Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1207.0067 · IEEE Transactions on Information Theory

A Decoupling Approach to Classical Data Transmission Over Quantum Channels

Frédéric Dupuis, Oleg Szehr, Marco Tomamichel

Published Jan 31, 2014• 20 citations
arXivHTMLPDF

Abstract

Most coding theorems in quantum Shannon theory can be proven using the decoupling technique. To send data through a channel, one guarantees that the environment gets no information about it. Uhlmann's theorem then ensures that the receiver must be able to decode. While a wide range of problems can be solved this way, one of the most basic coding problems remains impervious to a direct application of this method, sending classical information through a quantum channel. We will show that this problem can, in fact, be solved using decoupling ideas, specifically by proving a dequantizing theorem, which ensures that the environment is only classically correlated with the sent data. Our techniques naturally yield a generalization of the Holevo-Schumacher-Westmoreland theorem to the one-shot scenario, where a quantum channel can be applied only once.

DOI
10.1109/tit.2013.2295330

Related papers

A quantum version of Feinstein's Theorem and its application to channel coding

Nilanjana Datta, T. C. Dorlas

In this paper, a quantum version of Feinstein's theorem is developed. This is then used to give a completely self-contained proof of the direct channel coding theorem, for transmission of classical information through a quantum channel with Markovian correlated noise. Our proof does not rely on the Holevo-Schumacher-Westmoreland (HSW) theorem. In addition, for the case of memoryless channels, our method yields an alternative proof of the HSW Theorem

IEEE International Symposium on Information Theory (ISIT)•Jul 1, 2006•6 citations

The decoupling approach to quantum information theory

Frédéric Dupuis

Quantum information theory studies the fundamental limits that physical laws impose on information processing tasks such as data compression and data transmission on noisy channels. This thesis presents general techniques that allow one to solve many fundamental problems of quantum information theory in a unified framework. The central theorem of this thesis proves the existence of a protocol that transmits quantum data that is partially known to the receiver through a single use of an arbitrary noisy quantum channel. In addition to the intrinsic interest of this problem, this theorem has as immediate corollaries several central theorems of quantum information theory. The following chapters use this theorem to prove the existence of new protocols for two other types of quantum channels, namely quantum broadcast channels and quantum channels with side information at the transmitter. These protocols also involve sending quantum information partially known by the receiver with a single use of the channel, and have as corollaries entanglement-assisted and unassisted asymptotic coding theorems. The entanglement-assisted asymptotic versions can, in both cases, be considered as quantum versions of the best coding theorems known for the classical versions of these problems. The last chapter deals with a purely quantum phenomenon called locking. We demonstrate that it is possible to encode a classical message into a quantum state such that, by removing a subsystem of logarithmic size with respect to its total size, no measurement can have significant correlations with the message. The message is therefore "locked" by a logarithmic-size key. This thesis presents the first locking protocol for which the success criterion is that the trace distance between the joint distribution of the message and the measurement result and the product of their marginals be sufficiently small.

arXiv:1004.1641•Apr 9, 2010•90 citations

Quantum Communication

Mark M. Wilde

The quantum capacity theorem is one of the most important theorems in quantum Shannon theory. It is a fundamentally “quantum” theorem in that it demonstrates that a fundamentally quantum information quantity, the coherent information, is an achievable rate for quantum communication over a quantum channel. The fact that the coherent information does not have a strong analog in classical Shannon theory truly separates the quantum and classical theories of information. The no-cloning theorem (Section 3.5.4) provides the intuition behind the quantum capacity theorem. The goal of any quantum communication protocol is for Alice to establish quantum correlations with the receiver Bob. We know well now that every quantum channel has an isometric extension, so that we can think of another receiver, the environment Eve, who is at a second output port of a larger unitary evolution. Were Eve able to learn anything about the quantum information that Alice is attempting to transmit to Bob, then Bob could not be retrieving this information—otherwise, they would violate the no-cloning theorem. Thus, Alice should figure out some subspace of the channel input where she can place her quantum information such that only Bob has access to it, while Eve does not. That the dimensionality of this subspace is exponential in the coherent information is perhaps then unsurprising in light of the above no-cloning reasoning. The coherent information is an entropy difference H ( B ) -H ( E )—a measure of the amount of quantum correlations that Alice can establish with Bob less the amount that Eve can gain. We proved achievability of the coherent information for quantum data transmission in Corollary 22.2.1, but the roundabout path that we followed to prove achievability there perhaps does not give much insight into the structure of a quantum code that achieves the coherent information. Our approach in this chapter is different and should shed more light on this structure. Specifically, we show how to make coherent versions of the private classical codes from the previous chapter.

