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 it 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.

arXiv:1305.5970Physica ScriptaDec 1, 201411 citations

Randomizing Channels in Quantum Information Theory

정갑균

Jan 1, 20121 citation

arXiv:1412.0980 · IEEE Transactions on Information Theory

Approximate Degradable Quantum Channels

David Sutter, Volkher B. Scholz, Andreas Winter, Renato Renner

Published Sep 19, 2017• 73 citations

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Abstract

DOI: 10.1109/tit.2017.2754268

Related papers

Approximate degradable quantum channels

David Sutter, Volkher B. Scholz, Renato Renner

Degradable quantum channels are an important class of completely positive trace-preserving maps. Among other properties, they offer a single-letter formula for the quantum and the private classical capacity and are characterized by the fact that the complementary channel can be obtained from the channel by applying a degrading map. In this work we introduce the concept of approximate degradable channels, which satisfy this condition up to some finite ε ≥ 0. That is, there exists a degrading map which upon composition with the channel is ε-close in the diamond norm to the complementary channel. We show that for any fixed channel the smallest such ε can be efficiently determined via a semidefinite program. Moreover, these approximate degradable channels also approximately inherit all other properties of degradable channels. As an application, we derive improved upper bounds to the quantum and private classical capacity for certain channels of interest in quantum communication.Degradable quantum channels are an important class of completely positive trace-preserving maps. Among other properties, they offer a single-letter formula for the quantum and the private classical capacity and are characterized by the fact that the complementary channel can be obtained from the channel by applying a degrading map. In this work we introduce the concept of approximate degradable channels, which satisfy this condition up to some finite ε ≥ 0. That is, there exists a degrading map which upon composition with the channel is ε-close in the diamond norm to the complementary channel. We show that for any fixed channel the smallest such ε can be efficiently determined via a semidefinite program. Moreover, these approximate degradable channels also approximately inherit all other properties of degradable channels. As an application, we derive improved upper bounds to the quantum and private classical capacity for certain channels of interest in quantum communication.

IEEE International Symposium on Information Theory (ISIT)Jun 1, 201513 citations

Approaches for approximate additivity of the Holevo information of quantum channels

Felix Leditzky, Eneet Kaur, Nilanjana Datta +1 more

We study quantum channels that are close to another channel with weakly additive Holevo information, and we derive upper bounds on their classical capacity. Examples of channels with weakly additive Holevo information are entanglement-breaking channels, unital qubit channels, and Hadamard channels. Related to the method of approximate degradability, we define approximation parameters for each class above, which measure how close an arbitrary channel is to satisfying the respective property. This gives us upper bounds on the classical capacity in terms of functions of the approximation parameters, as well as an outer bound on the dynamic capacity region of a quantum channel. Since these parameters are defined in terms of the diamond distance, the upper bounds can be computed efficiently using semidefinite programming (SDP). We exhibit the usefulness of our method with two example channels: a convex mixture of amplitude damping and depolarizing noise and a composition of amplitude damping and dephasing noise. For both channels, our bounds perform well in certain regimes of the noise parameters in comparison to a recently derived SDP upper bound on the classical capacity. Along the way, we define the notion of a generalized channel divergence (which includes the diamond distance as an example), and we prove that for jointly covariant channels these quantities are maximized by purifications of a state invariant under the covariance group. This latter result may be of independent interest.We study quantum channels that are close to another channel with weakly additive Holevo information, and we derive upper bounds on their classical capacity. Examples of channels with weakly additive Holevo information are entanglement-breaking channels, unital qubit channels, and Hadamard channels. Related to the method of approximate degradability, we define approximation parameters for each class above, which measure how close an arbitrary channel is to satisfying the respective property. This gives us upper bounds on the classical capacity in terms of functions of the approximation parameters, as well as an outer bound on the dynamic capacity region of a quantum channel. Since these parameters are defined in terms of the diamond distance, the upper bounds can be computed efficiently using semidefinite programming (SDP). We exhibit the usefulness of our method with two example channels: a convex mixture of amplitude damping and depolarizing noise and a composition of amplitude damping and dephasing noise. For both channels, our bounds perform well in certain regimes of the noise parameters in comparison to a recently derived SDP upper bound on the classical capacity. Along the way, we define the notion of a generalized channel divergence (which includes the diamond distance as an example), and we prove that for jointly covariant channels these quantities are maximized by purifications of a state invariant under the covariance group. This latter result may be of independent interest.

