The Fidelity of Recovery is Multiplicative

I. INTRODUCTION

• The FoR measures local recoverability of global quantum states and is connected to geometric squashed entanglement and the study of topological order.The CQMI also underlies squashed entanglement and has an operational interpretation as the optimal quantum communication cost in quantum state redistribution.Source: §I · p. 1
• The authors generalize the FoR to compare bipartite states that may have different marginals on A.The generalized form allows comparison of quantum information on different-dimensional systems, supporting study of correlations independently of their representation.Source: §I · pp. 1–2
• Using an SDP dual formulation, the authors show that the generalized FoR is multiplicative for product states.The result implies that an optimal recovery map can also have product structure.Source: §I · p. 2
• Multiplicativity simplifies an information-theoretic proof of the Fawzi–Renner lower bound using quantum state redistribution and the ideas of Brandão et al..The proof avoids continuity and representation-theoretic arguments, including de Finetti reductions.Source: §I · p. 2

II. FIDELITY OF RECOVERY AS AN SDP

• For a maximally entangled state with d_B = d_A, the FoR reduces to conditional min-entropy by SDP duality.The SDP derivation follows the approach of Watrous’ lecture notes.Source: §II · p. 2
• Lemma 1 states that FC→B(ρAB∥σAC), as in (6), is the solution to a minimization problem over RAB > 0 and QAD > 0, subject to QAD ⊗ 1B ≥ RAB ⊗ 1D.Source: §II · p. 2
• The resulting dual program is described as the Alberti form of the FoR and simplifies to the fidelity form when B and C are trivial.The authors note that a related Alberti expression does not directly simplify to the displayed special-case expressions for maximally entangled and pure states.Source: §II · p. 2
• The Choi–Jamiołkowski representation replaces optimization over recovery channels with optimization over corresponding states.Every state satisfying the stated constraint corresponds to a CPTP map, yielding an expression for the FoR.Source: §II · p. 3
• The dual program is simplified by substitutions and an equivalent matrix-inequality condition, producing a minimization over positive R_AB and Q_AD.The simplified constraint is Q_AD ⊗ 1_B ≥ R_AB ⊗ 1_D.Source: §II · p. 4

• Multiplicativity also implies that an optimal recovery map can have product structure.This consequence is stated for the fidelity of recovery in the paper’s broader results.Source: §III · p. 4
• For product states, the fidelity of recovery equals the product of the individual fidelities of recovery.The proof combines the lower bound from product-structured recovery maps with a dual-SDP upper bound.Source: §III · p. 4
• The dual-SDP argument constructs a feasible pair for the product-state optimization from feasible pairs for each component.The construction uses positivity under tensoring to preserve the required operator inequalities.Source: §III · p. 4

• The proof simplifies Brandão et al.’s operational derivation of the Fawzi–Renner bound by using fidelity-of-recovery multiplicativity instead of de Finetti reductions.The earlier argument relates the CQMI to twice the quantum communication cost of state redistribution.Source: §IV · p. 5
• Quantum state redistribution transfers A while preserving the joint state, using local encoding and decoding operations and communication of Q.The communication cost is q = log dQ, and the protocol’s fidelity is traded off against that cost.Source: Figure 1 · p. 5, §IV · p. 5
• A one-shot redistribution protocol achieves fidelity 1 − 144ε^2 at a cost expressed using smooth conditional min- and max-entropies.Those smooth entropies asymptotically converge to the conditional von Neumann entropy.Source: §IV · p. 5
• The proof uses a high-fidelity redistribution protocol to bound decoder performance even when quantum communication is omitted.The cited argument invokes the operator inequality dX · 1X ⊗ ωY ≥ ωXY.Source: §IV · p. 6
• The proof’s second part differs from Brandão et al.’s and uses operator monotonicity of the square-root function.Source: §IV · p. 6
• Applying the asymptotic equipartition property and fidelity-of-recovery multiplicativity to tensor powers yields the lower bound on the CQMI.The argument takes the limit as n tends to infinity after dividing the inequality by n.Source: §IV · p. 7

V. CONCLUSION

• The generalized fidelity of recovery is multiplicative for product states, and this gives an information-theoretic proof of the Fawzi–Renner bound without de Finetti reductions.The authors also note that Brandão et al. obtain a potentially stronger bound using regularization and de Finetti reductions.Source: §V · p. 7
• Rephrased using the sandwiched quantum Rényi divergence of order 1/2, the multiplicativity result yields the Fawzi–Renner bound by monotonicity in the Rényi parameter.Additivity for every α in (1/2, 1] could yield stronger bounds, including a relative-entropy-distance bound asked about by Li and Winter [26].Source: §V · p. 7
• The authors identify estimating whether the optimal recovery map can be nearly matched by the Petz recovery map as an open question.Later work reports that the Petz map is not square-root optimal relative to the optimal map, though some near-optimality might still hold.Source: §V · pp. 7–8
• The results were extended to multiparty settings and to a lower bound on relative-entropy monotonicity under CPTP maps, while related SDP techniques were applied to quantum discord.The cited extensions and application are reported in [43], [6], and [30], respectively.Source: §V · p. 7
• After this work, improved lower bounds on CQMI and on relative-entropy monotonicity under CPTP maps were proven.The cited results are [20] [33] [34] [44].Source: §V · p. 7
• The main result was extended to additivity for more general relative entropies of recovery, and fidelity of recovery was shown to equal a quantum interactive proof system’s maximum acceptance probability.The latter connection implies multiplicativity by a result of Kitaev and Watrous.Source: §V · p. 8