Cambridge University Press eBooks•Nov 2, 2016•5 citations

Classical Communication

Mark M. Wilde

This chapter begins our exploration of “dynamic” information-processing tasks in quantum Shannon theory, where the term “dynamic” indicates that a quantum channel connects a sender to a receiver and their goal is to exploit this resource for communication. We specifically consider the scenario where a sender Alice would like to communicate classical information to a receiver Bob, and the capacity theorem that we prove here is one particular generalization of Shannon's noisy channel coding theorem from classical information theory (overviewed in Section 2.2). In later chapters, we will see other generalizations of Shannon's theorem, depending on what resources are available to assist their communication or depending on whether they are trying to communicate classical or quantum information. For this reason and others, quantum Shannon theory is quite a bit richer than classical information theory. The naive approach to communicate classical information over a quantum channel is for Alice and Bob simply to mimic the approach used in Shannon's noisy channel coding theorem. That is, they select a random classical code according to some distribution pX(x) , and Bob performs individual measurements of the outputs of a noisy quantum channel according to some POVM. The POVM at the output induces some conditional probability distribution pY∣X(y∣x) , which we can in turn think of as an induced noisy classical channel. The classical mutual information I(X; Y) of this channel is an achievable rate for communication, and the best strategy for Alice and Bob is to optimize the mutual information over all of Alice's inputs to the channel and over all measurements that Bob could perform at the output.

Cambridge University Press eBooks•Apr 18, 2013•0 citations

The Packing Lemma

Mark M. Wilde

The packing lemma is a general method for one party to “pack” or encode classical messages into a Hilbert space so that another party can distinguish the encoded messages. The first party can prepare an ensemble of quantum states, and the other party has access to a set of projectors using which he can form a quantum measurement. If the ensemble and the projectors satisfy the conditions of the packing lemma, then it guarantees the existence of a scheme by which the second party can distinguish the classical messages that the first party prepares. The statement of the packing lemma is quite general, and this approach has a great advantage because we can use it as a primitive for many coding theorems. Examples of coding theorems that we can prove using the packing lemma are the Holevo–Schumacher–Westmoreland (HSW) theorem for the transmission of classical information over a quantum channel and the entanglement-assisted classical capacity theorem for the transmission of classical information over an entanglement-assisted quantum channel (furthermore, Chapter 22 shows that these two protocols are sufficient to generate most known protocols in quantum Shannon theory). Combined with the covering lemma of the next chapter, the packing lemma gives a method for transmitting private classical information over a quantum channel, and this technique in turn gives a way to communicate quantum information over a quantum channel. As long as we can determine an ensemble and a set of projectors satisfying the conditions of the packing lemma, we can apply it in a straightforward way. For example, we prove the HSW coding theorem in Chapter 20 largely by relying on the properties of typical and conditionally typical subspaces that we proved in the previous chapter, and some of these properties are equivalent to the conditions of the packing lemma. The packing lemma is a “one-shot” lemma because it applies to a general scenario that is not limited only to i.i.d. uses of a quantum channel. This “oneshot” approach is part of the reason that we can apply it to a variety of situations. The technique of proving a “one-shot” result and applying it to the i.i.d.

