Qubit Channels Research Articles

Dense coding capacity of a quantum channel

We consider the fundamental protocol of dense coding of classical information assuming that noise affects both the forward and backward communication lines between Alice and Bob. Assuming that this noise is described by the same quantum channel, we define its dense coding capacity by optimizing over all adaptive strategies that Alice can implement, while Bob encodes the information by means of Pauli operators. Exploiting techniques of channel simulation and protocol stretching, we are able to establish the dense coding capacity of Pauli channels in arbitrary finite dimension, with simple formulas for depolarizing and dephasing qubit channels.

Detecting non-Markovianity via uncertainty relations

We present a formalism for detection of non-Markovianity (NM) through uncertainty relations. We show that when there is an information back-flow to the system from its environment through CP-divisibility breaking, the Choi-states corresponding to the reduced system evolution contain at least one negative eigenvalue. The consequent break down of uncertainty relations for such states can be used to witness non-Markovian dynamics. We present some relevant examples of the phenomenon for qubit channels. We further prove that square of the variance of a suitable Hermitian operator can act as a non-linear witness of NM. We finally show that NM is necessary in order to decrease the uncertainty of the states undergoing unital dynamics for qubits. This provides another method of certifying NM.

Quasi-inversion of qubit channels

Quantum operations, or quantum channels cannot be inverted in general. An arbitrary state passing through a quantum channel looses its fidelity with the input. Given a quantum channel ${\cal E}$, we introduce the concept of its quasi-inverse as a map ${\cal E}^{qi}$ which when composed with ${\cal E}$ increases its average input-output fidelity in an optimal way. The channel ${\cal E}^{qi}$ comes as close as possible to the inverse of a quantum channel. We give a complete classification of such maps for qubit channels and provide quite a few illustrative examples.

Duistermaat–Heckman measure and the mixture of quantum states

In this paper, we present a general framework to solve a fundamental problem in random matrix theory (RMT), i.e. the problem of describing the joint distribution of eigenvalues of the sum of two independent random Hermitian matrices and . Some considerations about the mixture of quantum states are basically subsumed into the above mathematical problem. Instead, we focus on deriving the spectral density of the mixture of adjoint orbits of quantum states in terms of the Duistermaat–Heckman measure, originated from the theory of symplectic geometry. Based on this method, we can obtain the spectral density of the mixture of independent random states. In particular, we obtain explicit formulas for the mixture of random qubits. We also find that, in the two-level quantum system, the average entropy of the equiprobable mixture of n random density matrices chosen from a random state ensemble (specified in the text) increases with the number n. Hence, as a physical application, our results quantitatively explain that the quantum coherence of the mixture monotonously decreases statistically as the number of components n in the mixture. Besides, our method may be used to investigate some statistical properties of a special subclass of unital qubit channels.

Data-driven inference of physical devices: theory and implementation

Given a physical device as a black box, one can in principle fully reconstruct its input–output transfer function by repeatedly feeding different input probes through the device and performing different measurements on the corresponding outputs. However, for such a complete tomographic reconstruction to work, full knowledge of both input probes and output measurements is required. Such an assumption is not only experimentally demanding, but also logically questionable, as it produces a circular argument in which the characterization of unknown devices appears to require other devices to have been already characterized beforehand. Here, we introduce a method to overcome such limitations present in usual tomographic techniques. We show that, even without any knowledge about the tomographic apparatus, it is still possible to infer the unknown device to a high degree of precision, solely relying on the observed data. This is achieved by employing a criterion that singles out the minimal explanation compatible with the observed data. Our method, that can be seen as a data-driven analog of tomography, is solved analytically and implemented as an algorithm for the learning of qubit channels.

The pairwise quantum correlations for teleported state via a symmetric multi-qubit system

We study the pairwise quantum correlations for teleported state via a symmetric multi-qubitsystem. In the other words, the proposed model is considered as a quantum channel. Using thequantum discord, super quantum discord and concurrence to quantify quantum correlations forteleported state, some analytical and numerical results are presented. Moreover, we compare thedynamical evolutions of quantum correlations and fidelity versus measurement strength and thenumber of qubit channel for teleported state via symmetric multi-qubit model. Our main goalnow is to study how we can increase the quantum correlations and the fidelity of the teleportedstate in the presence of decoherence. The results show that, measurement strength and thenumber of qubit can control the quantum information obtained through the quantum channel.Therefore, measurement strength can be a good option for measuring exchanged information inthe teleportation process. In addition to, quantum correlations can provide an effective role inquantum teleportation

