Qubit Channels Research Articles

Local two-qubit entanglement-annihilating channels

We address the problem of the robustness of entanglement of bipartite systems (qubits) interacting with dynamically independent environments. In particular, we focus on characterization of so-called local entanglement-annihilating two-qubit channels, which set the maximum permissible noise level allowing to perform entanglement-enabled experiments. The differences, but also subtle relations between entanglement-breaking and local entanglement-annihilating channels are emphasized. A detailed characterization of latter ones is provided for a variety of channels including depolarizing, unital, (generalized) amplitude-damping, and extremal channels. We consider also the convexity structure of local entanglement-annihilating qubit channels and introduce a concept of entanglement-annihilation duality.

Quantum gate fidelity in terms of Choi matrices

We provide new results for computing and comparing the quantum gate fidelity of quantum channels via their Choi matrices. We extend recent work that showed that there exist non-dual pairs of quantum channels with equal gate fidelity by providing an explicit characterization of all such channels. We use our characterization to show that when the dimension is 2 (or 3, under slightly stronger hypotheses), the gate fidelity of two channels is equal if and only if their difference equals the difference of some unital map and its dual—a fact that has been shown to be false when the dimension is 4 or larger. We also present a formula for the minimum gate fidelity of a channel in terms of a well-studied norm on a compression of its Choi matrix. As a consequence, several new ways of bounding and approximating the minimum gate fidelity follow, including a simple semidefinite program to compute it for qubit channels.

Memory-keeping effects and forgetfulness in the dynamics of a qubit coupled to a spin chain

Using recently proposed measures for non-Markovianity [H.-P. Breuer, E. M. Laine, and J. Piilo, Phys. Rev. Lett. 103, 210401 (2009)], we study the dynamics of a qubit coupled to a spin environment via an energy-exchange mechanism. We show the existence of a point, in the parameter space of the system, where the qubit dynamics is effectively Markovian and that such a point separates two regions with completely different dynamical behaviors. Indeed, our study demonstrates that the qubit evolution can in principle be tuned from a perfectly forgetful one to a deep non-Markovian regime where the qubit is strongly affected by the dynamical backaction of the environmental spins. By means of a theoretical quantum process tomography analysis, we provide a complete and intuitive characterization of the qubit channel.

Classical Representations of Qubit Channels

A set of qubit channels has a classical representation when it is isomorphic to the convex closure of a group of classical channels. We prove that there are five such groups, each being either a subgroup of the alternating group on four letters, or a subgroup of the symmetric group on three letters.

Comment on “Transitions in the Communication Capacity of Dissipative Qubit Channels”

A Comment on the Letter by D. Daems, Phys. Rev. Lett. 102, 180503 (2009). The authors of the Letter offer a Reply.Received 2 June 2009DOI:https://doi.org/10.1103/PhysRevLett.105.188902© 2010 The American Physical Society

Quantum finite-depth memory channels: Case study

We analyze the depth of the memory of quantum memory channels generated by a fixed unitary transformation describing the interaction between the principal system and internal degrees of freedom of the process device. We investigate the simplest case of a qubit memory channel with a two-level memory system. In particular, we explicitly characterize all interactions for which the memory depth is finite. We show that the memory effects are either infinite, or they disappear after at most two uses of the channel. Memory channels of finite depth can be to some extent controlled and manipulated by so-called reset sequences. We show that actions separated by the sequences of inputs of the length of the memory depth are independent and constitute memoryless channels.

ENTANGLEMENT-ASSISTED QUANTUM ERROR-CORRECTING CODES CONSTRUCTED FROM IRREGULAR REPEAT ACCUMULATE CODES

In some contexts, it would make sense to look for codes that can use the maximal amount of shared ebits between sender and receiver, in order to boost qubit channel yield. Also, the codes are also expected to have feat net rates, when there are not enough ebits shared between the sender and receiver. In this paper, entanglement-assisted irregular quantum error-correcting codes achieving this goal are constructed from quaternary classical irregular repeat accumulate codes. Because of their sparse tanner graphs, the constructed codes have relatively low polynomial time-encoding complexity, and can be efficiently decoded by using sum–product algorithm. Moreover, the simulation results show that they outperform their existing quantum regular counterparts which are based on binary classical regular codes.

