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Assisted distillation of quantum coherence
E. Chitambar, A. Streltsov, S. Rana, M. N. Bera, G. Adesso, M. Lewenstein
TL;DR
The paper proves that every state not quantum-incoherent has nonzero distillable coherence under collaboration. It uses repeated local quantum and incoherent operations with classical communication, and relates the resulting transformations to SLOCC maps, including conditions for deterministic implementation.
Problem
The paper addresses whether states that are not quantum-incoherent can yield distillable coherence through collaboration.
Method
The analysis uses local quantum operations on one party, incoherent operations on the other, classical communication, and monotonicity of QI relative entropy.
Results
Every non-QI state has nonzero distillable coherence of collaboration, and repeated procedures produce maximally coherent states at a nonzero rate.
Takeaways & Limitations
For single-qubit states, coherence of assistance equals S(∆(ρ)), while related LQICC transformations can always occur with nonzero probability and sometimes with probability one.
Takeaways & Limitations
The multipartite setting assumes arbitrary local operations for parties A1 through AN, incoherent operations for qubit party B, and classical communication among all parties.
Abstract
from arXiv · showhide
We introduce and study the task of assisted coherence distillation. This task arises naturally in bipartite systems where both parties work together to generate the maximal possible coherence on one of the subsystems. Only incoherent operations are allowed on the target system while general local quantum operations are permitted on the other, an operational paradigm that we call local quantum-incoherent operations and classical communication (LQICC). We show that the asymptotic rate of assisted coherence distillation for pure states is equal to the coherence of assistance, an analog of the entanglement of assistance, whose properties we characterize. Our findings imply a novel interpretation of the von Neumann entropy: it quantifies the maximum amount of extra quantum coherence a system can gain when receiving assistance from a collaborative party. Our results are generalized to coherence localization in a multipartite setting and possible applications are discussed.
Proof of Theorem 2
The proof shows that every bipartite state that is not quantum-incoherent has positive distillable coherence of collaboration. Alice’s local measurement can produce a coherent state for Bob with nonzero probability, after which Bob can distill coherence at a nonzero rate.
- Any state that is not quantum-incoherent has nonzero distillable coherence of collaboration.
- Alice’s measurement outcomes either include a non-incoherent post-measurement state or reveal off-diagonal terms in cross-block operators.
- For off-diagonal cross-block terms, Alice chooses a two-dimensional von Neumann measurement basis with a suitable angle θ.
- Some measurement outcome has nonzero probability and leaves Bob’s post-measurement state with off-diagonal elements.
- Repeating the procedure across copies gives many coherent states, from which Bob distills maximally coherent states at a nonzero rate.
Proof of Theorem 3
The proof establishes the QI relative entropy as an upper bound on the distillable coherence of collaboration. It uses continuity, additivity, and monotonicity under LQICC operations to transfer finite-copy bounds to the asymptotic rate.
- The QI relative entropy bounds the distillable coherence of collaboration from above.
- The argument represents the distillation task through an LQICC protocol that produces a maximally coherent state on an auxiliary subsystem.
- Continuity of the QI relative entropy controls changes between nearby states using trace distance and the total-system dimension.
- Additivity and monotonic nonincrease of the QI relative entropy under LQICC operations yield the asymptotic upper bound.
maximally correlated states
The proof connects coherence of assistance with entanglement of assistance for associated maximally correlated states. An optimal entanglement decomposition induces an optimal coherence-of-assistance decomposition because pure-state coherence equals the corresponding entanglement.
- An arbitrary state ρ defines a maximally correlated state ρmc with matrix elements ρij mapped to |ii⟩⟨jj|.
- An optimal decomposition of ρmc supplies a decomposition of ρ through the coefficients of its pure states.
- The equality follows from Cr(|φk⟩) = E(|ψk⟩) for each corresponding pair of pure states.
- The induced decomposition of ρ is optimal for coherence of assistance.
