive the single-shot classical channel capacity and show that entanglement is needed to achieve the channel best performance. We discuss the degradability properties of the channel and evaluate the quantum capacity for any value of the noise parameter. We finally compute the entanglement-assisted classical channel capacity.
We construct a class of quantum channels in arbitrary dimensions for which entanglement improves the performance of the channel. The channels have correlated noise and when the level of correlation passes a critical value we see a sharp transition in the optimal input states (states which minimize the output entropy) from separable to maximally entangled states. We show that for a subclass of channels with some extra conditions, including the examples which we consider, the states which minimize the output entropy are the ones which maximize the mutual information.
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…
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.
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 …
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…
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…
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.
One of the main figures of merit for quantum memories and quantum communication devices is their quantum capacity. It has been studied for arbitrary kinds of quantum channels, but its practical estimation has so far been limited to devices that implement independent and identically distributed (i.i.d.) quantum channels, where each qubit is affected by the same noise process. Real devices, however, typically exhibit correlated errors. Here, we overcome this limitation by presenting protocols that estimate a channel's one-shot quantum capacity for the case where the device acts on (an arbitrary …
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…
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.
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…
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…
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…
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.
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.
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.
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 process of quantum teleportation can be considered as a quantum channel. The exact classical capacity of the continuous variable teleportation channel is given. Also, the channel fidelity is derived. Consequently, the properties of the continuous variable quantum teleportation are discussed and interesting results are obtained.
The classical capacity of the lossy bosonic channel is calculated exactly. It is shown that its Holevo information is not superadditive, and that a coherent-state encoding achieves capacity. The capacity of far-field, free-space optical communications is given as an example.
We discuss the capacity of quantum channels for information transmission and storage. Quantum channels have dual uses: they can be used to transmit known quantum states which code for classical information, and they can be used in a purely quantum manner, for transmitting or storing quantum entanglement. We propose here a definition of the von Neumann capacity of quantum channels, which is a quantum mechanical extension of the Shannon capacity and reverts to it in the classical limit. As such, the von Neumann capacity assumes the role of a classical or quantum capacity depending on the usage o…
A quantum channel is conjugate degradable if the channel's environment can be simulated up to complex conjugation using the channel's output. For all such channels, the quantum capacity can be evaluated using a single-letter formula. In this article we introduce conjugate degradability and establish a number of its basic properties. We then use it to calculate the quantum capacity of N to N+1 and 1 to M universal quantum cloning machines as well as the quantum capacity of a channel that arises naturally when data is being transmitted to an accelerating receiver. All the channels considered tur…
When classical information is sent through a quantum channel of nonorthogonal states, there is a possibility that transmittable classical information exceeds a channel capacity in a single use of the initial channel by extending it into multi-product channel. In this paper, it is shown that this remarkable feature of a quantum channel, so-called superadditivity, appears even in as low as the third extended coding of the simplest binary input channel. A physical implementation of this channel is indicated based on cavity QED techniques.
We consider a nontrivial class of infinite dimensional quantum channels characterized by finiteness of the Holevo capacity. Some general properties of channels of this class are described. In particular, a special sufficient condition of existence of an optimal measure is obtained and examples of channels with no optimal measure are constructed. It is shown that each channel with finite Holevo capacity has a natural extension to the set of all positive normalized functionals on the algebra of all bounded operators. General properties of such an extension are described. The class of infinite di…
We study the effect of correlated Markovian noise channels on the quantum speed limit of an open system. This is done for correlated dephasing and amplitude damping channels for a two qubit atomic model. Our model serves as a platform for a detailed study of speed of quantum evolution in correlated open systems.
The capacity of quantum channel with product input states was formulated by the quantum coding theorem. However, whether entangled input states can enhance the quantum channel is still open. It turns out that this problem is reduced to aspecial case of the more general problem whether the capacity of product quantum channel exhibits additivity. In the present study, we apply one of the quantum Arimoto-Blahut type algorithms to the latter problem. The results suggest that the additivity of product quantum channel capacity always holds and that entangled input states cannot enhance the quantum c…
We develop a model for a noisy communication channel in which the noise affecting consecutive transmissions is correlated. This model is motivated by fluctuating birefringence of fiber optic links. We analyze the role of entanglement of the input states in optimizing the classical capacity of such a channel. Assuming a general form of an ensemble for two consecutive transmissions, we derive tight bounds on the classical channel capacity depending on whether the input states used for communication are separable or entangled across different temporal slots. This result demonstrates that by an ap…
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.