arXiv:1709.01111Physical Review AJan 25, 2018110 citations

Sharp Quantum Capacity Thresholds: Exponential Strong Converses for Degradable and Antidegradable Channels

Tulja Varun Kondra, Raphael Brinster, Hermann Kampermann +2 more

The quantum capacity of a noisy channel quantifies the maximum rate at which quantum information can be transmitted reliably. For general channels, its evaluation requires an optimization over arbitrarily many channel uses. Degradable channels form a central exception: their capacity is given by the single-letter coherent information, while antidegradable channels have zero capacity. Nevertheless, even for these fundamental classes, it has remained open whether communication above capacity becomes possible when a fixed non-maximal error is tolerated. Here we resolve this problem by proving an exponential strong converse for every finite-dimensional degradable and antidegradable channel: at any rate above capacity, the fidelity of every coding scheme decays exponentially with the number of channel uses. As an immediate consequence, we establish the first all-code exponential strong converse for the quantum erasure channel throughout its full parameter range, strengthening previous results that applied only to almost all codes. We also show that exponential strong-converse bounds are preserved under receiver post-processing. This yields efficiently computable semidefinite-programming bounds for arbitrary finite-dimensional channels, improved bounds for Pauli channels, and an exact exponential strong converse for a nondegradable multilevel amplitude-damping family.The quantum capacity of a noisy channel quantifies the maximum rate at which quantum information can be transmitted reliably. For general channels, its evaluation requires an optimization over arbitrarily many channel uses. Degradable channels form a central exception: their capacity is given by the single-letter coherent information, while antidegradable channels have zero capacity. Nevertheless, even for these fundamental classes, it has remained open whether communication above capacity becomes possible when a fixed non-maximal error is tolerated. Here we resolve this problem by proving an exponential strong converse for every finite-dimensional degradable and antidegradable channel: at any rate above capacity, the fidelity of every coding scheme decays exponentially with the number of channel uses. As an immediate consequence, we establish the first all-code exponential strong converse for the quantum erasure channel throughout its full parameter range, strengthening previous results that applied only to almost all codes. We also show that exponential strong-converse bounds are preserved under receiver post-processing. This yields efficiently computable semidefinite-programming bounds for arbitrary finite-dimensional channels, improved bounds for Pauli channels, and an exact exponential strong converse for a nondegradable multilevel amplitude-damping family.

