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Symmetries and Entanglement in Channel Coding Problems

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Symmetries and Entanglement in Channel Coding Problems University of Illinois at Urbana-Champaign March 24, 2020 Symmetries and entanglement in channel coding problems Felix Leditzky IQC, University of Waterloo Perimeter Institute partly based on arXiv:1910.00471 (joint work with J. Bausch) Channel coding in information theory Many communication tasks can be formalized as a channel coding problem: Senders Receivers Information Information encoding Channel decoding Classical, quantum, hybrid, ... Classical, quantum, ... Classical, quantum, eavesdroppers, environment, ... Central notions in information theory: Capacity of a channel: quantifies information-processing capabilities of a channel. Coding theorem: expresses capacity as (optimization over) entropic quantities. 1 Channel coding in information theory Prototypical example: classical point-to-point channel Sender Receiver Shannon's channel coding theorem [Shannon '48] Shannon entropy: Single-letter: bounded optimization problem Mutual information: Shannon's theorem can be phrased as a Capacity of a classical channel can be geometric program, a type of convex efficiently computed in time program. [Chiang, Boyd '04] [Arimoto '72; Blahut '72] 2 Channel coding in information theory More general settings Complications Network information theory Increased complexity of algorithms (anything beyond 1 sender 1 receiver) Non-convex optimization problems Quantum resources: quantum channels, quantum information, ... Unbounded optimization problems (multi-letter formulas) Focus of my research Non-convexity and multipartite entanglement are the main problems/objects of study. Use mathematical/numerical tools, in particular symmetries and optimization techniques, to study channel coding tasks. Example in this talk: Quantum capacity of a quantum channel 3 Talk outline Quantum information 101 Quantum capacity of quantum channels Pauli channels, stabilizer states and graph states Decoherence properties of graph states Exploiting graph symmetries Results: repetition codes Conclusion 4 Talk outline Quantum information 101 Quantum capacity of quantum channels Pauli channels, stabilizer states and graph states Decoherence properties of graph states Exploiting graph symmetries Results: repetition codes Conclusion 5 Quantum information terminology Quantum states Quantum channels Partial trace Von Neumann entropy 6 Entanglement Entangled states Prototypical example Multipartite entangled states 7 Talk outline Quantum information 101 Quantum capacity of quantum channels Pauli channels, stabilizer states and graph states Decoherence properties of graph states Exploiting graph symmetries Results: repetition codes Conclusion 8 Quantum capacity of a quantum channel Quantum channel models: Information theory Error correction Noisy communication link Environmental noise in a between quantum parties. quantum device. Quantum capacity characterizes: How much quantum information How much quantum information can be sent through channel? can be protected against noise? 9 Quantum capacity of a quantum channel Quantum information transmission is equivalent to entanglement generation. separatelabs Protocol: A Alice R B m Achievable rate /k : coherent information Bob [Devetak '05; Devetak, Winter '05] 10 Quantum capacity of a quantum channel Idea: separatelabs A1 Alice R A2 B1 A3 Superadditivity of coherent information B2 B3 A4 [Shor, Smolin '96; DiVincenzo et al. '98] B4 Bob 11 Quantum capacity of a quantum channel Quantum capacity of a quantum channel Quantum capacity is given by multi-letter formula and in general intractable to compute. Unbounded optimization problem Non-concave maximization problem (because of superadditivity) and for channels with superadditive known pathological behavior. coherent information. [Schumacher '96; Schumacher, Nielsen '96; Lloyd '97; Shor '02; Devetak '05; Watanabe '12; Cubitt et al. '15] 12 Quantum capacity of a quantum channel Quantum capacity Mathematical ansatz for approximate solution For specific quantum channels, consider symmetric codes and exploit symmetries to compute coherent information. By the above theorem, this yields lower bounds on the true capacity. 13 Talk outline Quantum information 101 Quantum capacity of quantum channels Pauli channels, stabilizer states and graph states Decoherence properties of graph states Exploiting graph symmetries Results: repetition codes Conclusion 14 Pauli channels and stabilizer states Qubit-qubit Pauli channels Important examples: [Shor '95; Steane '96] faithful quantum communication possible. perfect error-correcting code exists. 15 Pauli channels and stabilizer states 1 [Gottesman '97] 2 3 Graph states 4 5 6 16 Graph states Every stabilizer state is local unitary (LU) equivalent to a graph state: [van den Nest et al. '04] Important quantum codes as graph states: system qubits Ai reference qubit R Repetition code (GHZ state) Cat code (Shor code) Star graph 17 Talk outline Quantum information 101 Quantum capacity of quantum channels Pauli channels, stabilizer states and graph states Decoherence properties of graph states Exploiting graph symmetries Results: repetition codes Conclusion 18 Decoherence of graph states Goal 1 2 2 4 2 4 1 3 1 3 2 4 2 4 Observation 1 3 1 3 2 4 2 4 1 3 1 3 19 Decoherence of graph states Graph states subjected to Pauli noise [Hein et al. '05] PROBLEM 1 Exponential scaling: 2 3 4 20 Talk outline Quantum information 101 Quantum capacity of quantum channels Pauli channels, stabilizer states and graph states Decoherence properties of graph states Exploiting graph symmetries Results: repetition codes Conclusion 21 Exploiting graph symmetries 1 1 Graph automorphism 2 3 4 2 3 4 = 22 Exploiting graph symmetries system qubits reference qubits 2 2 2 Z 1 4 1 4 1 4 3 3 3 X Y 23 Exploiting graph symmetries Homomorphic group actions Symmetry-aware algorithm 24 Orbit representatives Group stabilizer chain Strong generating system Application to find orbit representatives Use lexicographical order within orbits. [Borie '12] 25 Algorithm overview 26 Talk outline Quantum information 101 Quantum capacity of quantum channels Pauli channels, stabilizer states and graph states Decoherence properties of graph states Exploiting graph symmetries Results: repetition codes Conclusion 27 Repetition code thresholds 0.1 z 0.075 0.05 0.5 0.025 2P 0 BB84 dep −0.25 x y −0.05 0 0 −0.075 −0.1 0.5 0.5 28 Talk outline Quantum information 101 Quantum capacity of quantum channels Pauli channels, stabilizer states and graph states Decoherence properties of graph states Exploiting graph symmetries Results: repetition codes Conclusion 29 Symmetries in channel coding problems Summary of arXiv:1910.00471 z We can use the graph state formalism and exploit 0.5 graph symmetries and tools from group theory 2P to approximate quantum capacity of interesting channels. BB84 dep x y Ideas for future work: 0 0 Generalize methods to handle more general quantum channels? 0.5 0.5 Improve upon/adopt more tools from computational group theory (CGT)? Can we use the framework of group actions and CGT to analyze concrete quantum error correction codes and their decoders, thresholds, etc? 30 Conclusion Channel capacities characterize fundamental concepts such as: Faithful communication using noisy resources Existence of error-correcting codes necessary for computations Secure key distribution Take-away message: Information-theoretic considerations Multipartite entanglement (Quantum) channels and their and its decoherence properties (quantum) capacities under quantum channels Mathematical methods 31 for your attention! 32.
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