Source-linked AI summary
Optimal quantum source coding with quantum information at the encoder and decoder
Jon Yard, Igor Devetak
TL;DR
The paper asks how to redistribute part of a multipartite pure quantum state while preserving its global purity. It proves an optimal protocol using communication and shared entanglement, and identifies operational consequences for quantum conditional mutual information and strong subadditivity.
Problem
The paper studies the asymptotic cost of transferring C from Alice to Bob while preserving the purity of a shared quadripartite state.
Source: §I · p. 1Method
Results
The protocol achieves optimal state-redistribution rates by matching the Luo-Devetak outer bound.
Source: §V · p. 9Takeaways & Limitations
The optimal qubit cost gives the first known operational interpretation of quantum conditional mutual information and leads to an operational proof of strong subadditivity.
Source: §I · p. 2
Abstract
from arXiv · showhide
Consider many instances of an arbitrary quadripartite pure state of four quantum systems ABCD. Alice holds the AC part of each state, Bob holds B, while D represents all other parties correlated with ABC. Alice is required to redistribute the C systems to Bob while asymptotically preserving the overall purity. We prove that this is possible using Q qubits of communication and E ebits of shared entanglement between Alice and Bob, provided that Q geq I(C;D|B)/2 and Q+E geq H(C|B), proving the optimality of the Luo-Devetak outer bound. The optimal qubit rate provides the first known operational interpretation of quantum conditional mutual information. We also show how our protocol leads to a fully operational proof of strong subadditivity and uncover a general organizing principle, in analogy to thermodynamics, that underlies the optimal rates.
I. INTRODUCTION
The introduction formulates quantum state redistribution: Alice transfers C to Bob while preserving the global state, using communication and shared entanglement. It states the optimality of the Luo-Devetak bound and highlights operational implications, including a connection to strong subadditivity.
I. INTRODUCTION
- Alice transfers C to Bob from a pure state ABCD while preserving global purity, with A and C initially held by Alice and B by Bob.D is inaccessible to both parties, and the task is considered asymptotically for many copies with fidelity approaching one.Source: §I · p. 1
- The protocol uses transmitted qubits and shared Bell-state entanglement, and the paper proves the Luo-Devetak outer bound is optimal.The entanglement cost may be negative, corresponding to generating entanglement.Source: §I · p. 1
- The optimal qubit cost gives the first known operational interpretation of quantum conditional mutual information and yields an operational proof of strong subadditivity.The proof follows from a direct coding theorem rather than a converse proof.Source: §I · p. 2
- State redistribution generalizes Schumacher compression, state merging and splitting, and entanglement concentration and dilution.The paper notes that these tasks arise when various subsystems are trivial.Source: §I · p. 2
A. Notational conventions
- Quantum conditional mutual information is nonnegative, whereas quantum conditional entropy can be negative and conditional quantities are not generally averages.Conditional quantities cannot generally be interpreted as averages unless the conditioning system is purely classical.Source: A. Notational conventions · p. 3
II. RESOURCE INEQUALITIES
The paper uses resource inequalities to express which communication and entanglement resources can simulate a target protocol. Finite inequalities give concrete examples of this resource accounting.
II. RESOURCE INEQUALITIES
- Resource inequalities provide a high-level notation for expressing the paper’s main result and intermediate protocols.The authors use an elementary formulation they consider sufficient for their purposes.Source: §II · p. 3
A. Finite resource inequalities
- A finite resource inequality means that the resources on its left can simulate those on its right.Available resources can be added, so both inputs may be treated as available together.Source: A. Finite resource inequalities · p. 3
- A qubit channel can distribute an ebit, while an ebit plus two classical bit channels can simulate a qubit channel.The passage presents these as finite resource inequalities for entanglement distribution and teleportation.Source: A. Finite resource inequalities · p. 3
B. Approximate resource inequalities
Approximate resource inequalities formalize simulation with error. The paper connects trace distance and fidelity as measures of approximation and defines the error by testing simulated resources on an entangled input.
B. Approximate resource inequalities
- Fidelity and trace distance are equivalent measures when approximating states to arbitrarily high accuracy.The passage invokes Uhlmann’s theorem to characterize fidelity through purifications.Source: B. Approximate resource inequalities · p. 3
- The choice between fidelity and trace distance does not affect the asymptotic analysis because the error can be made arbitrarily small.The paper treats the two measures as equivalent for this purpose.Source: B. Approximate resource inequalities · p. 3
- An approximate resource inequality requires the simulated and target global states to be ε-close in trace distance or fidelity.The test applies each target channel to half of a maximally entangled state, with target quantum states also included in the global state.Source: B. Approximate resource inequalities · p. 3
C. Asymptotic resource inequalities
Asymptotic resource inequalities describe achievable resource rates with arbitrarily small error for sufficiently large blocks. The notation also accommodates sublinear inputs and resources generated rather than consumed.