Cambridge University Press eBooks•Nov 2, 2016•0 citations

On the near-optimality of one-shot classical communication over quantum channels

Anurag Anshu, Rahul Jain, Naqueeb Ahmad Warsi

We study the problem of transmission of classical messages through a quantum channel in several network scenarios in the one-shot setting. We consider both the entanglement assisted and unassisted cases for the point to point quantum channel, the quantum multiple-access channel, the quantum channel with a state, and the quantum broadcast channel. We show that it is possible to near-optimally characterize the amount of communication that can be transmitted in these scenarios, using the position-based decoding strategy introduced in a prior study (A. Anshu, R. Jain, and N. Warsi, https://ieee.org/document/8399830). In the process, we provide a short and elementary proof of the converse for entanglement-assisted quantum channel coding in terms of the quantum hypothesis testing divergence [obtained earlier in the work of W. Matthews and S. Wehner, IEEE Trans. Inf. Theory 60, 7317–7329 (2014)]. Our proof has the additional utility that it naturally extends to various network scenarios mentioned above. Furthermore, none of our achievability results require a simultaneous decoding strategy, existence of which is an important open question in quantum Shannon theory.

arXiv:1804.09644•Journal of Mathematical Physics•Jan 1, 2019•27 citations
Most coding theorems in quantum Shannon theory can be proven using the decoupling technique. To send data through a channel, one guarantees that the environment gets no information about it. Uhlmann's theorem then ensures that the receiver must be able to decode. While a wide range of problems can be solved this way, one of the most basic coding problems remains impervious to a direct application of this method, sending classical information through a quantum channel. We will show that this problem can, in fact, be solved using decoupling ideas, specifically by proving a dequantizing theorem, which ensures that the environment is only classically correlated with the sent data. Our techniques naturally yield a generalization of the Holevo-Schumacher-Westmoreland theorem to the one-shot scenario, where a quantum channel can be applied only once.
Quantum information theory studies the fundamental limits that physical laws impose on information processing tasks such as data compression and data transmission on noisy channels. This thesis presents general techniques that allow one to solve many fundamental problems of quantum information theory in a unified framework. The central theorem of this thesis proves the existence of a protocol that transmits quantum data that is partially known to the receiver through a single use of an arbitrary noisy quantum channel. In addition to the intrinsic interest of this problem, this theorem has as immediate corollaries several central theorems of quantum information theory. The following chapters use this theorem to prove the existence of new protocols for two other types of quantum channels, namely quantum broadcast channels and quantum channels with side information at the transmitter. These protocols also involve sending quantum information partially known by the receiver with a single use of the channel, and have as corollaries entanglement-assisted and unassisted asymptotic coding theorems. The entanglement-assisted asymptotic versions can, in both cases, be considered as quantum versions of the best coding theorems known for the classical versions of these problems. The last chapter deals with a purely quantum phenomenon called locking. We demonstrate that it is possible to encode a classical message into a quantum state such that, by removing a subsystem of logarithmic size with respect to its total size, no measurement can have significant correlations with the message. The message is therefore "locked" by a logarithmic-size key. This thesis presents the first locking protocol for which the success criterion is that the trace distance between the joint distribution of the message and the measurement result and the product of their marginals be sufficiently small.
The quantum capacity theorem is one of the most important theorems in quantum Shannon theory. It is a fundamentally “quantum” theorem in that it demonstrates that a fundamentally quantum information quantity, the coherent information, is an achievable rate for quantum communication over a quantum channel. The fact that the coherent information does not have a strong analog in classical Shannon theory truly separates the quantum and classical theories of information. The no-cloning theorem (Section 3.5.4) provides the intuition behind the quantum capacity theorem. The goal of any quantum communication protocol is for Alice to establish quantum correlations with the receiver Bob. We know well now that every quantum channel has an isometric extension, so that we can think of another receiver, the environment Eve, who is at a second output port of a larger unitary evolution. Were Eve able to learn anything about the quantum information that Alice is attempting to transmit to Bob, then Bob could not be retrieving this information—otherwise, they would violate the no-cloning theorem. Thus, Alice should figure out some subspace of the channel input where she can place her quantum information such that only Bob has access to it, while Eve does not. That the dimensionality of this subspace is exponential in the coherent information is perhaps then unsurprising in light of the above no-cloning reasoning. The coherent information is an entropy difference H ( B ) -H ( E )—a measure of the amount of quantum correlations that Alice can establish with Bob less the amount that Eve can gain. We proved achievability of the coherent information for quantum data transmission in Corollary 22.2.1, but the roundabout path that we followed to prove achievability there perhaps does not give much insight into the structure of a quantum code that achieves the coherent information. Our approach in this chapter is different and should shed more light on this structure. Specifically, we show how to make coherent versions of the private classical codes from the previous chapter.