Proof of Theorem 4
The proof establishes the relation between regularized coherence of assistance for Bob’s reduced state and assisted coherence distillation from its purification. It combines an upper bound with a lower bound and invokes Theorem 3 for completion.
- The regularized coherence of assistance of Bob’s reduced state is bounded above by the distillable coherence of collaboration of its purification.
- A lower bound is obtained by combining the coherence-of-assistance relation with the corresponding main-text equation.
- The upper and lower bounds, together with Theorem 3, establish the claimed equality.
Proof of Theorem 5
The proof establishes that coherence of assistance equals the relative-entropy coherence for every single-qubit state, while higher-dimensional examples show nonadditivity.
- Single-qubit states: The proof proceeds by expanding an arbitrary purification in Bob’s incoherent basis and selecting orthogonal states that form a mutually unbiased basis for Alice’s measurement.This measurement produces the required post-measurement states on Bob’s system.
- Single-qubit states: For any single-qubit state ρ, the coherence of assistance satisfies Ca(ρ) = S(∆(ρ)) = C∞a(ρ).Alice measures in a mutually unbiased basis, producing Bob states with coherence S(∆(ρ)).
- Counterexample: The resulting dimension-4 inequality implies that coherence of assistance is not additive.The example uses a 2 ⊗ 4 state.
- Counterexample: For a particular state, every measurement outcome on Alice’s system gives Bob coherence strictly below the maximal value 2.Assuming maximal coherence forces the measurement operator’s relevant matrix elements to vanish, contradicting a nonzero outcome probability.
- Counterexample: The argument leaves open whether coherence of assistance is additive for qutrit states.The proof explicitly notes that additivity remains unclear in that dimension.
Proof of Theorem 6
The theorem extends assisted coherence localization to multipartite systems with a qubit target. Local operations and classical communication among assisting parties achieve the relative-entropy coherence, and joint operations cannot improve it.
- Multipartite localization: For a multipartite state with qubit B, an LOCC protocol among the assisting parties can generate Bob states with coherence S(∆(ρB)).The assisting systems are unrestricted locally, while B remains restricted to incoherent operations.
- Multipartite localization: Joint operations on all assisting systems cannot generate more coherence on B than the protocol achieves.Thus the achieved rate is optimal within the stated multipartite setting.
- Multipartite localization: The protocol expands the state in Bob’s incoherent basis and uses a mutually unbiased basis of orthogonal multipartite states for the assisting parties.The expansion supplies the measurement structure needed to obtain the desired conditional states on B.
- Multipartite localization: Any two orthogonal multipartite states used in the protocol can be perfectly distinguished by LOCC.This makes the required measurement implementable by the assisting parties.
Relating LQICC and tripartite SLOCC maps
LQICC transformations of bipartite states induce stochastic tripartite LOCC transformations of corresponding maximally correlated states. These transformations are always possible with nonzero probability and become deterministic for reversible incoherent Kraus maps.
- General correspondence: Any pair of bipartite states related by an LQICC map has corresponding maximally correlated states related by a stochastic tripartite LOCC map.The construction applies to arbitrary LQICC protocols decomposed into local operations and communicated outcomes.
- Protocol construction: Alice’s local measurements and classical communication can be simulated deterministically on the corresponding tripartite states.The nontrivial stochastic construction is required for Bob’s local incoherent operations.
- Protocol construction: Bob’s incoherent operation is implemented through a sequence of measurement, outcome broadcast, ancilla introduction, conditional unitary rotation, and generalized-Hadamard measurement.The desired maximally correlated state arises from the specified final measurement outcome.
- Success probability: The desired tripartite transformation succeeds with nonzero probability whenever the corresponding LQICC outcome has nonzero probability.The proof establishes pα > 0 implies qα > 0 and then shows the final measurement outcome also has nonzero probability.
- Deterministic cases: If every incoherent Kraus operator uses a reversible function fα, the corresponding tripartite transformation can be implemented with probability one.Charlie can then apply a unitary rotation that directly produces the desired maximally correlated state.