University of California at San Diego – Department of Physics – Prof. John McGreevy
Physics 239 Spring 2016
Assignment 4 –
Solutions
Due 11am Tuesday, May 3, 2016
1. Binary erasure channel.
Find the channel capacity of this channel:
Let p(a = 0) ≡ p determine the input distribution. Then the joint distribution is
p(1 − q)
pq
0
p(ab) = p(b|a)p(a) a,b =
0
(1 − p)q (1 − p)(1 − q) a=01,b=0e1
This has the same form as the binary symmetric channel with some extra zeros.
The distribution for the output variable is p(b) = p(1 − q) q (1 − p)(1 − q) b .
We find for the mutual information
I(A : B) = H(A) + H(B) − H(AB) = H2 (p)(1 − q)
which is maximized when p = 12 , and therefore the capacity is
maxp(A) I(A : B) = 1 − q.
Satisfyingly, this is also the probability that any bit we send is not erased.
2. Test of Landauer’s Principle.
Show that copying a known bit (say 0) onto an unkown bit by the method described in lecture with the double-well potential costs energy at least kB T ln 2.
Clearly this process cannot be reversible, since the initial state of the target bit
is lost in the process. In the physical realization of the bit as a column of gas, it
is clear that we must do work kB T ln 2 to accomplish this. I was hoping someone
would show how this constraint is manifested in the realization in terms of the
deformable double-well potential. In particular, it must be that a protocol for
deforming the potential which is guaranteed to adiabatically take the target bit
from one of the two states (say 0) onto the state of the given bit to be copied
must fail if employed with the target bit initialized in the other state (1).
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3. Making a Bell pair from a product state.
Find the output of the following quantum circuit:
|0i
|0i
H
•
Here H = √12 (X + Z) is a Hadamard gate, and the two-qbit gate is the CX gate
as in lecture.
Use 1 and 2 to label the top and bottom qbits respectively. The control X
distinguishes control and target bits:
CXct = |0i h0|c 1 t + |1i h1|c Xt .
Then the circuit is doing
CX21 · H2 |00i = CX21
|00i + |01i
√
2
1
= √ (|00i + |11i) .
2
4. Quantum Teleportation. Convince yourself that it is possible to transmit an
unknown state of a qbit by sending two classical bits to someone with whom you
share a Bell pair, using the following circuit:
•
|ψi
•
H
|0i
|0i
H
•
X
Z
|ψi
Time goes from left to right here; you should recognize the first two operations
from the previous problem. Imagine that the register on the bottom line is
separated in space from the top two after this point. The measurement boxes
indicate measurements of Z; the double lines indicate that the outcomes of these
measurements s = 1, 01 (this is the sending of the two classical bits) determine
whether or not (respectively) to act with the indicated gate.
A solution to this problem and the next can be found on pages 1-58 – 1-60 here.
5. Quantum Dense Coding. Find a circuit which does the reverse of the previous: by sending an unknown qbit to someone with whom you share a Bell
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note the correction.
2
pair, transmit two classical bits. (Hint: basically just reverse everything in the
previous problem.)
See the previous problem.
3