P1 = √1 (|0⟩+i|1⟩)· √1 (⟨0|−i⟨1|) and P2 = √1 (|0⟩−i|1⟩)· √1 (⟨0|+i⟨1|). 2222
Check that the above matrices are projection operators.
Check that the above matrices can be used for organizing a von Neumann measurement.
Find the classical and quantum outputs of the measurements of a qubit in 1. state |0⟩,
2. state |1⟩.
Find the classical and quantum outputs of the measurement of a qubit in 1. state √1 (|0⟩ + i|1⟩),
2. state √1 (|0⟩ − i|1⟩). 2
For problems 3 and 4, please present results in the following form:
State |v⟩ = …
Quantum outputs Classical outputs Probability
Homework 6
Write here q. out Write here cl. out Prob. of this event
We will use the following notations and assumptions.
1. Basis 0, bit 0, means that Alice sends quantum state |0⟩.
2. Basis 0, bit 1, means that Alice sends quantum state |1⟩.
3. Basis 1, bit 0, means that Alice sends quantum state |+⟩ = √1 (|0⟩ + |1⟩). 2
4. Basis 1, bit 1, means that Alice sends quantum state |−⟩ = √1 (|0⟩ − |1⟩). 2
5. Making measurement in basis B0 means to use P0 = |0⟩⟨0| and P1 = |1⟩⟨1|.
6. Making measurement in basis B1 means to use P0 = |+⟩⟨+| and P1 = |−⟩⟨−|.
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Let Alice choose the following bases and bits for sending qubits to Bob:
time instance: t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 basis: B0 B1 B1 B0 B1 B0 B1 B0 B0 B1
bit: 0 1 0 1 1 1 0 0 0 0 Let now Bob use the following bases for conducting measurements
time instance: t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 basis: B0 B0 B0 B0 B1 B1 B1 B0 B0 B1
Further Alice and Bob follow the protocol BB84 (which we discussed in the class). Find the secret key obtained by Alice and Bob.
6. For the same settings (bases and bits used by Alice and bases used by Bob) let us assume that Eve intercepts the first 5 qubits sent by Alice. Eve measures these qubits using bases
time instances: t1 t2 t3 t4 t5 basis: B1 B1 B1 B1 B1
and sends the qubits (after measurements) further to Bob.
Find possible classical outputs and the corresponding probabilities of the first 5 measure- ments conducted by Bob.
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