Divisibility of qubit channels and dynamical maps

The concept of divisibility of dynamical maps is used to introduce an analogous concept for quantum channels by analyzing the simulability of channels by means of dynamical maps. In particular, this is addressed for Lindblad divisible, completely positive divisible and positive divisible dynamical maps. The corresponding L-divisible, CP-divisible and P-divisible subsets of channels are characterized (exploiting the results by Wolf et al. \cite{cirac}) and visualized for the case of qubit channels. We discuss the general inclusions among divisibility sets and show several equivalences for qubit channels. To this end we study the conditions of L-divisibility for finite dimensional channels, especially the cases with negative eigenvalues, extending and completing the results of Ref.~\cite{Wolf2008}. Furthermore we show that transitions between every two of the defined divisibility sets are allowed. We explore particular examples of dynamical maps to compare these concepts. Finally, we show that every divisible but not infinitesimal divisible qubit channel (in positive maps) is entanglement breaking, and open the question if something similar occurs for higher dimensions.

Diagonal quantum discord

Quantum discord measures quantum correlation by comparing the quantum mutual information with the maximal amount of mutual information accessible to a quantum measurement. This paper analyzes the properties of diagonal discord, a simplified version of discord that compares quantum mutual information with the mutual information revealed by a measurement that correspond to the eigenstates of the local density matrices. In contrast to the optimized discord, diagonal discord is easily computable; it also finds connections to thermodynamics and resource theory. Here we further show that, for the generic case of non-degenerate local density matrices, diagonal discord exhibits desirable properties as a preferable discord measure. We employ the theory of resource destroying maps (Liu Z-W et al 2017 Phys. Rev. Lett. 118 060502) to prove that diagonal discord is monotonically nonincreasing under the operation of local discord nongenerating qudit channels, d > 2, and provide numerical evidence that such monotonicity holds for qubit channels as well. We also show that it is continuous, and derive a Fannes-like continuity bound. Our results hold for a variety of simple discord measures generalized from diagonal discord.

Qubit-channel metrology with very noisy initial states

We consider an arbitrary qubit channel depending on a single parameter, which is to be estimated by a physical process. Using the quantum Fisher information per channel invocation to quantify the estimation accuracy, we consider various estimation protocols when the available initial states are mixed with very low purity. We compare a protocol using a single channel invocation on one out of $n$ qubits prepared in a particular correlated input state to the optimal protocol using uncorrelated input states, with the same initial state purity. We show that, to lowest order in initial-state purity, for a unital channel this correlated state protocol enhances the estimation accuracy by a factor between $n-1$ and $n.$ We also show that to lowest order in initial-state purity, a broad class of non-unital channels yields no gain regardless of the input state.

Solving Quantum Channel Discrimination Problem With Quantum Networks and Quantum Neural Networks

Distinguishing one quantum channel from another becomes a next natural question following the advent of quantum state discrimination. Here, we address the problem of quantum channel discrimination with the separable states and the entangled states input. Two schemes, based on the simulator for quantum networks and channels framework (SQUANCH) and the quantum machine learning toolbox (QMLT), are proposed to perform the quantum channel discrimination task. In the first scheme, we give an example of several qubit channels, verify that the performance metrics of channel discrimination can be improved by the entangled states. In the second scheme, we train a quantum circuit learning model to classify quantum channels using the Adam stochastic optimization algorithm. Although we consider the distinguishability in the case of only two possible channels, this model can be extended to any number of channels. This method can provide a good solution for distinguishing some complex quantum channels.

Volume of the space of qubit-qubit channels and state transformations under random quantum channels

The simplest building blocks for quantum computations are the qubit-qubit quantum channels. In this paper, we analyze the structure of these channels via their Choi representation. The restriction of a quantum channel to the space of classical states (i.e. probability distributions) is called the underlying classical channel. The structure of quantum channels over a fixed classical channel is studied, the volume of the general and unital qubit channels with respect to the Lebesgue measure is computed and explicit formulas are presented for the distribution of the volume of quantum channels over given classical channels. We study the state transformation under uniformly random quantum channels. If one applies a uniformly random quantum channel (general or unital) to a given qubit state, the distribution of the resulted quantum states is presented.

Quantumness of quantum channels

Quantum coherence is a fundamental aspect of quantum physics and plays a central role in quantum information science. This essential property of the quantum states could be fragile under the influence of the quantum operations. The extent to which quantum coherence is diminished depends both on the channel and the incoherent basis. Motivated by this, we propose a measure of nonclassicality of a quantum channel as the average quantum coherence of the state space after the channel acts on, minimized over all orthonormal basis sets of the state space. Utilizing the squared $l_1$-norm of coherence for the qubit channels, the minimization can be treated analytically and the proposed measure takes a closed form of expression. If we allow the channels to act locally on a maximally entangled state, the quantum correlation is diminished making the states more classical. We show that the extent to which quantum correlation is preserved under local action of the channel cannot exceed the quantumness of the underlying channel. We further apply our measure to the quantum teleportation protocol and show that a nonzero quantumness for the underlying channel provides a necessary condition to overcome the best classical protocols.