C. Asymptotic resource inequalities
- An asymptotic resource inequality represents achievable output rates for sufficiently large n and every positive error tolerance.The formal statement permits rates arbitrarily close to the specified output rates.Source: C. Asymptotic resource inequalities · p. 3
- The formalism allows a sublinear amount o(n) of an input resource.The notation writes this as a resource with an o coefficient on the left side of an inequality.Source: C. Asymptotic resource inequalities · p. 3
- Negative rates on the left mean that the corresponding resources are generated instead of consumed.Formally, such resources are negated and moved to the right side.Source: C. Asymptotic resource inequalities · p. 3
- The cancellation lemma allows an input resource used at a greater rate than its output rate to be replaced by a sublinear input requirement.The stated implication applies when R_in > R_out ≥ 0.Source: C. Asymptotic resource inequalities · p. 3
D. Distributed states
The section sets notation for distributed states and their asymptotic resource inequalities. This notation describes Alice’s transfer of C to Bob while preserving its purification.
D. Distributed states
- State redistribution is represented by ψ_AC|B before the protocol and ψ_A|CB afterward, with rates evaluated on an arbitrary purification into D.Alice initially holds AC and ends holding A; Bob initially holds B and ends holding CB.Source: D. Distributed states · p. 4
- For Schumacher compression, ψ_C|∅ denotes Alice holding C from many i.i.d. copies of a fixed purification while Bob holds nothing.The corresponding ψ_∅|C notation places those same C systems with Bob.Source: D. Distributed states · pp. 3–4
- The coding theorem proves the resource region’s direct part at its corner point; the converse that rates must lie in the region is established in [4].The section states that [q →q] ≥ [qq] makes proving the corner point sufficient for the direct part.Source: D. Distributed states · p. 4
III. PROOF OF THEOREM 1
The proof of Theorem 1 uses an auxiliary protocol for transferring C, then combines coherent-channel simulation with cancellation. The cancellation step has sublinear resource costs in specified boundary cases.
III. PROOF OF THEOREM 1
- To prove Theorem 1, an auxiliary protocol is demonstrated that transfers Cn to Bob and has the desired net communication and entanglement cost.Source: §III · p. 4
- Cancellation requires sublinear entanglement when I(B; C) ≥ I(A; C), and sublinear communication when I(C; BD) = I(C; D).The latter equality is identified with saturation of strong subadditivity; the extra entanglement cost can be absorbed into communication because [q →q] ≥ [qq].Source: §III · p. 4
- Theorem 2 is proved using a protocol that simulates coherent channels, with repeated concatenation yielding the stated asymptotic resource inequality.The paper identifies [q →qq] as a quantum feedback channel that is an isometry from Alice to Alice and Bob.Source: §III · p. 4
A. Proof of Theorem 3
Theorem 3 is proved by applying typical-state bounds to many copies and using the robust one-shot redistribution protocol. The construction’s error bound can decrease exponentially with the number of copies.
A. Proof of Theorem 3
- Finally, for all of the above bounds, there is a δ-independent constant c > 0 such that one may take ϵ = 2^−ncδ².Source: A. Proof of Theorem 3 · p. 5
- The typical projection onto C^n_δ succeeds with probability at least 1 − ϵ, and its support dimension is bounded by 2^{n[H(T)±δ]} in the theorem’s typical-subspace statement.After a failed projection, Alice declares an error and aborts.Source: A. Proof of Theorem 3 · p. 5
- The one-shot protocol supplies encoding isometries and a decoding isometry so Alice can send the compressed system S and Bob can recover Cδ.Bob can identify Alice’s encoding by measuring Kout; a controlled isometry instead simulates a coherent channel from Kin to KinKout.Source: A. Proof of Theorem 3 · p. 5
- The proof bounds the overall fidelity below by 1 − 6ϵ^(1/8) when ϵ is sufficiently small.The asymptotic resource inequality of Theorem 3 follows because the bound holds for arbitrarily small ϵ, including exponentially vanishing ϵ.Source: A. Proof of Theorem 3 · p. 6
B. Proof of Theorem 4
Theorem 4’s proof builds a robust one-shot redistribution protocol from decoupling and Bob’s side information. Random encodings let Bob identify the encoding, and the construction can be made coherent.
B. Proof of Theorem 4
- The robust one-shot decoupling lemma underpins the proof, and applying the method of types to it gives the corresponding merging and splitting rates.The circuits are related by time reversal and swapping A and B.Source: B. Proof of Theorem 4 · p. 6, Figure 2 · p. 6
- When Alice’s random encoding decouples bA from the reference, Bob can use a local isometry to reconstruct C and obtain the other half of the generated entanglement.The paper relates bA being maximally entangled with BS to bB being approximately maximally mixed and decoupled from R.Source: B. Proof of Theorem 4 · p. 6
- Splitting is obtained by reversing a merging circuit and swapping A and B; the paper distinguishes these fully quantum tasks from merging with classical communication.The latter terminology is reserved for protocols in the sense of [7], [10].Source: B. Proof of Theorem 4 · p. 6
- Bob’s side information lets him distinguish among random encodings, so Alice can piggyback classical information on the transmitted qubits.A controlled isometry makes the construction coherent, allowing a superposition of encodings to simulate coherent channels.Source: B. Proof of Theorem 4 · p. 7, Figure 3 · p. 7
IV. AN OPERATIONAL PROOF OF STRONG SUBADDITIVITY
The redistribution result yields an operational proof of strong subadditivity: if conditional mutual information were negative, a fidelity bound would contradict the protocol’s achievable fidelity.