quant-pharXiv:2608.01308Aug 4, 20260 citations

Tightening Continuity Bounds for Entropies and Bounds on Quantum Capacities

Michael G. Jabbour, Nilanjana Datta

Uniform continuity bounds on entropies are generally expressed in terms of a single distance measure between probability distributions or quantum states, typically, the total variation- or trace distance. However, if an additional distance measure is known, the continuity bounds can be significantly strengthened. Here, we prove a tight uniform continuity bound for the Shannon entropy in terms of both the local- and total variation distances, sharpening an inequality in (Sason, 2013). We also obtain a uniform continuity bound for the von Neumann entropy in terms of both the operator norm- and trace distances. We then apply our results to compute upper bounds on channel capacities. We first refine the concept of approximate degradable channels by introducing $(\varepsilon ,\nu)$ –degradable channels. These are$\varepsilon $–close in diamond norm and$ \nu $–close in completely bounded spectral norm to their complementary channel when composed with a degrading channel. This leads to improved upper bounds to the quantum- and private classical capacities of such channels. Moreover, these bounds can be further improved by considering certain unstabilized versions of the above norms. We show that upper bounds on the latter can be efficiently expressed as semidefinite programs. As an application, we obtain a new upper bound on the quantum capacity of the qubit depolarizing channel.Uniform continuity bounds on entropies are generally expressed in terms of a single distance measure between probability distributions or quantum states, typically, the total variation- or trace distance. However, if an additional distance measure is known, the continuity bounds can be significantly strengthened. Here, we prove a tight uniform continuity bound for the Shannon entropy in terms of both the local- and total variation distances, sharpening an inequality in (Sason, 2013). We also obtain a uniform continuity bound for the von Neumann entropy in terms of both the operator norm- and trace distances. We then apply our results to compute upper bounds on channel capacities. We first refine the concept of approximate degradable channels by introducing$(\varepsilon ,\nu)$–degradable channels. These are$\varepsilon $–close in diamond norm and$\nu $–close in completely bounded spectral norm to their complementary channel when composed with a degrading channel. This leads to improved upper bounds to the quantum- and private classical capacities of such channels. Moreover, these bounds can be further improved by considering certain unstabilized versions of the above norms. We show that upper bounds on the latter can be efficiently expressed as semidefinite programs. As an application, we obtain a new upper bound on the quantum capacity of the qubit depolarizing channel.

arXiv:2310.17329IEEE Journal on Selected Areas in Information TheoryJan 1, 20243 citations

The private classical capacity of a partially degradable quantum channel

László Gyöngyösi

arXiv:1305.5970Physica ScriptaDec 1, 201411 citations

Randomizing Channels in Quantum Information Theory

정갑균

The randomization of quantum states in Quantum Information Theory is an important issue and has many applications on information processing. Such a method for randomizing quantum states is given by a completely positive and trace-preserving map known as an epsilon-randomizing map or a random unitary channel.\nIn this thesis, we represent a general formula for the\nepsilon-randomizing maps with respect to the p-norm of\nd dimensional quantum states and introduce some applications. Recently, the epsilon-randomizing channel together with its complementary channel gives a key technique for the proof of the additivity conjecture, that is, the additivity violation for the classical\ncapacity of quantum channel, which is the motivation of our work.\n\nThere are plenty of applications of the epsilon-randomizing channels in the view of reducing communication resources as well as the approximate encryption and decryption of quantum states. We explicitly investigate several applications such as\napproximate quantum state sharing, approximate state merging, and quantum sequential transmission protocol.The randomization of quantum states in Quantum Information Theory is an important issue and has many applications on information processing. Such a method for randomizing quantum states is given by a completely positive and trace-preserving map known as an epsilon-randomizing map or a random unitary channel.\nIn this thesis, we represent a general formula for the\nepsilon-randomizing maps with respect to the p-norm of\nd dimensional quantum states and introduce some applications. Recently, the epsilon-randomizing channel together with its complementary channel gives a key technique for the proof of the additivity conjecture, that is, the additivity violation for the classical\ncapacity of quantum channel, which is the motivation of our work.\n\nThere are plenty of applications of the epsilon-randomizing channels in the view of reducing communication resources as well as the approximate encryption and decryption of quantum states. We explicitly investigate several applications such as\napproximate quantum state sharing, approximate state merging, and quantum sequential transmission protocol.

Jan 1, 20121 citation

(\varepsilon ,\nu)

–degradable channels. These are$\varepsilon

–close in diamond norm and

\nu $–close in completely bounded spectral norm to their complementary channel when composed with a degrading channel. This leads to improved upper bounds to the quantum- and private classical capacities of such channels. Moreover, these bounds can be further improved by considering certain unstabilized versions of the above norms. We show that upper bounds on the latter can be efficiently expressed as semidefinite programs. As an application, we obtain a new upper bound on the quantum capacity of the qubit depolarizing channel.