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:0704.2513

Reaching the Holevo Capacity via von Neumann measurement, and its use

Lóránt Farkas

Published Apr 19, 2007• 1 citation
arXivHTMLPDF

Abstract

In this article we show that the Holevo capacity of a classical quantum channel, can be reached not only by a POVM but by von Neumann measurement too. Furthermore we show two use of this fact 1) We can generalize the classical compound channel to quantum environment and with this new measure technique we can give an optimal coding/decoding algorithm to it. 2) We give an algorithm, with it a classical system can be decoded by quantum apparatus in linear time, which is a surprising result, because best known optimal classical decoding algorithm need n*log(n) time. This means that if a quantum machine can perform arbitrarily von Neumann measurement with only two possible outcome, then it can solve non-polynomial classical problems in linear time.

DOI
10.48550/arxiv.0704.2513

Related papers

Detecting the normal quantum/Compound Quantum channel with von Neumann measurement

Lóránt Farkas

In this brief summary we show an alternate prove for the fundamental theorem of quantum channel coding. This new prove is stronger then the previously known results because it is used von Neumann measurement. Moreover, the compound quantum channel is defined, and its capacity deduced, with the help of this new measurement.

IEEE International Symposium on Information Theory (ISIT)•Jul 1, 2008•0 citations

Variational estimation of capacity bounds for quantum channels

Junaid ur Rehman, Seongjin Hong, Yong‐Su Kim +1 more

We propose a framework to variationally obtain detectable capacity bounds for quantum channels. The proposal of this framework is motivated by the difficulty of estimating the von Neumann entropy of an unknown quantum state. Instead of estimating the von Neumann entropy at the channel output, we propose to estimate the state purity---which can be measured by a single measurement setting---followed by bounding the von Neumann entropy from above and below. This procedure leads to new upper and lower bounds on various communication rates of quantum channels for some fixed input states. Then, by utilizing the variational method to find optimal input states we obtain lower bounds on (i) the quantum capacity of arbitrary channels, (ii) the entanglement-assisted classical capacity of arbitrary channels, and (iii) the classical capacity of covariant channels. Corresponding to these lower bounds, we also obtain upper bounds on (i) $N$-shot coherent information, (ii) the entanglement-assisted classical capacity, and (iii) the $N$-shot Holevo capacity of arbitrary quantum channels. All these bounds can be estimated by a single measurement setting without needing a full process tomography or any further a priori knowledge, e.g., preferred basis of the channel.

Physical Review A•Mar 28, 2022•5 citations

Achieving the Holevo capacity of a pure state classical-quantum channel via unambiguous state discrimination

Masahiro Takeoka, Hari Krovi, Saikat Guha

We prove that the ultimate channel capacity, the Holevo bound, for sending classical data on a quantum channel (the so-called classical-quantum, or cq channel) can be achieved for a pure-state cq channel by decoding codewords via a collective quantum measurement based on unambiguous state discrimination (USD). In cq communication theory, the channel decoder acts directly on the modulated codeword waveform in the quantum (viz., electromagnetic or optical) domain, and it is known that collective measurements on long codeword blocks are needed to attain the Holevo capacity, which is strictly larger than the Shannon capacity of the classical channel induced by any specific measurement choice on each channel use. The USD measurement based channel decoder we propose, can distinguish finite blocklength codeword quantum states unambiguously (i.e., an incorrect codeword is never chosen) provided one allows for a finite probability of obtaining an inconclusive (erasure) outcome. We show that the probability of the inconclusive outcome goes to zero for asymptotically long codewords whenever the code rate is below the Holevo bound. The USD channel decoder is an addition to a small list of other collective measurements known to achieve the Holevo capacity (such as, the square root measurement, the minimum probability of error measurement, the sequential decoding measurement, and the quantum successive cancellation decoder for the cq polar code). A structured optical receiver design is not known yet for any of these decoders. What makes the USD decoder special is that there is no classical analogue to truly unambiguous discrimination (say, of samples drawn from a set of probability distributions). Secondly, the erasures-only decoding of USD is likely to result in a better channel reliability function. Finally, the USD measurement seems more likely to lead naturally to a structured optical receiver design and implementation.

