Mixing (or quasirandom) properties of the natural transition matrix associated to a graph can be quantified by its distance to the complete graph. Different mixing properties correspond to different norms to measure this distance. For dense graphs, two such properties known as spectral expansion and uniformity were shown to be equivalent in seminal 1989 work of Chung, Graham and Wilson. Recently, Conlon and Zhao extended this equivalence to the case of sparse vertex transitive graphs using the famous Grothendieck inequality. Here we generalize these results to the non-commutative, or `quantum', case, where a transition matrix becomes a quantum channel. In particular, we show that for irreducibly covariant quantum channels, expansion is equivalent to a natural analog of uniformity for graphs, generalizing the result of Conlon and Zhao. Moreover, we show that in these results, the non-commutative and commutative (resp.) Grothendieck inequalities yield the best-possible constants.
Being attracted by the property of classical polar code, researchers are trying to find its analogue in quantum fields, which is called quantum polar code. The first step and the key to design quantum polar code is to find out for the quantity which can measure the quality of quantum channels, whether there is a polarization phenomenon which is similar to classical channel polarization. Coherent information is believed to be the quantum analogue of classical mutual information and the quantity to measure the capacity of quantum channel. In this paper, we define a class of quantum channels call…
We investigate the coherence of quantum channels and establish a resource theory for quantifying the coherence of quantum channels via Choi matrix. To this aim, we define the incoherent channels and incoherent superchannels. This theory recovers the case of quantum states when we view quantum states as a special case of quantum channels and also, this theory allows some analytical expressions for coherence measures.
Quantum capacity, as the key figure of merit for a given quantum channel, upper bounds the channel's ability in transmitting quantum information. Identifying different type of channels, evaluating the corresponding quantum capacity and finding the capacity-approaching coding scheme are the major tasks in quantum communication theory. Quantum channel in discrete variables has been discussed enormously involving various error models, while error model in the continuous variable channel has been less studied due to the infinite dimensional problem. In this paper, we investigate a general continuo…
A collection of quantum channels is called incompatible if they cannot be obtained as marginals from a single channel. No-cloning theorem is the most prominent instance of incompatibility of quantum channels. We show that every collection of incompatible channels can be more useful than compatible ones as preprocessings in a state discrimination task with multiple ensembles of states. This is done by showing that the robustness of channel incompatibility which is a measure for incompatibility of channels exactly quantifies the maximum advantage in the state discrimination. We also show that in…
Memory effects play an important role in the theory of open quantum systems. There are two completely independent insights about memory for quantum channels. In quantum information theory, the memory of the quantum channel is depicted by the correlations between consecutive uses of the channel on a set of quantum systems. In the theory of open quantum systems memory effects result from correlations which are created during the quantum evolution. Here, we study the quantum speed limit time for correlated quantum channel i.e. when there exist correlation between consecutive uses of quantum chann…
Based on the resource theory for quantifying the coherence of quantum channels, we introduce a new coherence quantifier for quantum channels via maximum relative entropy. We prove that the maximum relative entropy for coherence of quantum channels is directly related to the maximally coherent channels under a particular class of superoperations, which results in an operational interpretation of the maximum relative entropy for coherence of quantum channels. We also introduce the conception of sub-superchannels and sub-superchannel discrimination. For any quantum channels, we show that the adva…
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…
We study the general framework of quantum channel simulation, that is, the ability of a quantum channel to simulate another one using different classes of codes. First, we show that the minimum error of simulation and the one-shot quantum simulation cost under no-signalling assisted codes are given by semidefinite programs. Second, we introduce the channel's smooth max-information, which can be seen as a one-shot generalization of the mutual information of a quantum channel. We provide an exact operational interpretation of the channel's smooth max-information as the one-shot quantum simulatio…
We prove additivity of the minimum output entropy and the Holevo capacity for rotationally invariant quantum channels acting on spin-1/2 and spin-1 systems. The physical significance of these channels and their relations to other known channels is also discussed.
The essence of quantum error-correction is to use redundant Hilbert space to identify and correct errors, and the channel fidelity of the quantum channel does not affect which errors can be identified and corrected. Based on this, it is found that quantum error-correction can be used to classify the independent quantum channel into 5 types, and 4 of the 5 types can be purified. It is found in quantum error-correction, the decoherence of quantum state may be related to the degree of identification for the state under quantum noise, and the results of this work confirmed that the degree of purit…
Memory effects play a fundamental role in the dynamics of open quantum systems. There exist two different views on memory for quantum noises. In the first view, the quantum channel has memory when there exist correlations between successive uses of the channels on a sequence of quantum systems. These types of channels are also known as correlated quantum channels. In the second view, memory effects result from correlations which are created during the quantum evolution. In this work we will consider the first view and study the quantum speed limit time for a correlated quantum channel. Quantum…
Quantum resource theories have been widely studied to systematically characterize the non-classicality of quantum systems. Most resource theories focus on quantum states and study their interconversions. Although quantum channels are generally used as the tool for state manipulation, such a manipulation capability can be naturally regarded as a generalized quantum resource, leading to an open research direction in the resource theories of quantum channels. Various resource-theoretic properties of channels have been investigated, however, without treating channels themselves as operational reso…
We define the quantum-incoherent relative entropy of coherence ($\mathcal{QI}$ REC) of quantum channels in the framework of the resource theory by using the Choi-Jamiolkowsky isomorphism. Coherence-breaking channels are introduced as free operations and their corresponding Choi states as free states. We also show the relationship between the coherence of channel and the quantum discord and find that basis-dependent quantum asymmetric discord can never be more than the $\mathcal{QI}$ REC for any quantum channels. {Also}, we prove the $\mathcal{QI}$ REC is decreasing for any divisible quantum in…
Entanglement is a key issue in the quantum physics which gives rise to resources for achieving tasks that are not possible within the realm of classical physics. Quantum entanglement varies with the evolution of the quantum systems. It is of significance to investigate the entanglement dynamics in terms of quantum channels. We study the entanglement-breaking channels and present the necessary and sufficient conditions for a quantum channel to an entanglement-breaking one for qubit systems. Furthermore, a concept of strong entanglement-breaking channel is introduced. The amendment of entangleme…
Coherence is a basic notion for quantum states. Instead of quantum states, in this work, We establish a resource theory for quantifying the coherence of Gaussian channels. To do this, we propose the definitions of incoherent Gaussian channels and incoherent Gaussian superchannels.
The coexistence of quantum and classical signals over the same optical fiber with minimal degradation of the transmitted quantum information is critical for operating large-scale quantum networks over the existing communications infrastructure. Here, we systematically characterize the quantum channel that results from simultaneously distributing approximate single-photon polarization-encoded qubits and classical light of varying intensities through fiber-optic channels of up to 15~km. Using spectrally resolved quantum process tomography with a Bayesian reconstruction method we developed, we es…
The reliability of quantum channels for transmitting information is of profound importance from the perspective of quantum information. This naturally leads to the question as how well a quantum state is preserved when subjected to a quantum channel. We propose a measure of quantumness of channels based on non-commutativity of quantum states that is intuitive and easy to compute. We apply the proposed measure to some well known noise channels, both Markovian as well as non-Markovian and find that the results are in good agreement with those from a recently introduced $l_1$-norm coherence based…
