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Relating the Resource Theories of Entanglement and Quantum Coherence
Eric Chitambar, Min-Hsiu Hsieh
TL;DR
The paper addresses the unresolved optimal rate for distilling eCoBits from pure states under LIOCC while relating coherence and entanglement formation and distillation. It develops rate characterizations and an LIOCC monotone, and proves an equivalence between ordinary and LIOCC entanglement distillation for mixed states.
Problem
The optimal rate at which eCoBits can be distilled from a pure state using LIOCC remains unknown.
Method
The paper characterizes achievable and optimal coherence-entanglement rate triples for pure-state formation and distillation and introduces the function CL as an LIOCC monotone.
Results
A mixed state ρAB has distillable entanglement if and only if entanglement can be distilled using LIOCC.
Takeaways & Limitations
The results establish LIOCC as sufficient for entanglement distillation from mixed states and provide operational constraints for pure-state coherence-entanglement resource conversion.
Takeaways & Limitations
The optimal eCoBit distillation rate for pure states under LIOCC remains unresolved.
Abstract
from arXiv · showhide
Quantum coherence and quantum entanglement represent two fundamental features of non-classical systems that can each be characterized within an operational resource theory. In this paper, we unify the resource theories of entanglement and coherence by studying their combined behavior in the operational setting of local incoherent operations and classical communication (LIOCC). Specifically we analyze the coherence and entanglement trade-offs in the tasks of state formation and resource distillation. For pure states we identify the minimum coherence-entanglement resources needed to generate a given state, and we introduce a new LIOCC monotone that completely characterizes a state's optimal rate of bipartite coherence distillation. This result allows us to precisely quantify the difference in operational powers between global incoherent operations, LIOCC, and local incoherent operations \textit{without} classical communication. Finally, a bipartite mixed state is shown to have distillable entanglement if and only if entanglement can be distilled by LIOCC, and we strengthen the well-known Horodecki criterion for distillability.
Distance Measures
The paper defines trace-distance-based error and fidelity measures, then uses standard continuity bounds to control entropy changes and gentle measurements.
- Distance and approximation: The trace distance is defined from the trace norm and bounds approximation errors between quantum states.The paper writes ρ ≈_ε σ when their trace distance is at most ε.
- Distance and approximation: Fidelity provides an alternative comparison between states and is related to trace distance through standard inequalities.For pure σ, the paper notes that a stronger lower bound is available.
- Entropy continuity: Fannes–Audenaert continuity bounds quantify entropy differences using trace distance and the binary entropy function.The bound is |S(ρ) − S(σ)| ≤ Dtr(ρ,σ) log(d−1) + h(Dtr(ρ,σ)).
- Gentle measurements: The gentle measurement lemma shows that a measurement with high acceptance probability causes only limited disturbance to a subnormalized state.The paper states the lemma for 0 ≤ X ≤ I and tr(ρX) ≥ 1−ε.
Types, Typical Sequences, Channel Coding
This section introduces types and typical sequences, then develops classical–quantum channel notation and coding quantities used in the paper’s asymptotic constructions.
- Types and typicality: An empirical type records symbol frequencies in a sequence, and its type class contains all sequences with that frequency distribution.Typical sequences are defined by requiring their empirical type to be sufficiently close to a target distribution.
- Types and typicality: For i.i.d. samples, typical sequences occur with high probability as the block length grows.The paper also records standard cardinality and entropy properties of typical sets.
- Quantum typicality: The computational basis is taken as the incoherent basis, and empirical types induce corresponding type projectors on tensor-product quantum systems.These projectors connect classical sequence types with quantum block structure.
- Channel models: CQ and CC channels respectively map classical basis states to quantum states or classical mixtures, with associated Holevo and mutual-information quantities.The paper denotes the corresponding channels by W_CQ and W_CC.
- Channel coding: A channel code assigns codewords to messages and evaluates decoding performance through the induced output states.The construction uses tensor-product codewords for n channel uses.
Coding Theorems
The coding theorems establish large families of channel codes that cover almost all elements of each type class, providing the combinatorial foundation for later resource protocols.
- Channel-code covering: Randomly sampled codebooks can be organized into families whose sizes are governed by H(X) and I(X:B).The construction uses L = ⌈2^n(H(X)−I(X:B)+2δ)⌉ codebooks, each with C = ⌊2^n(I(X:B)−δ)⌋ codewords.
- Channel-code covering: For sufficiently large n, a fraction 1−3ε of a type class can be covered by channel-code families.The result follows by combining random coding with an evenness argument controlling codeword multiplicities.
- Channel-code covering: With probability approaching one, most randomly generated codebooks satisfy the required channel-coding condition.The stated probability is that a fraction 1−2ε of the codebooks are valid as n → ∞.
- Obfuscation-set covering: A type class can also be partitioned into obfuscation sets of size approximately 2^n(I(X:B)+δ), with most subsets being good.At least a fraction 1−ε of the subsets are good for sufficiently large n.
- Combined construction: The paper combines channel-code and obfuscation-set decompositions to obtain the basic structure used by its formation and distillation codes.The resulting construction is summarized in four decompositions and illustrated in Figure 2.
CODE STRUCTURE
The code structure decomposes n-copy pure states by typical types and several nested partitions, then uses channel decoding and incoherent transformations to support resource protocols.