IEEE International Symposium on Information Theory (ISIT)•Jul 1, 2013•12 citations

von Neumann capacity of noisy quantum channels

Christoph Adami, Nicolas J. Cerf

We discuss the capacity of quantum channels for information transmission and storage. Quantum channels have dual uses: they can be used to transmit known quantum states which code for classical information, and they can be used in a purely quantum manner, for transmitting or storing quantum entanglement. We propose here a definition of the von Neumann capacity of quantum channels, which is a quantum-mechanical extension of the Shannon capacity and reverts to it in the classical limit. As such, the von Neumann capacity assumes the role of a classical or quantum capacity depending on the usage of the channel. In analogy to the classical construction, this capacity is defined as the maximum von Neumann mutual entropy processed by the channel, a measure which reduces to the capacity for classical information transmission through quantum channels (the ``Kholevo capacity'') when known quantum states are sent. The quantum mutual entropy fulfills all basic requirements for a measure of information, and observes quantum data-processing inequalities. We also derive a quantum Fano inequality relating the quantum loss of the channel to the fidelity of the quantum code. The quantities introduced are calculated explicitly for the quantum depolarizing channel. The von Neumann capacity is interpreted within the context of superdense coding, and an extended Hamming bound is derived that is consistent with that capacity.

arXiv:quant-ph/9609024•Physical Review A•Nov 1, 1997•209 citations

Von Neumann Mutual Information in Nonequilibrium Quantum Dynamics

Masashi Ban, Sachiko Kitajima, Fumiaki Shibata

Information transmission via a quantum channel is one of the most important processes considered by the quantum information theory. Quantum channels obey nonequilibrium quantum dynamics. The task of the quantum information theory investigates properties of information transmission by means of quantum channels. The von Neumann mutual information and Holevo capacity are useful entropic quantities for this purpose. This short note discusses the relation between the von Neumann mutual information and the Holevo function and also provides a useful formula for calculating these quantities. The result is applied to the quantum dense coding under the influence of an external environment, which is the generalization of the results of ref. 4 Suppose that Alice sends a signal state carrying classical information to Bob through a quantum channel described by an arbitrary trace-preserving completely positive map L. The signal states are represented by orthonormal quantum states j ki (k 1⁄4 1; 2; . . .) which are assumed to be realized with prior probabilities k (k 1⁄4 1; 2; . . .). Then the quantum state of Alice’s system, i.e., the input system of the quantum channel, is given by in 1⁄4 P k k ðkÞ in , where  ðkÞ in 1⁄4 j kih kj. In this case, we have the eigenvalue equation, inj ki 1⁄4 kj ki. After receiving the signal, Bob obtains the quantum state ðkÞ out 1⁄4 Lj kih kj with the probability k and thus the quantum state of the output system becomes out 1⁄4 Lin 1⁄4 P k k ðkÞ out. Then we consider the eigenvalue problems, outj ni 1⁄4 nj ni and ðkÞ outj ðkÞ n i 1⁄4 ðkÞ n j ðkÞ n i. To obtain the von Neumann mutual information between the input and output systems of the quantum channel L, it is necessary to introduce a composite state of the input and output systems, which is not uniquely determined. Since we now consider the transmission of classical information, we use a separable state in{out as a composite state, 5,6) in{out 1⁄4 X