Private information via the Unruh effect

In a relativistic theory of quantum information, the possible presence of horizons is a complicating feature placing restrictions on the transmission and retrieval of information. We consider two inertial participants communicating via a noiseless qubit channel in the presence of a uniformly accelerated eavesdropper. Owing to the Unruh effect, the eavesdropper's view of any encoded information is noisy, a feature the two inertial participants can exploit to achieve perfectly secure quantum communication. We show that the associated private quantum capacity is equal to the entanglement-assisted quantum capacity for the channel to the eavesdropper's environment, which we evaluate for all accelerations.

A Strong Converse for Classical Channel Coding Using Entangled Inputs

A fully general strong converse for channel coding states that when the rate of sending classical information exceeds the capacity of a quantum channel, the probability of correctly decoding goes to zero exponentially in the number of channel uses, even when we allow code states which are entangled across several uses of the channel. Such a statement was previously only known for classical channels and the quantum identity channel. By relating the problem to the additivity of minimum output entropies, we show that a strong converse holds for a large class of channels, including all unital qubit channels, the d-dimensional depolarizing channel and the Werner-Holevo channel. This further justifies the interpretation of the classical capacity as a sharp threshold for information transmission.

Transitions in the Communication Capacity of Dissipative Qubit Channels

The information transmission is studied for quantum channels in which the noise includes dissipative effects, more specifically, nonunitality. Noise is usually a nuisance but can sometimes be helpful. For these channels, the communication capacity is shown to increase with the dissipative component of the noise and may exhibit transitions beyond which it increases faster. The optimal states are constructed analytically as well as the pertaining "phase" diagram.

Random unitary qubit channels: entropy relations,private quantum channels and non-malleability

Channels encrypting quantum bits by the application of randomly chosen unitary operators are studied. Quantities based on averages of linear entropies which characterize certain aspects of the encoding quality and the non-malleability of the channels are introduced. The relation between the entropy of the classical key and the choice of the encryption operators with the behaviour of these properties is discussed. The extension of exact private quantum channels in order to improve non-malleability via additional encryption operators is considered.

The structure of degradable quantum channels

Degradable quantum channels are among the only channels whose quantum and private classical capacities are known. As such, determining the structure of these channels is a pressing open question in quantum information theory. We give a comprehensive review of what is currently known about the structure of degradable quantum channels, including a number of new results as well as alternate proofs of some known results. In the case of qubits, we provide a complete characterization of all degradable channels with two dimensional output, give a new proof that a qubit channel with two Kraus operators is either degradable or anti-degradable, and present a complete description of anti-degradable unital qubit channels with a new proof. For higher output dimensions we explore the relationship between the output and environment dimensions (dB and dE, respectively) of degradable channels. For several broad classes of channels we show that they can be modeled with an environment that is “small” in the sense of ΦC. Such channels include all those with qubit or qutrit output, those that map some pure state to an output with full rank, and all those which can be represented using simultaneously diagonal Kraus operators, even in a non-orthogonal basis. Perhaps surprisingly, we also present examples of degradable channels with “large” environments, in the sense that the minimal dimension dE>dB. Indeed, one can have dE>14dB2. These examples can also be used to give a negative answer to the question of whether additivity of the coherent information is helpful for establishing additivity for the Holevo capacity of a pair of channels. In the case of channels with diagonal Kraus operators, we describe the subclasses that are complements of entanglement breaking channels. We also obtain a number of results for channels in the convex hull of conjugations with generalized Pauli matrices. However, a number of open questions remain about these channels and the more general case of random unitary channels.

