Multiplicativity of completely bounded p-norms implies a new additivity result
arXiv:quant-ph/0506196 · doi:10.1007/s00220-006-0034-0
Abstract
We prove additivity of the minimal conditional entropy associated with a quantum channel Phi, represented by a completely positive (CP), trace-preserving map, when the infimum of S(gamma_{12}) - S(gamma_1) is restricted to states of the form gamma_{12} = (I \ot Phi)(| psi >< psi |). We show that this follows from multiplicativity of the completely bounded norm of Phi considered as a map from L_1 -> L_p for L_p spaces defined by the Schatten p-norm on matrices; we also give an independent proof based on entropy inequalities. Several related multiplicativity results are discussed and proved. In particular, we show that both the usual L_1 -> L_p norm of a CP map and the corresponding completely bounded norm are achieved for positive semi-definite matrices. Physical interpretations are considered, and a new proof of strong subadditivity is presented.
arXiv commentFinal version for Commun. Math. Physics. Section 5.2 of previous version deleted in view of the results in quant-ph/0601071 Other changes minor
References in corpus (10)
- The capacity of the noisy quantum channel
- Entanglement-Assisted Classical Capacity of Noisy Quantum Channels
- General Entanglement Breaking Channels
- Quantum information can be negative
- Information transmission through a noisy quantum channel
- Inequalities for Quantum Entropy: A Review with Conditions for Equality
- Entanglement of assistance and multipartite state distillation
- Counterexample to an additivity conjecture for output purity of quantum channels
- Qubit Entanglement Breaking Channels
- On entanglement-assisted classical capacity
Cited by in corpus (49)
- Advances in Quantum Cryptography
- Random Quantum Circuits are Approximate 2-designs
- Quantum Reverse Shannon Theorem
- Quantum channels and their entropic characteristics
- Converse bounds for private communication over quantum channels
- A Matrix Convexity Approach to Some Celebrated Quantum Inequalities
- Unbounded violation of tripartite Bell inequalities
- Contractivity of positive and trace preserving maps under norms
- Strong converse exponents for a quantum channel discrimination problem and quantum-feedback-assisted communication
- Counterexamples to the maximal p-norm multiplicativity conjecture for all p > 1
- Amortized Channel Divergence for Asymptotic Quantum Channel Discrimination
- Continuity of quantum channel capacities
- Application of the Resource Theory of Channels to Communication Scenarios
- Statistical speed of quantum states: Generalized quantum Fisher information and Schatten speed
- Quantum Error Correction of Observables
- A Minkowski Type Trace Inequality and Strong Subadditivity of Quantum Entropy II: Convexity and Concavity
- Thermodynamic Capacity of Quantum Processes
- Continuous-variable quantum repeater based on quantum scissors and mode multiplexing
- Entropy of a quantum channel
- Amortized entanglement of a quantum channel and approximately teleportation-simulable channels
- Testing product states, quantum Merlin-Arthur games and tensor optimisation
- Continuous-variable entanglement distillation over a pure loss channel with multiple quantum scissors
- Assessing the performance of quantum repeaters for all phase-insensitive Gaussian bosonic channels
- Versatile relative entropy bounds for quantum networks
- Upper bounds on secret key agreement over lossy thermal bosonic channels
- Computing Quantum Channel Capacities
- Nonadditivity effects in classical capacities of quantum multiple-access channels
- Impossibility of Local State Transformation via Hypercontractivity
- Distributed Private Randomness Distillation
- Capacity Estimates via comparison with TRO channels
- Uniform Additivity in Classical and Quantum Information
- Hypercontractivity and the logarithmic Sobolev inequality for the completely bounded norm
- Rate limits in quantum networks with lossy repeaters
- Tight bounds for private communication over bosonic Gaussian channels based on teleportation simulation with optimal finite resources
- Quantum-proof randomness extractors via operator space theory
- Thermodynamic Implementations of Quantum Processes
- Estimating the output entropy of a tensor product of two quantum channels
- Operator-valued Schatten spaces and quantum entropies
- What is Entropy? A new perspective from games of chance
- Capacity Bounds via Operator Space Methods
- Entanglement properties of a measurement-based entanglement distillation experiment
- Energy-constrained LOCC-assisted quantum capacity of bosonic dephasing channel
- Strong Converse Exponent for Entanglement-Assisted Communication
- Information-theoretic aspects of the generalized amplitude damping channel
- Fundamental limitations on the recoverability of quantum processes
- Adaptive Channel Reshaping for Improved Entanglement Distillation
- Potential output purity of completely positive maps
- Asymptotic Equipartition Theorems in von Neumann algebras
- Additivity and chain rules for quantum entropies via multi-index Schatten norms