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

International Journal of Quantum Information

HOW LONG CAN IT TAKE FOR A QUANTUM CHANNEL TO FORGET EVERYTHING?

André Ahlbrecht, FLORIAN RICHTER, Reinhard F. Werner

Published Aug 1, 2012• 2 citations
Publisher

Abstract

We investigate quantum channels, which after a finite number k of repeated applications erase all input information, i.e. channels whose kth power (but no smaller power) is a completely depolarizing channel. We show that on a system with Hilbert space dimension d, the order is bounded by k ≤ d2 - 1, and give an explicit construction scheme for such channels. We also consider strictly forgetful memory channels, i.e. channels with an additional input and output in every step, which after exactly k steps retain no information about the initial memory state. We establish an explicit representation for such channels showing that the same bound applies for the memory depth k in terms of the memory dimension d.

DOI
10.1142/s0219749912500578

Related papers

How long can it take for a quantum channel to forget everything?

André Ahlbrecht, Richter, Florian, Reinhard F. Werner

We investigate quantum channels, which after a finite number $k$ of repeated applications erase all input information, i.e., channels whose $k$-th power (but no smaller power) is a completely depolarizing channel. We show that on a system with Hilbert space dimension $d$, the order is bounded by $k\leq d^2-1$, and give an explicit construction scheme for such channels. We also consider strictly forgetful memory channels, i.e., channels with an additional input and output in every step, which after exactly $k$ steps retain no information about the initial memory state. We establish an explicit representation for such channels showing that the same bound applies for the memory depth $k$ in terms of the memory dimension $d$.

arXiv:1205.0693•May 3, 2012•0 citations

Quantum channels with memory

Dennis Kretschmann, Reinhard F. Werner

We present a general model for quantum channels with memory and show that it is sufficiently general to encompass all causal automata: any quantum process in which outputs up to some time $t$ do not depend on inputs at times ${t}^{\ensuremath{'}}>t$ can be decomposed into a concatenated memory channel. We then examine and present different physical setups in which channels with memory may be operated for the transfer of (private) classical and quantum information. These include setups in which either the receiver or a malicious third party have control of the initializing memory. We introduce classical and quantum channel capacities for these settings and give several examples to show that they may or may not coincide. Entropic upper bounds on the various channel capacities are given. For forgetful quantum channels, in which the effect of the initializing memory dies out as time increases, coding theorems are presented to show that these bounds may be saturated. Forgetful quantum channels are shown to be open and dense in the set of quantum memory channels.

arXiv:quant-ph/0502106•Physical Review A•Dec 16, 2005•235 citations

Classical and entanglement-assisted capacity of a qubit depolarizing memory channel

Jaideep Mulherkar

We study the classical and entanglement-assisted capacity of a forgetful quantum memory channel that randomly switches between two qubit depolarizing channels. We show that when the input consists of two qubits then depending on channel parameters either the maximally entangled input states or product input states achieve the two-use classical capacity. We conjecture that as the number of input qubits is increased the classical capacity approaches the product state capacity for all values of the parameters. We also derive an expression for the entanglement-assisted classical capacity of this quantum memory channel in terms of the entropy rate of a Markov chain.

arXiv:1503.07075•International Journal of Quantum Information•Apr 1, 2016•3 citations