A NEW APPROACH TO CHARACTERIZE QUBIT CHANNELS

We analyze qubit channels by exploiting the possibility of representing two-level quantum systems in terms of characteristic functions. To do so, we use functions of non-commuting variables (Grassmann variables), defined in terms of generalized displacement operators, following an approach which resemble the one adopted for continuous–variable (Bosonic) systems. It allows us to introduce the notion of qubit Gaussian channels and to show that they share similar properties with the corresponding continuous–variable counterpart. Some examples of qubit channels are investigated using this approach.

Publisher's Note: Qubit channels with small correlations [Phys. Rev. A77, 052323 (2008)

Publisher's Note: Qubit channels with small correlations [Phys. Rev. A<b>77</b>, 052323 (2008)

Qubit channels with small correlations

We introduce a class of quantum channels with correlations acting on pairs of qubits, where the correlation takes the form of a shift operator onto a maximally entangled state. We optimize the output purity and show that below a certain threshold the optimum is achieved by partially entangled states whose degree of entanglement increases monotonically with the correlation parameter. Above this threshold, the optimum is achieved by the maximally entangled state characterizing the shift. Although, a full analysis can only be done for the 2-norm, both numerical and heuristic arguments indicate that this behavior and the optimal inputs are independent of $pg1$ when the optimal output purity is measured using the $p$-norm.

Three methods to distill multipartite entanglement over bipartite noisy channels

We first assume that there are only bipartite noisy qubit channels in a given multipartite system, and present three methods to distill the general Greenberger–Horne–Zeilinger state. By investigating the methods, we show that multipartite entanglement distillation by bipartite entanglement distillation has higher yield than ones in the previous multipartite entanglement distillations.

Dividing Quantum Channels

We investigate the possibility of dividing quantum channels into concatenations of other channels, thereby studying the semigroup structure of the set of completely-positive trace-preserving maps. We show the existence of ‘indivisible’ channels which can not be written as non-trivial products of other channels and study the set of ‘infinitesimal divisible’ channels which are elements of continuous completely positive evolutions. For qubit channels we obtain a complete characterization of the sets of indivisible and infinitesimal divisible channels. Moreover, we identify those channels which are solutions of time-dependent master equations for both positive and completely positive evolutions. For arbitrary finite dimension we prove a representation theorem for elements of continuous completely positive evolutions based on new results on determinants of quantum channels and Markovian approximations.

On the discrimination of qubit channels

We discuss the discrimination of arbitrary qubit channels and the discrimination of qubit channels with the aid of entanglement. Nevertheless, the maximally entangled states might decrease the distinguishability, even if the channel is not entanglement-breaking one.

Pauli diagonal channels constant on axes

We define and study the properties of channels which are analogous to unital qubit channels in several ways. A full treatment can be given only when the dimension d = pm a prime power, in which case each of the d + 1 mutually unbiased bases (MUB) defines an axis. Along each axis the channel looks like a depolarizing channel, but the degree of depolarization depends on the axis. When d is not a prime power, some of our results still hold, particularly in the case of channels with one symmetry axis. We describe the convex structure of this class of channels and the subclass of entanglement breaking channels. We find new bound entangled states for d = 3. For these channels, we show that the multiplicativity conjecture for maximal output p-norm holds for p = 2. We also find channels with behaviour not exhibited by unital qubit channels, including two pairs of orthogonal bases with equal output entropy in the absence of symmetry. This provides new numerical evidence for the additivity of minimal output entropy.

Simulating a single-qubit channel using a mixed-state environment

We analyze the class of single qubit channels with the environment modeled by a one-qubit mixed state. The set of affine transformations for this class of channels is computed analytically, employing the canonical form for the two-qubit unitary operator. We demonstrate that, 3/8 of the generalized depolarizing channels can be simulated by the one-qubit mixed state environment by explicitly obtaining the shape of the volume occupied by this class of channels within the tetrahedron representing the generalized depolarizing channels. Further, as a special case, we show that the two-Pauli Channel cannot be simulated by a one-qubit mixed state environment.