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The questions for the m. Phil. Exam in quantum information theory, covering topics such as subadditivity of von neumann entropy, concavity of entropy, relative entropy, quantum information sources, memoryless sources, depolarizing channel, holevo χ quantity, and maximally entangled states.
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Thursday 7 June 2007 9.00 to 12.
Attempt FOUR questions. There are FIVE questions in total.
The questions carry equal weight.
Cover sheet None Treasury Tag Script paper
(a) Prove that the von Neumann entropy is subadditive, i.e.
S (ρAB ) 6 S (ρA) + S (ρB ) , (1)
where ρAB is the density matrix of a bipartite system AB and ρA, ρB are the reduced density matrices of the two subsystems A and B respectively.
(b) Using the bound (1) or otherwise, prove the concavity of the von Neumann entropy
( (^) ∑r
i=
pi ρi
∑^ r
i=
pi S (ρi) ,
where pi > 0 ,
∑r i=1 pi^ = 1 and^ ρi^ ,^ (i^ = 1,... , r) are density matrices.
(c) Consider a quantum system A which is in a state ρi with probability pi , and let σ be some other density matrix acting on the Hilbert Space HA of the system A. Prove that (^) ∑
i
pi S ( ρi || σ) =
i
pi S ( ρi || ρ¯) + S ( ¯ρ || σ). (2)
In the above, ¯ρ :=
i piρi^ and the notation^ S^ (ω^ ||^ σ) denotes the relative entropy of the states ω and σ.
Quantum Information Theory
3 The action of the depolarizing channel on the state ρ of a qubit is given by
Φ(ρ) = (1 − p)ρ +
p 3
(σxρxσx + σy ρσy + σz ρσz ), (6)
where 0 < p < 1 and σx, σy and σz are the Pauli matrices.
(a) Prove that the depolarizing channel can alternatively be expressed as follows, for some 0 < q < 1: Φ(ρ) = (1 − q)ρ + q
where I is the identity operator acting on the single qubit Hilbert space. Hence find the relation between p and q. (b) Derive the effect of the depolarizing channel on the Bloch sphere, hence justifying its name.
(c) Write an expression for the Holevo χ quantity for an ensemble of quantum states E := {pi, ρi}. Express χ(E) in terms of the relative entropy and prove that it can never increase under a quantum operation.
(d) State the Holevo–Schumacher–Westmoreland (HSW) Theorem and use it to derive the product state capacity of a qubit depolarizing channel with parameter q, defined by (7).
Quantum Information Theory
(a) Let HA, HB be two Hilbert Spaces, each of dimension d. Write an expression for a maximally entangled state |ΨAB 〉, of size d, in the Hilbert Space HA ⊗ HB and explain why it is said to be maximally entangled.
(b) Prove that any arbitrary vector |φA〉 ∈ HA can be expressed in terms of the maximally entangled state |ΨAB 〉, as follows:
|φA〉 = 〈φ∗ B | Ψ˜AB 〉 , (8)
via the relative state method. Here | Ψ˜AB 〉 :=
d|ΨAB 〉, and |φ∗ B 〉 is the index state in HB that yields |φA〉.
(c) Prove that the pure state resulting from the action of any arbitrary operator MA on a state vector |φA〉 ∈ HA can be obtained as a relative state from the state (MA ⊗ IB )| Ψ˜AB 〉.
(d) It can be shown that if ΦA : B(HA) 7 → B(HA) is a linear, completely positive trace–preserving (CPT) map, then
ΦA(|φA〉〈φA|) = 〈φ∗ B |(ΦA ⊗ idB )(| Ψ˜AB 〉〈 Ψ˜AB |)|φ∗ B 〉. (9)
Using this result, prove that any linear CPT map, ΦA, can be written in the Kraus form, i.e., ΦA(ρ) =
k
AkρA† k ,
for any ρ ∈ B(HA), where the Ak are linear operators in B(HA), satisfying ∑
k
A† kAk = IA ,
with IA being the identity operator in B(HA).
Quantum Information Theory [TURN OVER