- CODE STRUCTURE: The construction uses four decompositions of |Ψ⟩AB, all defined for sufficiently large n.The decompositions organize typical sequences by types, blocks, and codewords.
- Decomposition 1: The first decomposition partitions each typical type class into good sets whose Bob-side average states are approximately uniform within a set.Sequences are labeled by a typical type, set index, and within-set position.
- Decomposition 1: The first decomposition’s set and block rates are linked to conditional entropy and mutual information quantities.The paper states that 1/n log S_t is approximately I(X:B)_Ψ and gives rate relations for S_t and M_t.
- Decompositions 2 and 3: The second and third decompositions use channel codes for W_CC and W_CQ, respectively, following entanglement-assisted and GHZ distillation constructions.Their codebooks are indexed by typical type, code, and codeword position.
- Decompositions 2 and 3: The CC decoding is implemented by incoherent isometries, whereas the CQ decoding is generally not incoherent.This distinction is explicit in the construction of the decoding maps.
- Decomposition 4: The fourth decomposition hybridizes the first and second constructions by restricting to good sets and replacing their representation with W_CC channel codes.Uhlmann’s theorem supplies the corresponding transformations for good blocks.
Proof of Theorem 1
Theorem 1 characterizes achievable coherence–entanglement formation rates for pure bipartite states, with matching converse bounds and explicit LIOCC protocols.
- Theorem 1 gives achievable coherence–entanglement formation-rate triples for every pure state |Ψ⟩AB.
- The achievable points and their A↔B counterparts are optimal because every achievable rate triple satisfies the stated lower bounds.
- The lower bounds require Eco ≥ E(Ψ), RA + RB ≥ S(XY)∆(Ψ), and RB + Eco ≥ S(XY)∆(Ψ).
- Formation protocols use shared eCoBits or CoBits, local incoherent operations, measurements, classical announcements, and conditional incoherent unitaries.
- The protocols asymptotically consume rates approaching Eqs. (51) and (52), while every eCoBit can deterministically become one CoBit for Bob.
Proof of Lemmas 2 and 3
The lemmas provide two operational ingredients: implementing arbitrary unitaries with coherence and converting bipartite pure states under a majorization condition.
- An arbitrary d × d unitary can be implemented using incoherent operations and ⌈log d⌉ CoBits.
- The unitary construction introduces maximally entangled auxiliary resources, performs incoherent operations, and obtains the desired post-measurement state.
- Controlled unitaries whose target operators act on a d-dimensional system also require ⌈log d⌉ CoBits.
- The unitary lemma yields an alternative pure-state coherence-dilution protocol approaching rate S(X)∆(ψ).
- If both incoherent bases are Schmidt bases and τ(φ) majorizes τ(ψ), an LIOCC protocol deterministically transforms |ψ⟩AB into |φ⟩AB.
Proof of Theorem 4
Theorem 4 establishes that CL is an LIOCC monotone by combining relative-entropy coherence quantities with convex-roof and round-by-round arguments.
- Theorem 4 states that CL is an LIOCC monotone.
- For pure states, the relevant relative-entropy quantities reduce to S(A)∆(ϕ) − E(ϕ), S(B)∆(ϕ) − E(ϕ), and S(B)∆(ϕ).
- Monotonicity is proved for pure-state transformations and then extended through convex-roof reasoning and successive measurement rounds.
- The quantity CA|B r is identified with the optimal asymptotic coherence-distillation rate on Bob’s side when Alice helps.
Proof of Theorem 5
Theorem 5 gives achievable pure-state coherence–entanglement distillation rates, proves near-complete corner-point optimality, and identifies the optimal eCoBit rate through optimized mutual information.
- Theorem 5 provides achievable coherence–entanglement distillation-rate triples and their A↔B counterparts, with optimality at the stated corner points.
- Any achievable triple must satisfy RA + RB ≤ CL(Ψ) and RB + Eco ≤ S(Y)∆(Ψ).
- The protocol begins with a typical-type measurement, followed by incoherent measurements, classical communication, and conditional incoherent unitaries.
- The resulting coherence-distillation rates asymptotically approach the rates in Eq. (65), including Bob’s rate S(Y|X)∆(Ψ).
- Rate triple (66) is not generally eCoBit-optimal because LIOCC measurements can increase I(X : Y)∆(Ψ), even from a state where it initially equals zero.
- The optimal pure-state eCoBit distillation rate is obtained by optimizing I(X : Y)∆(Ψ) over all LIOCC protocols.
Proof of Theorem 6
The proof uses a typical-type measurement and codebook decomposition to construct a maximally entangled state whose size approaches Ct→E(Ψ) as n grows.
- The protocol is based on decomposition (42c).
- Alice measures the typical type |t⟩A1 and codebook |l⟩.
- The resulting state is maximally entangled, with size approaching Ct→E(Ψ) as n →∞.
Proof of Theorem 7
The proof shows that any entanglement distillable by LOCC is also distillable by LIOCC, by first distilling local coherence and then implementing the LOCC protocol incoherently.
- A mixed state ρAB has distillable entanglement if and only if entanglement can be distilled using LIOCC.
- Any LOCC protocol producing ΦAB can be converted into an LIOCC protocol by consuming finite local coherence.
- Alice and Bob distill sufficient local coherence from additional copies of ρAB before implementing the converted protocol.
- The required projective measurement can be implemented incoherently without changing Bob’s post-measurement state.