Single Qubit CP Map Geometry

1
Single Qubit CP Map Geometry

• Artur Ekert, Patrick Hayden,
• Hitoshi Inamori, Daniel Oi
• Clarendon Laboratory/Wolfson College

daniel.oi_at_qubit.org
2
Talk Outline

• CP Maps
• Representation of Single Qubit CP Maps
• Unital CP Maps
• Simulating Single Qubit Channels
• Channel Dynamics
• Approximating Positive Maps and QC
• Stormer-Woronowicz Result

3
CP Maps

• Allowable Physical Operations on State

• Maps Density Operators to Density Operators
• Positive, Linear, Trace Preserving

4
Complete Positivity

• Physically Motivated Condition

• Eg. Transpose is P but not CP

5
Single Qubit CP Maps

• Contraction Maps on the Bloch Ball

6
Unital CP Maps

• Fixes the Maximally Mixed State

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Conditions for CPness

• Necessary and Sufficient Condition

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Unital CP Map Geometry

• Tetrahedron in Parameter Space

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Simulating CP Channel

• AsPR, sDet(A), P(AAT)1/2
• QDQTR

10
Dynamical Simulation

• Unitary Evolution on Extended System

11
CP Dynamics Cont.
12
Animation
13
Approximating Positive Maps

• Euclidean Metric in Parameter Space

14
Eavesdropping
15
Eavesdropping Cont.

• Symmetric Incoherent Attack

16
Eavesdropping Cont.
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Stormer-Woronowicz Result

• Horodecki Proof of Peres Separability Criterion
  for 2x2, 2x3 Bipartite Systems

Equivalently
18
Conclusion

• Simple Tetrahedral Structure to Unital CP Maps on
  Single Qubits
• Applicable to Previous Problems
• Dynamical CP-Map Generation
• Extension to General Non-Unital Case (Higher
  Dimensional Parameter Space)?