Entropy of a quantum channel Gilad Gour 1 Mark M. Wilde 2 1 - PowerPoint PPT Presentation

Entropy of a quantum channel Gilad Gour 1 Mark M. Wilde 2 1 Department of Mathematics and Statistics Institute for Quantum Science and Technology University of Calgary Alberta, Canada T2N 1N4 2 Hearne Institute for Theoretical Phyiscs Department of Physics and Astronomy Center for Computation and Technology Louisiana State University Baton Rouge, Louisiana 70803, USA Available as arXiv:1808.06980 ISIT 2020 (virtual) Gilad Gour, Mark M. Wilde Entropy of a quantum channel 1 / 11

Motivation von Neumann entropy is a central concept in physics and information theory, having a number of compelling physical interpretations. The most fundamental notion in quantum mechanics is a quantum channel, as quantum states, unitary evolutions, measurements, and discarding of quantum systems can each be regarded as certain kinds of quantum channels. An important goal is thus to define a consistent and meaningful notion of the entropy of a quantum channel. Gilad Gour, Mark M. Wilde Entropy of a quantum channel 2 / 11

von Neumann entropy Entropy of a quantum state For a quantum state ρ A of a system A , the entropy is defined as H ( A ) ρ ≡ − Tr { ρ A log 2 ρ A } . It has operational interpretations in terms of quantum data compression and optimal entanglement manipulation rates of pure bipartite quantum states. Entropy in terms of quantum relative entropy By defining the quantum relative entropy of ρ A and σ A as D ( ρ A � σ A ) = Tr { ρ A [log 2 ρ A − log 2 σ A ] } , we can rewrite the formula for quantum entropy as follows: H ( A ) ρ = log 2 | A | − D ( ρ A � π A ) , where | A | is dimension of A and π A ≡ I A / | A | denotes the maximally mixed state. Gilad Gour, Mark M. Wilde Entropy of a quantum channel 3 / 11

Entropy of a quantum channel The relative entropy of quantum channels N A → B and M A → B is defined as D ( N�M ) ≡ sup D ( N A → B ( ρ RA ) �M A → B ( ρ RA )) , ρ RA where the optimization is with respect to bipartite states ρ RA of a reference system R of arbitrary size and the channel input system A . Completely depolarizing channel generalizes the maximally mixed state: R A → B ( X A ) = Tr { X A } π B , where X A is an arbitrary operator for system A and π B is maximally mixed. Entropy of a quantum channel Let N A → B be a quantum channel. Then its entropy is defined as H ( N ) ≡ log 2 | B | − D ( N�R ) . See also [Yuan, arXiv:1807.05958]. Gilad Gour, Mark M. Wilde Entropy of a quantum channel 4 / 11

Quantum superchannels Most general physical transformation of a quantum channel is a superchannel , which accepts as input a quantum channel and outputs a quantum channel The superchannel Θ ( A → B ) → ( C → D ) takes as input a quantum channel N A → B and outputs a quantum channel K C → D , which we denote by Θ ( A → B ) → ( C → D ) ( N A → B ) = K C → D . Gilad Gour, Mark M. Wilde Entropy of a quantum channel 5 / 11

Physical realizations of quantum superchannels Superchannel has a physical realization in terms of pre- and post-processing quantum channels: Θ ( A → B ) → ( C → D ) ( N A → B ) = D BM → D ◦ N A → B ◦ E C → AM , where E C → AM and D BM → D are pre- and post-processing channels B A N C D E D M Gilad Gour, Mark M. Wilde Entropy of a quantum channel 6 / 11

Properties of the entropy of a channel Non-decrease under the action of a uniformity preserving superchannel: Let N A → B be a quantum channel, and let Θ be a uniformity preserving superchannel (one that preserves the completely randomizing channel). Then H (Θ( N )) ≥ H ( N ) . Additivity: Let N and M be quantum channels. Then the channel entropy is additive in the following sense: H ( N ⊗ M ) = H ( N ) + H ( M ) . Reduction to states and normalization: Let the channel N A → B be a replacer channel, defined such that N A → B ( ρ A ) = σ B for all states ρ A and some state σ B . Then the following equality holds H ( N ) = H ( B ) σ . Gilad Gour, Mark M. Wilde Entropy of a quantum channel 7 / 11

R´ enyi entropy of a quantum channel Recall that the sandwiched R´ enyi relative entropy of ρ and σ is defined as �� σ (1 − α ) / 2 α ρσ (1 − α ) / 2 α � α � 1 D α ( ρ � σ ) ≡ α − 1 log 2 Tr . This leads to the sandwiched R´ enyi divergence of channels N A → B and M A → B : D α ( N�M ) ≡ sup D α ( N A → B ( ρ RA ) �M A → B ( ρ RA )) . ρ RA R´ enyi entropy of a channel Let N A → B be a quantum channel. For α ∈ [1 / 2 , 1) ∪ (1 , ∞ ), the R´ enyi entropy of the channel N is defined as H α ( N ) ≡ log 2 | B | − D α ( N�R ) , where R A → B is the completely randomizing channel. It also obeys non-decrease under a uniformity-preserving superchannel, additivity, and reduction to states. Gilad Gour, Mark M. Wilde Entropy of a quantum channel 8 / 11

Asymptotic Equipartition Property Define the sine / purified channel distance of two channels N A → B and M A → B as P ( N , M ) ≡ sup P ( N A → B ( ρ RA ) , M A → B ( ρ RA )) , ρ RA � 1 − �√ ω √ τ � 2 where P ( ω, τ ) ≡ 1 is the sine / purified distance of states. Define the smoothed min-entropy of a channel for ε ∈ (0 , 1) as H ε H min ( � min ( N ) ≡ sup N ) . P ( N , � N ) ≤ ε Asymptotic equipartition property: For all ε ∈ (0 , 1), the following inequality holds 1 n H ε min ( N ⊗ n ) ≥ H ( N ) . lim n →∞ We also have that 1 n H ε min ( N ⊗ n ) ≤ H ( N ) . ε → 0 lim lim n →∞ Gilad Gour, Mark M. Wilde Entropy of a quantum channel 9 / 11

Operational interpretation in terms of quantum channel merging B n B n A n A n Φ K Φ L N N V Vs. V P B n B n E n E n E n Goal: for Bob to merge his share of the channel with Eve’s. Given channel N A → B , let V N ≡ ( U N A → BE ) ⊗ n , where U N A → BE extends N A → B . By consuming a maximally entangled state Φ K of Schmidt rank K and applying a one-way LOCC protocol P , Bob and Eve can distill a maximally entangled state Φ L of Schmidt rank L and transfer Bob’s systems B n to Eve, in such a way that any third party having access to the inputs A n and the outputs B n and E n would not be able to distinguish the difference between the ideal situation on the left and the simulation on the right. The optimal asymptotic rate of entanglement gain is equal to the entropy of the channel N . Gilad Gour, Mark M. Wilde Entropy of a quantum channel 10 / 11

Conclusion and future directions Defined entropy of a quantum channel as a fundamental notion in QIT and established that it satisfies several desirable axioms Showed that it satisfies an AEP and has an operational meaning in terms of quantum channel merging In future work, it is desired to establish its operational meaning in the context of a resource theory Gilad Gour, Mark M. Wilde Entropy of a quantum channel 11 / 11