arXiv:quant-ph/0202001 · Physical Review A

Quantum universal variable-length source coding

Masahito Hayashi, Keiji Matsumoto

Published Aug 26, 2002• 89 citations

Abstract

We construct an optimal quantum universal variable-length code that achieves the admissible minimum rate, i.e., our code is used for any probability distribution of quantum states. Its probability of exceeding the admissible minimum rate exponentially goes to 0. Our code is optimal in the sense of its exponent. In addition, its average error asymptotically tends to 0.

DOI: 10.1103/physreva.66.022311

Related papers

Simple construction of quantum universal variable-length source coding

Masahito Hayashi, Keiji Matsumoto

arXiv:quant-ph/0209124Sep 24, 20023 citations

Simple construction of quantum universal variable-length source coding

Masahito Hayashi, Koji Matsumoto

Quantum Information and ComputationNov 1, 200222 citations

Simple construction of quantum universal variable-length source coding

Masahito Hayashi, Koji Matsumoto

IEEE International Symposium on Information Theory (ISIT)Jan 1, 20034 citations

Exponents of quantum fixed-length pure-state source coding

Masahito Hayashi

We derive the optimal exponent of the error probability of the quantum fixed-length pure-state source coding in both cases of blind coding and visible coding. The optimal exponent is universally attained by Jozsa et al.'s [Phys. Rev. Lett. 81, 1714 (1998)] universal code. In the direct part, a group representation theoretical type method is essential. In the converse part, Nielsen and Kempe's [Phys. Rev. Lett. 86, 5184 (2001)] lemma is essential.

arXiv:quant-ph/0202002Physical Review ASep 27, 200248 citations

Optimal lower bound for lossless quantum block encoding

George Androulakis, Rabins Wosti

arXiv:2305.18748Quantum Information and ComputationNov 1, 20240 citations

Optimal Lower Bound of the Average Indeterminate Length Lossless Quantum Block Encoding

George Androulakis, Rabins Wosti

IEEE International Symposium on Information Theory (ISIT)Jul 7, 20240 citations

arXiv:quant-ph/0202001 · Physical Review A

Quantum universal variable-length source coding

Masahito Hayashi, Keiji Matsumoto

Published Aug 26, 2002• 89 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physreva.66.022311

Related papers

Simple construction of quantum universal variable-length source coding

Masahito Hayashi, Keiji Matsumoto

We simply construct a quantum universal variable-length source code in which, independent of information source, both of the average error and the probability that the coding rate is greater than the entropy rate $H(rho_p)$ , tend to 0. If $H(rho_p)$ is estimated, we can compress the coding rate to the admissible rate $H(rho_p)$ with a probability close to 1. However, when we perform a naive measurement for the estimation of $H(rho_p)$ , the input state is demolished. By smearing the measurement, we successfully treat the trade-off between the estimation of $H(rho_p)$ and the non-demolition of the input state. Our protocol can be used not only for the Schumacher's scheme but also for the compression of entangled states.We simply construct a quantum universal variable-length source code in which, independent of information source, both of the average error and the probability that the coding rate is greater than the entropy rate $H(rho_p)$, tend to 0. If $H(rho_p)$ is estimated, we can compress the coding rate to the admissible rate $H(rho_p)$ with a probability close to 1. However, when we perform a naive measurement for the estimation of $H(rho_p)$, the input state is demolished. By smearing the measurement, we successfully treat the trade-off between the estimation of $H(rho_p)$ and the non-demolition of the input state. Our protocol can be used not only for the Schumacher's scheme but also for the compression of entangled states.

