REVIEW
The private classical capacity and quantum capacity of a quantum channel
Quantum Physics · Submitted 18 Apr 2003 · 42 references
Pith's verdict: A landmark result in quantum Shannon theory: exact private, secret-key, entanglement-generation, and quantum capacities, with proofs that hold up under scrutiny.
T0 · AI review
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Referee report
The paper
The paper claims
Every capacity of a memoryless quantum channel — private classical, secret-key, entanglement, and quantum — is claimed to equal one regularized coherent-information or private-information expression, and forward public classical talk is not
The reading
What carries the argument
The coherent information $I_c(ho,N) = H(N(ho)) - H(ho_E)$, computed with the environment $E$ of an isometric extension of the channel, is the object that carries everything. It is the quantum analogue of a mutual information between input and output with the eavesdropper's share subtracted, and it is non-increasing under post-processing, which is what makes the converses short. The private information $I_p$ is the coherent information with a worst-case penalty for the ensemble decomposition of the input, capturing the fact that Eve may hold correlations with the state preparation itself. The technical engine for the direct coding theorem is an operator Chernoff bound plus Winter's gentle-operator lemma, applied to conditionally typical subspaces, producing codes whose outputs are nearly indistinguishable to Eve while Bob decodes with high probability.
Extended reading notes
Core claim
The paper proves two matching formulas. For private classical transmission and secret-key generation over a quantum channel $N$ it gives $C_p(N) = K(N) = lim_{l o infty} (1/l) max_ho I_p(ho, N^{otimes l})$, where the private information $I_p$ is the coherent information minus the smallest average coherent information over decompositions of the input state into an ensemble; and it shows $K(N) ge C_p(N)$ with the reverse direction supplied by an explicit code that never uses the public channel. For entanglement generation and quantum information transmission it gives $E(N) = Q(N) = ilde{Q}(N) = lim_{l o infty} (1/l) max_ho I_c(ho, N^{otimes l})$. The bridge between the two is a coherent version of the private code: take a pure-state eigen-decomposition of the input density operator, replace the classical message index by an entangled superposition, and undo the decoder's measurement unitarily, so that Bob's identification of the message becomes Bob's share of a maximally entangled state. The converse for the quantum capacity then follows in a few lines from the quantum data processing inequality plus a continuity bound on conditional entropy.
Load-bearing premise
The load-bearing premise is that a private classical code built from a pure-state decomposition of the input density operator can be converted, by replacing labels with superpositions and the decoder's measurement with a unitary, into a genuine entanglement-generating code; the paper itself notes this conversion appears to work only for pure decompositions, since mixed ones would require the decoder to act jointly on systems held by different parties.
Editorial extensions
If this is right
- Secret-key generation and private classical transmission over a quantum channel have the same capacity, so a channel useful for one is useful for the other at the same rate.
- A free forward public classical channel from Alice to Bob does not raise either the private or the secret-key capacity, in contrast with the two-way case where it is known to help.
- The quantum capacity theorem gets a new direct-coding proof and a substantially shorter converse, since entanglement generation is squeezed between private coding and teleportation.
- Entanglement generation and quantum information transmission have identical capacities, so any channel that generates pure entanglement at rate $R$ can transmit quantum states at rate $R$ via teleportation.
- The capacity formulas are multi-letter, so even though they are exact they are not directly computable for a general channel.
Reading between the lines
- If the private capacity can be strictly larger than the entanglement capacity for some channel, that channel would be operationally useful for key distribution but useless for quantum communication — a clean separation between secrecy and coherence that the paper leaves open as a question rather than settles.
- The coherent-conversion step is the template for a general trick: any coding theorem phrased in terms of labels and mixtures can be re-read as one phrased in terms of superpositions, provided the ensemble is pure. That suggests a systematic route from classical-quantum coding results to entanglement results, not just this one.
- Because the formulas are regularized, the interesting follow-up question is which channel classes collapse the limit to a single letter; the paper notes degradable channels do, and that the same is known for the private capacity of classical-quantum wire-tap channels but not for the private capacity of quantum channels.
- The proof's reliance on a pure-state decomposition hints that for channels where the optimal input has an essentially mixed structure the coherent conversion may need extra quantum resources, which would show up as a gap between the private and entanglement capacities.