Journal of the Physical Society of Japan•Oct 15, 2004•0 citations

Towards Efficient Decoding of Classical-Quantum Polar Codes

Mark M. Wilde, Olivier Landon-Cardinal, Patrick M. Hayden

Known strategies for sending bits at the capacity rate over a general channel with classical input and quantum output (a cq channel) require the decoder to implement impractically complicated collective measurements. Here, we show that a fully collective strategy is not necessary in order to recover all of the information bits. In fact, when coding for a large number N uses of a cq channel W, N*I(W_{acc}) of the bits can be recovered by a non-collective strategy which amounts to coherent quantum processing of the results of product measurements, where I(W_{acc}) is the accessible information of the channel W. In order to decode the other N(I(W)-I(W_{acc})) bits, where I(W) is the Holevo rate, our conclusion is that the receiver should employ collective measurements. We also present two other results: 1) collective Fuchs-Caves measurements (quantum likelihood ratio measurements) can be used at the receiver to achieve the Holevo rate and 2) we give an explicit form of the Helstrom measurements used in small-size polar codes. The main approach used to demonstrate these results is a quantum extension of Arikan's polar codes.

arXiv:1302.0398•Civil War Book Review•Jan 1, 2013•14 citations
In this article we show that the Holevo capacity of a classical quantum channel, can be reached not only by a POVM but by von Neumann measurement too. Furthermore we show two use of this fact 1) We can generalize the classical compound channel to quantum environment and with this new measure technique we can give an optimal coding/decoding algorithm to it. 2) We give an algorithm, with it a classical system can be decoded by quantum apparatus in linear time, which is a surprising result, because best known optimal classical decoding algorithm need n*log(n) time. This means that if a quantum machine can perform arbitrarily von Neumann measurement with only two possible outcome, then it can solve non-polynomial classical problems in linear time.
We propose a framework to variationally obtain detectable capacity bounds for quantum channels. The proposal of this framework is motivated by the difficulty of estimating the von Neumann entropy of an unknown quantum state. Instead of estimating the von Neumann entropy at the channel output, we propose to estimate the state purity---which can be measured by a single measurement setting---followed by bounding the von Neumann entropy from above and below. This procedure leads to new upper and lower bounds on various communication rates of quantum channels for some fixed input states. Then, by utilizing the variational method to find optimal input states we obtain lower bounds on (i) the quantum capacity of arbitrary channels, (ii) the entanglement-assisted classical capacity of arbitrary channels, and (iii) the classical capacity of covariant channels. Corresponding to these lower bounds, we also obtain upper bounds on (i) NNN-shot coherent information, (ii) the entanglement-assisted classical capacity, and (iii) the NNN-shot Holevo capacity of arbitrary quantum channels. All these bounds can be estimated by a single measurement setting without needing a full process tomography or any further a priori knowledge, e.g., preferred basis of the channel.
We prove that the ultimate channel capacity, the Holevo bound, for sending classical data on a quantum channel (the so-called classical-quantum, or cq channel) can be achieved for a pure-state cq channel by decoding codewords via a collective quantum measurement based on unambiguous state discrimination (USD). In cq communication theory, the channel decoder acts directly on the modulated codeword waveform in the quantum (viz., electromagnetic or optical) domain, and it is known that collective measurements on long codeword blocks are needed to attain the Holevo capacity, which is strictly larger than the Shannon capacity of the classical channel induced by any specific measurement choice on each channel use. The USD measurement based channel decoder we propose, can distinguish finite blocklength codeword quantum states unambiguously (i.e., an incorrect codeword is never chosen) provided one allows for a finite probability of obtaining an inconclusive (erasure) outcome. We show that the probability of the inconclusive outcome goes to zero for asymptotically long codewords whenever the code rate is below the Holevo bound. The USD channel decoder is an addition to a small list of other collective measurements known to achieve the Holevo capacity (such as, the square root measurement, the minimum probability of error measurement, the sequential decoding measurement, and the quantum successive cancellation decoder for the cq polar code). A structured optical receiver design is not known yet for any of these decoders. What makes the USD decoder special is that there is no classical analogue to truly unambiguous discrimination (say, of samples drawn from a set of probability distributions). Secondly, the erasures-only decoding of USD is likely to result in a better channel reliability function. Finally, the USD measurement seems more likely to lead naturally to a structured optical receiver design and implementation.