Causal quantum-channel simulation: memory beyond entropy

Seth Douglas, Nidhal Mghirbi

How much memory does it take to apply the same quantum channel many times, when each output must be released before the next input arrives and every retained qubit counts? One-use invariants do not decide it. Two explicit channels on dimension 1664 have identical normalized Choi spectra (32 eigenvalues 1/32), the same maximal complementary entropy 5, and zero asymptotic purity cost. At the same optimal exchange rate, one is implemented exactly with five memory qubits for any number of uses, while the other needs Omega(n^(1/(2 log_2(107)+2))) memory qubits for n uses, at every fixed error at most 1/16 and whatever purity is available; memory O(n^(1/3) log^(4/3) n) suffices for it. The hard channel is built from Houghton's group H_3. Behind the example is a general theory: the achievable exchange and purity rates, the purity cost kappa, which vanishes exactly on the closure of finite tracial factorizations, and memory bounds from how well a channel's environment is approximated by finite ones, and, for channels built from finite presentations, from the approximation profile of the group. Version 1.1 adds where the expensive channels lie: on the relative boundary of the closure; any fixed depolarizing admixture brings them down to polylogarithmic memory at the rank rate; on a face containing the Houghton channel a single moment of a trace on the group decides expense; and one free fermion per site already forces polynomial memory. Version 1.1.1 adds a map of what the main separation depends on (published inputs, companion results and new arguments), sets finite-budget and asymptotic-rate statements side by side, cites the companion papers at their published versions, pins the tested software versions, and makes every check refuse to run under optimized Python. The mathematical results are unchanged. Version 1.1.2 corrects attributions and makes two headline statements precise. The separation of Theorem A is stated at the exchange budget 5n/2, and the channels that need polynomial memory at every exchange rate are those with purity limited to O(log n). Credits added or corrected: Bisio, D'Ariano, Perinotti and Chiribella, and Bisio, D'Ariano, Perinotti and Sedlak; Boes et al. for the square-root dephasing construction; Lie, Son, Boes, Ng and Wilming for the exact case of the tracialization lemma; Faist, Berta and Brandao for the relation to the thermodynamic capacity; Haagerup, Musat and Rordam for the moment criteria; Thoma, Johnson, Watrous, Horodecki-Oppenheim-Winter and Lie-Jeong; Cornulier for the Baumslag-Solitar bound. Known results are moved out of numbered statements. The mathematical results and the numbering are unchanged. All operational bounds use complete-experiment error against adaptive observers with quantum references. The general rate-region achievability result uses a cited closed-device theorem; the explicit separation does not. This preprint archive includes the PDF, TeX source, finite verification scripts, and scoped Lean arithmetic proofs. The artifacts do not formally verify the entire manuscript; their scope and external dependencies are documented in verification/README.md.

•Sep 29, 2026•0 citations

Forgetfulness of continuous Markovian quantum channels

Cosmo Lupo, Laleh Memarzadeh, Stefano Mancini

The notion of forgetfulness, used in discrete quantum memory channels, is slightly weakened in order to be applied to the case of continuous channels. This is done in the context of quantum memory channels with Markovian noise. As a case study, we apply the notion of weak forgetfulness to a bosonic memory channel with additive noise. A suitable encoding and decoding unitary transformation allows us to unravel the effects of the memory, hence the channel capacities can be computed using known results from the memoryless setting.

arXiv:0907.1544•Physical Review A•Oct 27, 2009•15 citations

Lower bounds on the quantum capacity and highest error exponent of general memoryless channels

M. Hamada

Tradeoffs between the information rate and fidelity of quantum error-correcting codes are discussed. Quantum channels to be considered are those subject to independent errors and modeled as tensor products of copies of a general completely positive (CP) linear map, where the dimension of the underlying Hilbert space is a prime number. On such a quantum channel, the highest fidelity of a quantum error-correcting code of length n and rate R is proven to be lower-bounded by 1-exp[-nE(R)+o(n)] for some function E(R). The E(R) is positive below some threshold R/sub 0/, a direct consequence of which is that R/sub 0/ is a lower bound on the quantum capacity. This is an extension of the author's earlier result. While the earlier work states the result for the depolarizing channel and a slight generalization of it (Pauli channels), the result of this work applies to general discrete memoryless channels, including channel models derived from a physical law of time evolution.