IV. AN OPERATIONAL PROOF OF STRONG SUBADDITIVITY
- The proof starts from the redistribution theorem and discards resources on its right-hand side to obtain the inequality used for the operational argument.The section then considers an attempted simulation of an identity channel, assisted by prior entanglement.Source: §IV · pp. 8–9
- The protocol implies I(C; D|B) ≥ 0, since a negative value would force entanglement fidelity toward zero despite the protocol attaining fidelity arbitrarily close to 1.The contradiction holds for sufficiently large n.Source: §IV · pp. 8–9
- If strong subadditivity failed, the fidelity upper bound would decay exponentially with n, whereas the protocol guarantees fidelity arbitrarily close to 1 for sufficiently large n.This contradiction establishes nonnegativity without relying on the entropy inequality itself.Source: §IV · pp. 8–9
V. DISCUSSION
The discussion frames state redistribution as a general fully quantum source-coding task with optimal qubit and entanglement rates. It connects those rates to correlations, strong subadditivity, and time-reversal symmetry, while noting open questions about sublinear costs and broader applications.
V. DISCUSSION
- State redistribution is the most general unidirectional two-terminal fully quantum source-coding problem, and its optimal protocol matches the Luo-Devetak outer bound [4].The optimal communication rate gives an operational interpretation of half the conditional mutual information; with classical communication and teleportation, the paper describes an interpretation without the factor of one half.Source: §V · p. 9
- The protocol’s essential reversibility explains why the communication cost is the same in both directions and why I(C; D|A) = I(C; D|B) for a quadripartite pure state.The Schumacher compression step succeeds with high probability and is essentially reversible.Source: §V · p. 9
- The discussion interprets quantum conditional mutual information as correlations between C and D from either A’s or B’s perspective.The protocol constructed for Theorem 3 consists entirely of isometries, while later steps for Theorem 1 add at most a sublinear amount of nonunitarity.Source: §V · p. 10
- With both parties lacking side information, redistribution recovers Schumacher compression; removing Alice’s or Bob’s side information gives optimal state-merging or state-splitting protocols, respectively.The new protocol is self-dual under time reversal, and successive redistribution can also be performed optimally using the optimal protocol.Source: §V · p. 10, Figure 4 · p. 10
- When D is trivial, strong subadditivity is saturated and any positive communication rate is achievable; further trivializing A or B recovers entanglement concentration or dilution.When both A and B are nontrivial, the protocol gives an alternate route through concentrating and then diluting entanglement, with net cost H(A)−H(B).Source: §V · p. 10
- The paper leaves sublinear resource costs unaccounted for; exponentially small error may not be possible when only sublinear resources are used.The authors note that finer accounting could clarify transformations between non-maximally entangled states.Source: §V · p. 10
- The authors suggest that one-shot techniques may apply beyond i.i.d. sources, including to ground states of many-body Hamiltonians, and connect redistribution to topological entanglement entropy.They describe these connections as topics for future work.Source: §V · p. 10
- The dynamic-potential difference gives the optimal qubit rate, interpreted as the reference correlations Alice must transfer to Bob; the static-potential difference gives the optimal ebit rate.The static difference is interpreted as the excess correlation between Alice and Bob involved in moving between configurations.Source: §V · p. 11
NOTE ADDED
After the preprint appeared, a one-shot version of the main result was found along the lines of Figure 5 [30], [31].
NOTE ADDED
- A one-shot version of the main result was found after the preprint appeared [30], [31].The passage says it follows the lines of Figure 5.Source: NOTE ADDED · p. 11
- The reported one-shot version follows the lines of Figure 5 [30], [31].The passage does not state further results about that version.Source: NOTE ADDED · p. 11
- The passage attributes the one-shot version to references [30] and [31].It notes that this finding came after the preprint became available.Source: NOTE ADDED · p. 11
APPENDIX
The appendix collects auxiliary proofs for Theorem 4, including a robust decoupling proof that relies on a non-robust version from [12].
APPENDIX
- The robust decoupling lemma’s proof relies on the non-robust version from [12].The appendix places this among auxiliary results used in proving Theorem 4.Source: APPENDIX · p. 11
- The one-shot decoupling lemma starts with a density matrix on CE and a unitary decomposition of C into S, b, and B.For each unitary U on C, it defines the transformed state as WU applied to the input state and its adjoint.Source: APPENDIX · p. 11
- The robust decoupling proof bounds a Haar-measure integral using convexity, trace-norm properties, and the assumed closeness of the states.The appendix then obtains the theorem by applying the one-shot decoupling lemma to that integral.Source: APPENDIX · pp. 11–12