This chapter begins our exploration of “dynamic” information-processing tasks in quantum Shannon theory, where the term “dynamic” indicates that a quantum channel connects a sender to a receiver and their goal is to exploit this resource for communication. We specifically consider the scenario where a sender Alice would like to communicate classical information to a receiver Bob, and the capacity theorem that we prove here is one particular generalization of Shannon's noisy channel coding theorem from classical information theory (overviewed in Section 2.2). In later chapters, we will see other generalizations of Shannon's theorem, depending on what resources are available to assist their communication or depending on whether they are trying to communicate classical or quantum information. For this reason and others, quantum Shannon theory is quite a bit richer than classical information theory. The naive approach to communicate classical information over a quantum channel is for Alice and Bob simply to mimic the approach used in Shannon's noisy channel coding theorem. That is, they select a random classical code according to some distribution pX(x) , and Bob performs individual measurements of the outputs of a noisy quantum channel according to some POVM. The POVM at the output induces some conditional probability distribution pY∣X(y∣x) , which we can in turn think of as an induced noisy classical channel. The classical mutual information I(X; Y) of this channel is an achievable rate for communication, and the best strategy for Alice and Bob is to optimize the mutual information over all of Alice's inputs to the channel and over all measurements that Bob could perform at the output.
The packing lemma is a general method for one party to “pack” or encode classical messages into a Hilbert space so that another party can distinguish the encoded messages. The first party can prepare an ensemble of quantum states, and the other party has access to a set of projectors using which he can form a quantum measurement. If the ensemble and the projectors satisfy the conditions of the packing lemma, then it guarantees the existence of a scheme by which the second party can distinguish the classical messages that the first party prepares. The statement of the packing lemma is quite general, and this approach has a great advantage because we can use it as a primitive for many coding theorems. Examples of coding theorems that we can prove using the packing lemma are the Holevo–Schumacher–Westmoreland (HSW) theorem for the transmission of classical information over a quantum channel and the entanglement-assisted classical capacity theorem for the transmission of classical information over an entanglement-assisted quantum channel (furthermore, Chapter 22 shows that these two protocols are sufficient to generate most known protocols in quantum Shannon theory). Combined with the covering lemma of the next chapter, the packing lemma gives a method for transmitting private classical information over a quantum channel, and this technique in turn gives a way to communicate quantum information over a quantum channel. As long as we can determine an ensemble and a set of projectors satisfying the conditions of the packing lemma, we can apply it in a straightforward way. For example, we prove the HSW coding theorem in Chapter 20 largely by relying on the properties of typical and conditionally typical subspaces that we proved in the previous chapter, and some of these properties are equivalent to the conditions of the packing lemma. The packing lemma is a “one-shot” lemma because it applies to a general scenario that is not limited only to i.i.d. uses of a quantum channel. This “oneshot” approach is part of the reason that we can apply it to a variety of situations. The technique of proving a “one-shot” result and applying it to the i.i.d.
We study the problem of transmission of classical messages through a quantum channel in several network scenarios in the one-shot setting. We consider both the entanglement assisted and unassisted cases for the point to point quantum channel, the quantum multiple-access channel, the quantum channel with a state, and the quantum broadcast channel. We show that it is possible to near-optimally characterize the amount of communication that can be transmitted in these scenarios, using the position-based decoding strategy introduced in a prior study (A. Anshu, R. Jain, and N. Warsi, https://ieee.org/document/8399830). In the process, we provide a short and elementary proof of the converse for entanglement-assisted quantum channel coding in terms of the quantum hypothesis testing divergence [obtained earlier in the work of W. Matthews and S. Wehner, IEEE Trans. Inf. Theory 60, 7317–7329 (2014)]. Our proof has the additional utility that it naturally extends to various network scenarios mentioned above. Furthermore, none of our achievability results require a simultaneous decoding strategy, existence of which is an important open question in quantum Shannon theory.