We discuss the capacity of quantum channels for information transmission and storage. Quantum channels have dual uses: they can be used to transmit known quantum states which code for classical information, and they can be used in a purely quantum manner, for transmitting or storing quantum entanglement. We propose here a definition of the von Neumann capacity of quantum channels, which is a quantum-mechanical extension of the Shannon capacity and reverts to it in the classical limit. As such, the von Neumann capacity assumes the role of a classical or quantum capacity depending on the usage of the channel. In analogy to the classical construction, this capacity is defined as the maximum von Neumann mutual entropy processed by the channel, a measure which reduces to the capacity for classical information transmission through quantum channels (the ``Kholevo capacity'') when known quantum states are sent. The quantum mutual entropy fulfills all basic requirements for a measure of information, and observes quantum data-processing inequalities. We also derive a quantum Fano inequality relating the quantum loss of the channel to the fidelity of the quantum code. The quantities introduced are calculated explicitly for the quantum depolarizing channel. The von Neumann capacity is interpreted within the context of superdense coding, and an extended Hamming bound is derived that is consistent with that capacity.
Information transmission via a quantum channel is one of the most important processes considered by the quantum information theory. Quantum channels obey nonequilibrium quantum dynamics. The task of the quantum information theory investigates properties of information transmission by means of quantum channels. The von Neumann mutual information and Holevo capacity are useful entropic quantities for this purpose. This short note discusses the relation between the von Neumann mutual information and the Holevo function and also provides a useful formula for calculating these quantities. The result is applied to the quantum dense coding under the influence of an external environment, which is the generalization of the results of ref. 4 Suppose that Alice sends a signal state carrying classical information to Bob through a quantum channel described by an arbitrary trace-preserving completely positive map L. The signal states are represented by orthonormal quantum states j ki (k 1⁄4 1; 2; . . .) which are assumed to be realized with prior probabilities k (k 1⁄4 1; 2; . . .). Then the quantum state of Alice’s system, i.e., the input system of the quantum channel, is given by in 1⁄4 P k k ðkÞ in , where  ðkÞ in 1⁄4 j kih kj. In this case, we have the eigenvalue equation, inj ki 1⁄4 kj ki. After receiving the signal, Bob obtains the quantum state ðkÞ out 1⁄4 Lj kih kj with the probability k and thus the quantum state of the output system becomes out 1⁄4 Lin 1⁄4 P k k ðkÞ out. Then we consider the eigenvalue problems, outj ni 1⁄4 nj ni and ðkÞ outj ðkÞ n i 1⁄4 ðkÞ n j ðkÞ n i. To obtain the von Neumann mutual information between the input and output systems of the quantum channel L, it is necessary to introduce a composite state of the input and output systems, which is not uniquely determined. Since we now consider the transmission of classical information, we use a separable state in{out as a composite state, 5,6) in{out 1⁄4 X
Known strategies for sending bits at the capacity rate over a general channel with classical input and quantum output (a cq channel) require the decoder to implement impractically complicated collective measurements. Here, we show that a fully collective strategy is not necessary in order to recover all of the information bits. In fact, when coding for a large number N uses of a cq channel W, N*I(W_{acc}) of the bits can be recovered by a non-collective strategy which amounts to coherent quantum processing of the results of product measurements, where I(W_{acc}) is the accessible information of the channel W. In order to decode the other N(I(W)-I(W_{acc})) bits, where I(W) is the Holevo rate, our conclusion is that the receiver should employ collective measurements. We also present two other results: 1) collective Fuchs-Caves measurements (quantum likelihood ratio measurements) can be used at the receiver to achieve the Holevo rate and 2) we give an explicit form of the Helstrom measurements used in small-size polar codes. The main approach used to demonstrate these results is a quantum extension of Arikan's polar codes.