# COMP-761 Quantum information theory Assignment5 Optional Due

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```					                     COMP-761: Quantum information theory
Assignment 5: Optional
Due: Tuesday, 21 April 2009

NOTE: If you intend to submit this assignment, let me know by the last day of classes at the latest.
Otherwise, I will submit your grade without including this assignment.

1. Some useful POVM measurements. Let |ϕ and |ψ be states in C2 such that ϕ|ψ = α. Choose
states |ϕ⊥ , |ψ ⊥ ∈ C2 such that ϕ|ϕ⊥ = ψ|ψ ⊥ = 0. For λ ∈ R, consider the triple of operators
E1 = λ|ϕ⊥ ϕ⊥ |,    E2 = λ|ψ ⊥ ψ ⊥ |,   E 3 = I − E1 − E2 .                   (1)
a) For which values of λ does {E1 , E2 , E3 } form a POVM?
b) Consider the POVM with the largest such λ. If this POVM is applied to an unknown state
which is either |ϕ or |ψ , outcome E1 implies the state must have been |ψ while outcome E2
implies the state must have been |ϕ . Explain why this does not provide a counterexample
to the theorem on the impossibility of perfectly distinguishing nonorthogonal states proved
in class.
c) Suppose you are given a quantum state chosen from a set {|ϕ1 , |ϕ2 , . . . , |ϕm } of linearly
independent states. Construct a POVM {E1 , E2 , . . . , Em+1 } such that if outcome Ej occurs
for 1 ≤ j ≤ m, then you can conclude with certainty that you were given state |ϕj . Your
POVM must be such that ϕj |Ej |ϕj > 0 for 1 ≤ j ≤ m.

2. Optimality of teleportation. You will show that any protocol for teleporting a d-dimensional state
must make use of at least 2 log d bits of classical communication. Deﬁne a teleportation protocol
for sending the quantum system A to consist of a bipartite entangled state |ψ CB , a measurement
{Fk ; 1 ≤ k ≤ M } on AC and a set of decoding operations (quantum channels) Dk on B. Alice
will perform the measurement on AC and send the outcome k to Bob classically, who will then
perform the decoding operation Dk . The protocol is exact if, for all pure input states |ϕ on A,
whenever the input state on A is |ϕ , the outcome state on B is also |ϕ .
a) Show that an exact teleportation protocol can be exploited to send a classical message drawn
from the set {1, 2, . . . , (dim A)2 } without errors by consuming entanglement and log M bits
of classical communication.
b) Show that M ≥ d2 must hold. Hint: Assume that M < d2 is possible and invoke the con-
struction of part (a). Then, instead of having Alice send Bob k, let him guess it, which he can
do correctly with probability at least 1/M > 1/d2 . Show that this leads to a contradiction in
the form of the ability to send classical messages with high reliability and no communication.

3. Mutual information and measurement channels. Let {Mj } be a POVM and consider the channel
N A→B (ρ) = j Tr(ρMj )|j j|. This is the channel that measures the POVM and records the out-
come. Suppose that |ϕ AA is a puriﬁcation of ρ and set σ = (N ⊗ idA )(ϕ). Recall that the quantity
I(A ; B)σ played an important role in evaluating the entanglement-assisted capacity of N .

1
You will show that for this particular channel, I(A ; B)σ has an interesting form. You may
assume that ρ is invertible for the remainder of the question.

a) Let pj = Tr ρ1/2 Mj ρ1/2 and ρj = p−1 ρ1/2 Mj ρ1/2 . (Set ρj = 0 if pj = 0, however.) Show that
j
{(pj , ρj )} is an ensemble of density operators. That is, j pj = 1 and for all j, pj ≥ 0 and ρj
is a density operator.

b) Show that I(A ; B)σ = H                j   p j ρj −   j   pj H(ρj ).

4. Classical state transfer. In this problem you’ll develop a classical analogue of the quantum state
transfer results we proved in class. Let > 0.
(n)
a) Recall that we used the notation A (X, Y ) for the set of sequences in X n × Y n that are
jointly -typical with respect to a joint probability density p(x, y). If y n ∈ Y n , let

A(n) (X|y n ) = {xn ∈ X n | (xn , y n ) ∈ A(n) }.
(n)
If y n ∈ A         (Y ), that is, if y n is -typical, show that

A(n) (X|y n ) ≤ 2n[H(X|Y )+2 ] .                     (2)

b) Let (X n , Y n ) be an i.i.d. sequence with (Xj , Yj ) ∼ p(x, y). Imagine that Alice holds X n ,
Bob holds Y n and Alice wishes to communicate X n to Bob. Obviously, she could compress
X n at the rate H(X), but that strategy wouldn’t exploit the fact that Bob already has some
partial information about X n through its correlations with Y n . Instead, they will employ a
strategy reminiscent of the approach used in our proof of Shannon’s noisy coding theorem.
To compress to a rate R, Alice will encode using a function E : X n → [2nR ] := {1, 2, . . . , 2nR }
chosen at random. More precisely, each E(xn ) is selected independently and uniformly at
random from [2nR ]. Both Alice and Bob are assumed to know the function E. When Bob
receives a transmission z, he checks to see whether there is a unique xn such that E(xn ) = z
and (xn , y n ) is jointly -typical. If yes, he concludes that Alice’s message was xn . Otherwise
he fails.
Consider the error event

E = {∃xn = X n | E(xn ) = E(X n ) and (xn , Y n ) ∈ A(n) (X, Y )}.

Show that the probability of E goes to zero as n goes to inﬁnity for any choice of R >
H(X|Y ) + 3 . Hint: You’ll want to use part (a).

c) By also considering any other types of errors that may occur, argue that the coding strategy
of the previous paragraph can be made to succeed at any rate R > H(X|Y ).

This gives a clear interpretation of the classical conditional entropy as nothing other than the
number of bits Alice needs to send Bob if we wants to learn X n when he already knows Y n .

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