Lower and Upper Bounds on Nonunital Qubit Channel Capacities

Classical capacity of unital qubit channels is well known, whereas that of nonunital qubit channels is not. We find lower and upper bounds on classical capacity of nonunital qubit channels by using a recently developed decomposition technique relating nonunital and unital positive qubit maps.

Optimal decomposition of incoherent qubit channel

We show that any incoherent qubit channel could be decomposed into four incoherent Kraus operators. The proof consists in showing existence of four incoherent Kraus operators by decomposing the corresponding Choi–Jamiołkowski–Sudarshan matrix. We mention some applications of this optimal decomposition. We also show that the Kraus rank and incoherent rank are different even for qubit channel.

Quantum channel discrimination without entanglement

We address the problem of unambiguous discrimination of quantum channels (UDQC) without entanglement. As our main result, we show that even in the absence of entanglement, partial UDQC (PUDQC) can still be performed—depending on the unknown given channel. We provide a necessary and sufficient condition for PUDQC and put forth a method to perform the PUDQC once the said condition is met. We propose the performance metrics that capture the expected performance of the PUDQC independent of the specific channels to be distinguished. Finally, we perform PUDQC on several qubit channel pairs as concrete examples and derive the proposed performance metrics for these channel pairs.

Coherence of quantum channels

We investigate the coherence of quantum channels using the Choi–Jamiołkowski isomorphism. The relation between the coherence and the purity of the channel respects a duality relation. It characterizes the allowed values of coherence when the channel has certain purity. This duality has been depicted via the coherence–purity (Co–Pu) diagrams. In particular, we study the quantum coherence of the unital and non-unital qubit channels and find out the allowed region of coherence for a fixed purity. We also study coherence of different incoherent channels, namely, incoherent operation (IO), strictly incoherent operation (SIO), physical incoherent operation (PIO) etc. Interestingly, we find that the allowed region for different incoherent operations maintains the relationPIO⊂SIO⊂IO. In fact, we find that if PIOs are coherence preserving operations (CPO), its coherence is zero otherwise it has unit coherence and unit purity. Interestingly, different kinds of qubit channels can be distinguished using the Co–Pu diagram. The unital channels generally do not create coherence whereas some nonunital can. All coherence breaking channels are shown to have zero coherence, whereas, this is not usually true for entanglement breaking channels. It turns out that the coherence preserving qubit channels have unit coherence. Although the coherence of the Choi matrix of the incoherent channels might have finite values, its subsystem contains no coherence. This indicates that the incoherent channels can either be unital or nonunital under some conditions.

Two-qubit causal structures and the geometry of positive qubit-maps

We study quantum causal inference in a setup proposed by Ried et al (2015 Nat. Phys. 11 414) in which a common cause scenario can be mixed with a cause–effect scenario, and for which it was found that quantum mechanics can bring an advantage in distinguishing the two scenarios: whereas in classical statistics, interventions such as randomized trials are needed, a quantum observational scheme can be enough to detect the causal structure if the common cause results from a maximally entangled state. We analyze this setup in terms of the geometry of unital positive but not completely positive qubit-maps, arising from the mixture of qubit channels and steering maps. We find the range of mixing parameters that can generate given correlations, and prove a quantum advantage in a more general setup, allowing arbitrary unital channels and initial states with fully mixed reduced states. This is achieved by establishing new bounds on signed singular values of sums of matrices. Based on the geometry, we quantify and identify the origin of the quantum advantage depending on the observed correlations, and discuss how additional constraints can lead to a unique solution of the problem.

Activation of zero-error classical capacity in low-dimensional quantum systems

Channel capacities of quantum channels can be nonadditive even if one of two quantum channels has no channel capacity. We call this phenomenon 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-dimensional systems.

All unital qubit channels are 4-noisy operations

We show that any unital qubit channel can be implemented by letting the input system interact unitarily with a $4$-dimensional environment in the maximally mixed state and then tracing out the environment. We also provide an example where the dimension of such an environment has to be at least $3$.

On S-mixing entropy of quantum channels

In this paper, an S-mixing entropy of quantum channels is introduced as a generalization of Ohya’s S-mixing entropy. We investigate several properties of the introduced entropy. Moreover, certain relations between the S-mixing entropy and the existing map and output entropies of quantum channels are investigated as well. These relations allowed us to find certain connections between separable states and the introduced entropy. Hence, there is a sufficient condition to detect entangled states. Moreover, several properties of the introduced entropy are investigated. Besides, entropies of qubit and phase-damping channels are calculated.