Original abstract
A formula for the capacity of a quantum channel for transmitting private classical information is derived. This is shown to be equal to the capacity of the channel for generating a secret key, and neither capacity is enhanced by forward public classical communication. Motivated by the work of Schumacher and Westmoreland on quantum privacy and quantum coherence, parallels between private classical information and quantum information are exploited to obtain an expression for the capacity of a quantum channel for generating pure bipartite entanglement. The latter implies a new proof of the quantum channel coding theorem and a simple proof of the converse. The coherent information plays a role in all of the above mentioned capacities.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
A formula for the capacity of a quantum channel for transmitting private classical information is derived. This is shown to be equal to the capacity of the channel for generating a secret key, and neither capacity is enhanced by forward public classical communication. Motivated by the work of Schumacher and Westmoreland on quantum privacy and quantum coherence, parallels between private classical information and quantum information are exploited to obtain an expression for the capacity of a quantum channel for generating pure bipartite entanglement. The latter implies a new proof of the quantum channel coding theorem and a simple proof of the converse. The coherent information plays a role in all of the above mentioned capacities.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | A formula for the capacity of a quantum channel for transmitting private classical information is derived. | result 0.66 | 0.94 | 0.41 | asserted 1.00 |
| s1 | This is shown to be equal to the capacity of the channel for generating a secret key, and neither capacity is enhanced by forward public classical communication. | result 1.00 | 0.88 | 0.30 | asserted 1.00 |
| s2 | Motivated by the work of Schumacher and Westmoreland on quantum privacy and quantum coherence, parallels between private classical information and quantum information are exploited to obtain an expression for the capacity of a quantum channel for generating pure bipartite entanglement. | contribution 0.58 | 0.90 | 0.15 | asserted 0.99 |
| s3 | The latter implies a new proof of the quantum channel coding theorem and a simple proof of the converse. | result 0.42 | 0.90 | 0.11 | asserted 1.00 |
| s4 | The coherent information plays a role in all of the above mentioned capacities. | result 0.55 | 0.70 | 0.16 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v6
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
References
How later papers used it
73 later papers cite it
2 later papers by other authors are listed below: 2 used its method or tool.
Uses its method or tool 2 of 4 papers with this label
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Constructed a quantum polar code achieving the coherent information of a degradable quantum channel by exploiting this paper's ideas for quantum coding.
Polar codes for degradable quantum channels (2011) · 0.80 · paraphrased: the licence does not allow quoting
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Operated its PPM scheme for private communication in a coherent fashion analogous to that of this paper.
Quantum Channel Capacities Per Unit Cost (2017) · 0.94 · paraphrased: the licence does not allow quoting
How the literature uses it
72 arXiv papers cite this in 135 sentences
Papers citing it · 76 sentences without a confident label
Builds on it 1 paper
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This kind of task was defined and developed in [ref] , albeit with the assistance of a particular kind of $k$-extendible channel and only the case $k=2$ was considered there, generalizing the usual notion of entanglement distillation and quantum communication protocols from [ref] .
Extendibility limits the performance of quantum processors · Introduction · 0.85
Uses its method or tool 4 papers
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From such a scheme for private classical communication over a quantum wiretap channel with classical environment, we can readily construct a quantum polar code achieving the coherent information of a degradable quantum channel by exploiting Devetak's ideas for quantum coding [ref] and the recent quantum successive cancellation decoder from Ref. [ref] .
Polar codes for degradable quantum channels · Quantum polar codes · 0.80
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To establish the theorem, it thus suffices to prove the following converse inequality [equation] To do so, we make use of several ideas from [ref] .
Energy-constrained private and quantum capacities of quantum channels · Energy-constrained quantum and private capacity of degradable channels · 0.94
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To do this, we will need the following lower bound for the quantum capacity [ref] , called the coherent information: [equation] where the entropies are evaluated on the state $(I)(ϕ)$.
Extensive nonadditivity of privacy · Introduction · 0.98
Compares with it 1 paper
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Note that the above definition of secret-key transmission combines the error probability and the security parameter into a single parameter $ε$, in contrast to the definitions from [ref] .
Converse bounds for private communication over quantum channels · Secret-key transmission codes · 0.98
Cites it as background 41 papers
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Consider a quantum wiretap channel $𝒩^A^'→BE$ [ref] with a qubit input system $A^'$ for the sender Alice, an output system $B$ for the legitimate receiver Bob, and an output system $E$ for the wiretapper Eve.
Polar codes for private classical communication · Classical-quantum channels for complementary variables · 0.97
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With $H(X)_ω$ the von Neumann entropy of the reduced state on $X$, the coherent information is [equation] the second equality holding because $ω_RBE$ is pure, and [equation] The Lloyd-Shor-Devetak theorem gives $Q(𝒩)=lim_k1/k Q^(1)(𝒩^⊗k)$ [ref] , which collapses to $Q(𝒩)=Q^(1)(𝒩)$ for degradable channels [ref] and to $Q(𝒩)=0$ for antidegradable ones.
Sharp Quantum Capacity Thresholds: Exponential Strong Converses for Degradable and Antidegradable Channels · Coherent information · 0.88
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The private capacity of a quantum wiretap channel was defined and characterized in [ref] .
Converse bounds for private communication over quantum channels · Introduction · 0.90
Discussion
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