Quantum Information Theory
Entanglement and Measurement
Pouya Bastani
pbastani@math.sfu.ca
CMPT 881 - Quantum Computing
Quantum Information Theory – p.1/20
Information
Information - classical or quantum - is measured by the minimum
amount of communication needed to convey it
How much is there to know?
Shannon:
the amount of information from a source is the memory
required to represent its output and is given by its ‘entropy’
How much do we already know?
the amount of information we already have is given by ‘mutual entropy
How large is our ignorance?
Slepian & Wolf:
the amount of information needed to obtained full
knowledge is given by a quantity called ‘conditional entropy’
Quantum Information Theory – p.2/20
Shannon Entropy
Entropy: amount of uncertainty in the state of a physical system
Shannon entropy of a random variable X:
average amount of uncertainty about X before we learn its value
average amount of information gain after we learn the value of X
Less probable events convey more information
Entropy of independent events is the sum of entropies
Entropy of an even with probability p is H(p) ≡ − lg p
Entropy associated with probability distribution p(x) of X:
H(X) ≡ −
X
p(x) lg p(x)
0 lg 0 ≡ 0
x
Entropy of a bit is 2
Quantum Information Theory – p.3/20
Entropy Types
Joint Entropy: total uncertainty about the pair (X, Y )
P
H(X, Y ) ≡ − x,y p(x, y) lg p(x, y)
Conditional Entropy: average uncertainty about X given Y
H(X|Y ) ≡ H(X, Y ) − H(Y )
Mutual Entropy: common information between X and Y
H(X:Y ) ≡ H(X) + H(Y ) − H(X, Y )
Conditional entropy is always non-negative
Quantum Information Theory – p.4/20
Definitions
The density operator for quantum system {|ψi i , pi } is defined by:
ρ≡
X
pi |ψi i hψi |
i
A quantum system is said to be in a pure state if its state is known
exactly. A pure state can be written as ρ = |ψi hψ|
Let ρAB be the density operator of a quantum system AB. The reduced
density operator for
system A is defined by ρA ≡ TrB (ρAB )
The partial trace TrB over system B is defined by
TrB (|a1 i ha2 | ⊗ |b1 i hb2 | = |a1 i ha2 | hb2 |b1 i
where |ai i and |bi i are any two vectors in the state space of A and B
Quantum Information Theory – p.5/20
Von Neumann Entropy
Qubits exist in superposition state, unlike classical bits
QIT must reflect this fact by including phase information
Von Neumann defined the entropy of a quantum state as
S(ρ) ≡ Tr(ρ lg ρ)
where ρ is the density operator of the system
P
Ifρ is expressed in a diagonal basis, ρ = x λx |xi hx|, then the von
Neumann entropy reduces to the classical Shannon entropy
S(ρ) = −
X
λx lg λx
0 lg 0 ≡ 0
x
An n × n matrix ρ is diagonalizable iff ρ has n independent eigenvectors
Quantum Information Theory – p.6/20
Entropy Types
Joint Entropy:
S(AB) ≡ −Tr(ρAB lg(ρAB ))
Conditional Entropy:
S(A|B) ≡ S(AB) − A(B)
Mutual Entropy:
S(A:B) ≡ S(A) + S(B) − S(AB)
Quantum Information Theory – p.7/20
Examples 1
Independent particles:
ρA = ρB = 21 (|0i h0| + |1i h1|)
ρAB = ρA ⊗ ρB
S(A) = 1
S(AB) = 2
S(A:B) = 0
S(A|B) = 1
Quantum Information Theory – p.8/20
Example 2
Fully correlated Particles:
ρAB = 21 (|00i h00| + |11i h11|)
ρA = TrB (ρAB ) = 21 (|0i h0| + |1i h1|)
S(A) = 1
S(AB) = 1
S(A:B) = 1
S(A|B) = 0
Quantum Information Theory – p.9/20
Example 3
Entangled (super-correlated) Particles:
|ψAB i =
√1 (|00i
2
+ |11i)
ρAB = |ψAB i hψAB |
ρA = TrB (ρAB ) = 12 (|0i h0| + |1i h1|)
S(A) = 1
S(AB) = 0
S(A:B) = 2
S(A|B) = −1
Quantum Information Theory – p.10/20
Observation
In QIT, conditional entropies can be negative. In fact,
isolated entangled systems have negative conditional entropy
Case III is forbidden is classical (Shannon) information theory
the entropy of the entire system AB cannot be smaller than the
entropy of one of its subparts: S(AB) ≮ S(B)
S(AB) = 0 in case III because the EPR-pair AB is created via a unitary
transformation of a zero entropy pure state. In fact,
any unitary transformation conserves entropy in a quantum system
CN OT H |00i =
1
(|00i + |11i) = |EP Ri
2
Quantum Information Theory – p.11/20
Example
Greenberger-Horne-Zeilinger (GHZ) state (EPR-triplet):
|ψABC i =
√1 (|000i
2
+ |111i)
ρABC = |ψABC i hψABC |
ρAB = 12 (|00i h00| + |11i h11|)
S(ABC) = 0
S(C|AB) = −1
Quantum entanglement between any part and the rest of the system
Ignoring one part creates classical correlation between the rest
Quantum Information Theory – p.12/20
Von Neumann Measurement
Measurement: correlation between the observer and the observed
