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!
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