arXiv:quant-ph/0112103•IEEE Transactions on Information Theory•Sep 1, 2002•24 citations
We investigate quantum channels, which after a finite number k of repeated applications erase all input information, i.e. channels whose kth power (but no smaller power) is a completely depolarizing channel. We show that on a system with Hilbert space dimension d, the order is bounded by k ≤ d2 - 1, and give an explicit construction scheme for such channels. We also consider strictly forgetful memory channels, i.e. channels with an additional input and output in every step, which after exactly k steps retain no information about the initial memory state. We establish an explicit representation for such channels showing that the same bound applies for the memory depth k in terms of the memory dimension d.
We investigate quantum channels, which after a finite number kkk of repeated applications erase all input information, i.e., channels whose kkk-th power (but no smaller power) is a completely depolarizing channel. We show that on a system with Hilbert space dimension ddd, the order is bounded by k≤d2−1k\leq d^2-1k≤d2−1, and give an explicit construction scheme for such channels. We also consider strictly forgetful memory channels, i.e., channels with an additional input and output in every step, which after exactly kkk steps retain no information about the initial memory state. We establish an explicit representation for such channels showing that the same bound applies for the memory depth kkk in terms of the memory dimension ddd.
We present a general model for quantum channels with memory and show that it is sufficiently general to encompass all causal automata: any quantum process in which outputs up to some time ttt do not depend on inputs at times ${t}^{\ensuremath{'}}>t$ can be decomposed into a concatenated memory channel. We then examine and present different physical setups in which channels with memory may be operated for the transfer of (private) classical and quantum information. These include setups in which either the receiver or a malicious third party have control of the initializing memory. We introduce classical and quantum channel capacities for these settings and give several examples to show that they may or may not coincide. Entropic upper bounds on the various channel capacities are given. For forgetful quantum channels, in which the effect of the initializing memory dies out as time increases, coding theorems are presented to show that these bounds may be saturated. Forgetful quantum channels are shown to be open and dense in the set of quantum memory channels.
We study the classical and entanglement-assisted capacity of a forgetful quantum memory channel that randomly switches between two qubit depolarizing channels. We show that when the input consists of two qubits then depending on channel parameters either the maximally entangled input states or product input states achieve the two-use classical capacity. We conjecture that as the number of input qubits is increased the classical capacity approaches the product state capacity for all values of the parameters. We also derive an expression for the entanglement-assisted classical capacity of this quantum memory channel in terms of the entropy rate of a Markov chain.
How much memory does it take to apply the same quantum channel many times, when each output must be released before the next input arrives and every retained qubit counts? One-use invariants do not decide it. Two explicit channels on dimension 1664 have identical normalized Choi spectra (32 eigenvalues 1/32), the same maximal complementary entropy 5, and zero asymptotic purity cost. At the same optimal exchange rate, one is implemented exactly with five memory qubits for any number of uses, while the other needs Omega(n^(1/(2 log_2(107)+2))) memory qubits for n uses, at every fixed error at most 1/16 and whatever purity is available; memory O(n^(1/3) log^(4/3) n) suffices for it. The hard channel is built from Houghton's group H_3. Behind the example is a general theory: the achievable exchange and purity rates, the purity cost kappa, which vanishes exactly on the closure of finite tracial factorizations, and memory bounds from how well a channel's environment is approximated by finite ones, and, for channels built from finite presentations, from the approximation profile of the group. Version 1.1 adds where the expensive channels lie: on the relative boundary of the closure; any fixed depolarizing admixture brings them down to polylogarithmic memory at the rank rate; on a face containing the Houghton channel a single moment of a trace on the group decides expense; and one free fermion per site already forces polynomial memory. Version 1.1.1 adds a map of what the main separation depends on (published inputs, companion results and new arguments), sets finite-budget and asymptotic-rate statements side by side, cites the companion papers at their published versions, pins the tested software versions, and makes every check refuse to run under optimized Python. The mathematical results are unchanged. Version 1.1.2 corrects attributions and makes two headline statements precise. The separation of Theorem A is stated at the exchange budget 5n/2, and the channels that need polynomial memory at every exchange rate are those with purity limited to O(log n). Credits added or corrected: Bisio, D'Ariano, Perinotti and Chiribella, and Bisio, D'Ariano, Perinotti and Sedlak; Boes et al. for the square-root dephasing construction; Lie, Son, Boes, Ng and Wilming for the exact case of the tracialization lemma; Faist, Berta and Brandao for the relation to the thermodynamic capacity; Haagerup, Musat and Rordam for the moment criteria; Thoma, Johnson, Watrous, Horodecki-Oppenheim-Winter and Lie-Jeong; Cornulier for the Baumslag-Solitar bound. Known results are moved out of numbered statements. The mathematical results and the numbering are unchanged. All operational bounds use complete-experiment error against adaptive observers with quantum references. The general rate-region achievability result uses a cited closed-device theorem; the explicit separation does not. This preprint archive includes the PDF, TeX source, finite verification scripts, and scoped Lean arithmetic proofs. The artifacts do not formally verify the entire manuscript; their scope and external dependencies are documented in verification/README.md.
The notion of forgetfulness, used in discrete quantum memory channels, is slightly weakened in order to be applied to the case of continuous channels. This is done in the context of quantum memory channels with Markovian noise. As a case study, we apply the notion of weak forgetfulness to a bosonic memory channel with additive noise. A suitable encoding and decoding unitary transformation allows us to unravel the effects of the memory, hence the channel capacities can be computed using known results from the memoryless setting.
Tradeoffs between the information rate and fidelity of quantum error-correcting codes are discussed. Quantum channels to be considered are those subject to independent errors and modeled as tensor products of copies of a general completely positive (CP) linear map, where the dimension of the underlying Hilbert space is a prime number. On such a quantum channel, the highest fidelity of a quantum error-correcting code of length n and rate R is proven to be lower-bounded by 1-exp[-nE(R)+o(n)] for some function E(R). The E(R) is positive below some threshold R/sub 0/, a direct consequence of which is that R/sub 0/ is a lower bound on the quantum capacity. This is an extension of the author's earlier result. While the earlier work states the result for the depolarizing channel and a slight generalization of it (Pauli channels), the result of this work applies to general discrete memoryless channels, including channel models derived from a physical law of time evolution.