Copenhagen interpretation: measurement is the interaction of a classical
apparatus with a quantum system, that collapses the wave-function
Von Neumann: the measuring ancilla must be quantum mechanical
The qantum system Q and ancilla A are initially in state
|Q, 0i =
X
αi |ai , 0i
i
Measurement introduces correlation between quantum system Q ancilla A
U
|Q, 0i −→ |Q, Ai =
X
αi |ai , ii
i
Observation collapses
this wavefunction to |ai , ii with probability αi
Quantum Information Theory – p.13/20
Flaw
No-cloning theorem: it is possible to copy a quantum state only if it
belongs to a set of orthogonal states
Attempt to copy an arbitrary quantum state generates entanglement
between quantum system Q and ancilla A
An arbitrary state cannot be measured without creating entanglement
In general |Q, Ai is entangled not just correlated
Entangled states have a description distinct from correlated states
Quantum Information Theory – p.14/20
So Far
Classical information corresponds to correlation between measured
system and measuring device (only differences are measurable)
Von Neumann measurement creates entanglement between quantum
system Q and measurement ancilla A rather than correlations
Entangled system QA cannot any reveal classical correlation
Quantum Information Theory – p.15/20
Resolution
Interact another ancilla A′ with entangled system QA:
′
|QAA i =
X
αi |ai , i, ii
i
State of the entire ancilla AA′ unconditional on Q yields:
ρAA′ = TrQ [ρQAA′ ] =
X
|αi |2 |i, ii hi, i|
i
A and A′ are maximally correlated:
S(A:A′ ) = S(A) = S(A′ ) = S(AA′ )
Measure the correlation between A and A′
Correlations between A and Q are unobservable
Quantum Information Theory – p.16/20
Pure vs Mixed State
No (Shannon) information about Q can be obtained from the ancilla:
S(Q:A:A′ ) = S(A:A′ ) − S(A:A′ |Q) = 0
Consequence of the fact that Q is initially in a pure state, S(Q) = 0
No information can be obtained in any measurement of a pure state
Pure state has vanishing von Neumann entropy, thus there is no
uncertainty about it, so nothing can be learned from it
Only mixed-state measurements can yield information (mutual entropy)
If a system is prepared in a mixed state described by {ρi , pi },
information I gained by a measurement is limited by the Kholevo bound:
I≤S
X
i
pi ρi
!
−
X
pi S(ρi )
i
Quantum Information Theory – p.17/20
“... We must abandon at least one cherished notion of orthodox measurement
theory: that the apparatus necessarily reflects the state of the system it was built to
measure, by being correlated with it in the sense of Shannon. Rather, it is the
correlations between the ancillae that create the illusion of measurement.”
“...Intuition dictates that performing a measurement means somehow observing
the state of Q. Rather, a measurement is constructed such as to infer the state of Q
from that of the ancilla – by ignoring Q itself. The correlations (in AA′ ) which emerge
from the fact that a part (Q) of an entangled state (QAA′ ) is ignored give rise to the
classical idea of a measurement. ”
C. Adami and N. J. Cerf
Quantum Information Theory – p.18/20
Negative Entropy Interpretation
Alice and Bob hold some information X and Y , respectively
Bob has some information about X due to correlation between X and Y
Additional information Alice needs to send Bob for him to fully learn X:
S(X|Y ) = S(XY ) − S(Y )
What if S(X|Y ) is negative?
Alice can transfer X using only classical communication
Alice and Bob have the potential to transfer additional quantum
information in the future at no further cost* to them
* Communicating classical information is far easier that quantum information
Quantum Information Theory – p.19/20
References
1. N.J. Cerf, C. Adami. Quantum mechanics of measurement, eprint
quant-ph/9605002
2. N.J. Cerf, C. Adami. Negative entropy and information in quantum
mechanics, PRL
79 (1997), pp. 5194-519
3. N.J. Cerf, C. Adami. Negative entropy in quantum information theory,
Proceedings of the 2nd International Symposium on Fundamental
Problems in Quantum Physics, Oviedo, 1996
4. N.J. Cerf, C. Adami. What Information Theory Can Tell Us About Quantum
Reality, Lecture
Notes in Computer Science 1509, 1999
5. M.A. Nielsen, I.L. Chuang. Quantum Computation and Quantum
Information.
Cambridge University Press, 2000
6. M. Horodecki, Oppenheim, A. Winter. Partial Quantum Information, Nature
436 (August 2005), pp. 673-676
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