arXiv:quant-ph/0209124Sep 24, 20023 citations

Simple construction of quantum universal variable-length source coding

Masahito Hayashi, Koji Matsumoto

We simply construct a quantum universal variable-length source code in which, independent of information source, both of the average error and the probability that the coding rate is greater than the entropy rate H(\overline{\rho}_p), tend to 0. If H(\overline{\rho}_p) is estimated, we can compress the coding rate to the admissible rate H(\overline{\rho}_p) with a probability close to 1. However, when we perform a naive measurement for the estimation of H(\overline{\rho}_p), the input state is demolished. By smearing the measurement, we successfully treat the trade-off between the estimation of H(\overline{\rho}_p) and the non-demolition of the input state. Our protocol can be used not only for the Schumacher's scheme but also for the compression of entangled states.We simply construct a quantum universal variable-length source code in which, independent of information source, both of the average error and the probability that the coding rate is greater than the entropy rate H(\overline{\rho}_p), tend to 0. If H(\overline{\rho}_p) is estimated, we can compress the coding rate to the admissible rate H(\overline{\rho}_p) with a probability close to 1. However, when we perform a naive measurement for the estimation of H(\overline{\rho}_p), the input state is demolished. By smearing the measurement, we successfully treat the trade-off between the estimation of H(\overline{\rho}_p) and the non-demolition of the input state. Our protocol can be used not only for the Schumacher's scheme but also for the compression of entangled states.

Quantum Information and ComputationNov 1, 200222 citations

Simple construction of quantum universal variable-length source coding

Masahito Hayashi, Koji Matsumoto

This paper describes the construction of a quantum universal variable-length source code in which, independent of information source, both of the average error and the probability that the coding rate is greater than the entropy rate H(/spl rho/~/sub p/), tend to 0. If H(/spl rho/~/sub p/) is estimated, and compress the coding rate to the admissible rate H(/spl rho/~/sub p/) with a probability close to 1. However, when we perform a naive measurement for the estimation of H(/spl rho/~/sub p/), the input state is demolished. By smearing the measurement, we successfully treat the trade-off between the estimation of H(/spl rho/~/sub p/) and the nondemolition of the input state. Our protocol can be used not only for the Schumacher's compression but also for the compression of entangled states.This paper describes the construction of a quantum universal variable-length source code in which, independent of information source, both of the average error and the probability that the coding rate is greater than the entropy rate H(/spl rho/~/sub p/), tend to 0. If H(/spl rho/~/sub p/) is estimated, and compress the coding rate to the admissible rate H(/spl rho/~/sub p/) with a probability close to 1. However, when we perform a naive measurement for the estimation of H(/spl rho/~/sub p/), the input state is demolished. By smearing the measurement, we successfully treat the trade-off between the estimation of H(/spl rho/~/sub p/) and the nondemolition of the input state. Our protocol can be used not only for the Schumacher's compression but also for the compression of entangled states.

IEEE International Symposium on Information Theory (ISIT)Jan 1, 20034 citations

Exponents of quantum fixed-length pure-state source coding

Masahito Hayashi

arXiv:quant-ph/0202002Physical Review ASep 27, 200248 citations

Optimal lower bound for lossless quantum block encoding

George Androulakis, Rabins Wosti

Consider a general quantum stochastic source that emits at discrete time steps quantum pure states which are chosen from a finite alphabet according to some probability distribution which may depend on the whole history. Also, fix two positive integers $m$ and $l$ . We encode any tensor product of $m$ many states emitted by the quantum stochastic source by breaking the tensor product into $m$ many blocks where each block has length $l$ , and considering sequences of $m$ many isometries so that each isometry encodes one of these blocks into the Fock space, followed by the concatenation of their images. We only consider certain sequences of such isometries that we call \lq\lq special block codes\rq\rq\ in order to ensure that the string of encoded states is uniquely decodable. We compute the minimum average codeword length of these encodings which depends on the quantum source and the integers $m$ , $l$ , among all possible special block codes. Our result extends the result of [Bellomo, Bosyk, Holik and Zozor, Scientific Reports 7.1 (2017): 14765] where the minimum was computed for one block, i.e.\ for $m=1$ . Lastly, we give a simplified non-adaptive compression technique based on constrained special block codes for general quantum stochastic sources. For quantum stationary sources in particular, we show that the minimum average codeword length per symbol computed over all constrained special block codes is equal to the von-Neumann entropy rate of the source for an asymptotically long block size.Consider a general quantum stochastic source that emits at discrete time steps quantum pure states which are chosen from a finite alphabet according to some probability distribution which may depend on the whole history. Also, fix two positive integers $m$ and $l$. We encode any tensor product of $m$ many states emitted by the quantum stochastic source by breaking the tensor product into $m$ many blocks where each block has length $l$, and considering sequences of $m$ many isometries so that each isometry encodes one of these blocks into the Fock space, followed by the concatenation of their images. We only consider certain sequences of such isometries that we call \lq\lq special block codes\rq\rq\ in order to ensure that the string of encoded states is uniquely decodable. We compute the minimum average codeword length of these encodings which depends on the quantum source and the integers $m$, $l$, among all possible special block codes. Our result extends the result of [Bellomo, Bosyk, Holik and Zozor, Scientific Reports 7.1 (2017): 14765] where the minimum was computed for one block, i.e.\ for $m=1$. Lastly, we give a simplified non-adaptive compression technique based on constrained special block codes for general quantum stochastic sources. For quantum stationary sources in particular, we show that the minimum average codeword length per symbol computed over all constrained special block codes is equal to the von-Neumann entropy rate of the source for an asymptotically long block size.

