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

Entropy of a quantum channel

Gilad Gour1 Mark M. Wilde2

1Department of Mathematics and Statistics

Institute for Quantum Science and Technology University of Calgary Alberta, Canada T2N 1N4

2Hearne 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)

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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.

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von Neumann entropy

Entropy of a quantum state

For a quantum state ρA of a system A, the entropy is defined as H(A)ρ ≡ − Tr{ρA log2 ρA}. 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 ρA and σA as D(ρAσA) = Tr{ρA [log2 ρA − log2 σA]}, we can rewrite the formula for quantum entropy as follows: H(A)ρ = log2 |A| − D(ρAπA), where |A| is dimension of A and πA ≡ IA/ |A| denotes the maximally mixed state.

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Entropy of a quantum channel

The relative entropy of quantum channels NA→B and MA→B is defined as D(NM) ≡ sup

ρRA

D(NA→B(ρRA)MA→B(ρRA)), where the optimization is with respect to bipartite states ρRA of a reference system R of arbitrary size and the channel input system A. Completely depolarizing channel generalizes the maximally mixed state: RA→B(XA) = Tr{XA}πB, where XA is an arbitrary operator for system A and πB is maximally mixed.

Entropy of a quantum channel

Let NA→B be a quantum channel. Then its entropy is defined as H(N) ≡ log2 |B| − D(NR). See also [Yuan, arXiv:1807.05958].

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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 NA→B and

• utputs a quantum channel KC→D, which we denote by

Θ(A→B)→(C→D)(NA→B) = KC→D.

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Physical realizations of quantum superchannels

Superchannel has a physical realization in terms of pre- and post-processing quantum channels: Θ(A→B)→(C→D)(NA→B) = DBM→D ◦ NA→B ◦ EC→AM, where EC→AM and DBM→D are pre- and post-processing channels

C D B

N

A

D

M

E

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Properties of the entropy of a channel

Non-decrease under the action of a uniformity preserving superchannel: Let NA→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 NA→B be a replacer channel, defined such that NA→B(ρA) = σB for all states ρA and some state σB. Then the following equality holds H(N) = H(B)σ.

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R´ enyi entropy of a quantum channel

Recall that the sandwiched R´ enyi relative entropy of ρ and σ is defined as Dα(ρσ) ≡ 1 α − 1 log2 Tr

• σ(1−α)/2αρσ(1−α)/2αα

. This leads to the sandwiched R´ enyi divergence of channels NA→B and MA→B: Dα(NM) ≡ sup

ρRA

Dα(NA→B(ρRA)MA→B(ρRA)).

R´ enyi entropy of a channel

Let NA→B be a quantum channel. For α ∈ [1/2, 1) ∪ (1, ∞), the R´ enyi entropy of the channel N is defined as Hα(N) ≡ log2 |B| − Dα(NR), where RA→B is the completely randomizing channel. It also obeys non-decrease under a uniformity-preserving superchannel, additivity, and reduction to states.

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Asymptotic Equipartition Property

Define the sine / purified channel distance of two channels NA→B and MA→B as P(N, M) ≡ sup

ρRA

P(NA→B(ρRA), MA→B(ρRA)), where P(ω, τ) ≡

• 1 − √ω√τ2

1 is the sine / purified distance of states.

Define the smoothed min-entropy of a channel for ε ∈ (0, 1) as Hε

min(N) ≡

sup

P(N , N )≤ε

Hmin( N). Asymptotic equipartition property: For all ε ∈ (0, 1), the following inequality holds lim

n→∞

1 n Hε

min(N ⊗n) ≥ H(N).

We also have that lim

ε→0 lim n→∞

1 n Hε

min(N ⊗n) ≤ H(N).

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Operational interpretation in terms of quantum channel merging

En An

N

V

Bn En An

N

V

Bn

P

ΦK ΦL

Bn En Bn Vs. Goal: for Bob to merge his share of the channel with Eve’s. Given channel NA→B, let VN ≡ (U N

A→BE)⊗n, where UN A→BE extends NA→B.

By consuming a maximally entangled state ΦK of Schmidt rank K and applying a

• ne-way LOCC protocol P , Bob and Eve can distill a maximally entangled state ΦL
• f Schmidt rank L and transfer Bob’s systems Bn to Eve, in such a way that any

third party having access to the inputs An and the outputs Bn 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.

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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

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