arXiv:2305.18748Quantum Information and ComputationNov 1, 20240 citations

Optimal Lower Bound of the Average Indeterminate Length Lossless Quantum Block Encoding

George Androulakis, Rabins Wosti

Consider a general quantum stochastic source that emits at discrete time steps quantum pure states which are chosen from a finite alphabet according to some probability distribution which may depend on the whole history. Also, fix two positive integers $m$ and l. We encode any tensor product of ml many states emitted by the given quantum source by breaking it into $m$ many blocks where each block has length l, and considering sequences of $m$ many isometries so that each isometry encodes one of these blocks into the Fock space, followed by the concatenation of their images. We only consider certain sequences of such isometries that we call “special block codes” in order to ensure that the string of encoded states is uniquely decodable. We compute the minimum average codeword length of these encodings which depends on the quantum source and the integers m, l, among all possible special block codes. Our result extends the result of [Bellomo, Bosyk, Holik and Zozor, Scientific Reports 7.1 (2017): 14765] where the minimum was computed for one block, i.e. for m = 1.Consider a general quantum stochastic source that emits at discrete time steps quantum pure states which are chosen from a finite alphabet according to some probability distribution which may depend on the whole history. Also, fix two positive integers$m$and l. We encode any tensor product of ml many states emitted by the given quantum source by breaking it into$m$many blocks where each block has length l, and considering sequences of$m$many isometries so that each isometry encodes one of these blocks into the Fock space, followed by the concatenation of their images. We only consider certain sequences of such isometries that we call “special block codes” in order to ensure that the string of encoded states is uniquely decodable. We compute the minimum average codeword length of these encodings which depends on the quantum source and the integers m, l, among all possible special block codes. Our result extends the result of [Bellomo, Bosyk, Holik and Zozor, Scientific Reports 7.1 (2017): 14765] where the minimum was computed for one block, i.e. for m = 1.

IEEE International Symposium on Information Theory (ISIT)Jul 7, 20240 citations

H(rho_p)

, tend to 0. If

H(rho_p)

is estimated, we can compress the coding rate to the admissible rate

H(rho_p)

with a probability close to 1. However, when we perform a naive measurement for the estimation of

H(rho_p)

, the input state is demolished. By smearing the measurement, we successfully treat the trade-off between the estimation of

H(rho_p)

and the non-demolition of the input state. Our protocol can be used not only for the Schumacher's scheme but also for the compression of entangled states.

m

and

l

. We encode any tensor product of

m

many states emitted by the quantum stochastic source by breaking the tensor product into

m

many blocks where each block has length

l

, and considering sequences of

m

m

l

, among all possible special block codes. Our result extends the result of [Bellomo, Bosyk, Holik and Zozor, Scientific Reports 7.1 (2017): 14765] where the minimum was computed for one block, i.e.\ for

m=1

. Lastly, we give a simplified non-adaptive compression technique based on constrained special block codes for general quantum stochastic sources. For quantum stationary sources in particular, we show that the minimum average codeword length per symbol computed over all constrained special block codes is equal to the von-Neumann entropy rate of the source for an asymptotically long block size.

m

and l. We encode any tensor product of ml many states emitted by the given quantum source by breaking it into

m

many blocks where each block has length l, and considering sequences of

m

many isometries so that each isometry encodes one of these blocks into the Fock space, followed by the concatenation of their images. We only consider certain sequences of such isometries that we call “special block codes” in order to ensure that the string of encoded states is uniquely decodable. We compute the minimum average codeword length of these encodings which depends on the quantum source and the integers m, l, among all possible special block codes. Our result extends the result of [Bellomo, Bosyk, Holik and Zozor, Scientific Reports 7.1 (2017): 14765] where the minimum was computed for one block, i.e. for m = 1.