Download Partial Trace and State Purification

Preamble
Measurements
Schmidt decomposition
Marginals
State purification
Postscript
Introduction to quantum information processing
Partial Trace and State Purification
Brad Lackey
27 October 2016
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Preamble
Measurements
Schmidt decomposition
Marginals
State purification
Postscript
O UTLINE
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Measurements
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Schmidt decomposition
3
Marginals
4
State purification
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Preamble
Measurements
Schmidt decomposition
Marginals
State purification
Postscript
L AST TIME ...
We found quantum states were more general than first thought.
States of the form |ψi are “pure” states.
Ensembles and states with external information are “mixed” states.
Mixed states are density operators ρ.
Pure states can be written as a mixed state via ρ = |ψihψ|.
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Preamble
Measurements
Schmidt decomposition
Marginals
State purification
Postscript
O UTLINE
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Measurements
2
Schmidt decomposition
3
Marginals
4
State purification
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Preamble
Measurements
Schmidt decomposition
Marginals
State purification
Postscript
P ROJECTION - VALUED MEASURES
Projective, or orthogonal, measurements are derived from observables.
Basic idea: in the lab an observation produces an “expected value.”
An observable is represented by a Hermitian operator.
Observing A when the system is in state |ψi is hAi = hψ|A|ψi.
Pr
Let’s use the spectral decomposition A = j=1 λj |φj ihφj | to expand this:
hAi = hψ|A|ψi =
r
X
λj hψ|Πj |ψi.
j=1
We have a probability of the event j happening: hψ|Πj |ψi.
The eigenvalue λj is the number that gets observed in the experiment.
Then hAi is just the mathematical formula for “expectation.”
The spectral projections {Πj }rj=1 define our “measurement basis.”
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Preamble
Measurements
Schmidt decomposition
Marginals
State purification
Postscript
G LEASON ’ S THEOREM
What should a state satisfy?
It should give us probabilities. So if Π is a projection then p(Π) ∈ [0, 1].
PN
If {Πj }Nj=1 is a projection-valued measure, then j=1 p(Πj ) = 1.
So can there be even things beyond densities that could be “states?”
Yes and no. Gleason Theorem addresses this precisely.
If dim H = 2, then some funny things can happen.
Otherwise density operators are precisely the set of states.
Theorem (Gleason (1957))
Suppose dim H ≥ 3. Let p : P(H) → [0, 1] be any function satisfying
PN
N
j=1 p(Πj ) = 1 whenever {Πj }j=1 is a projection-valued measure. Then
there exists a unique density operator ρ such that p(Π) = tr(Πρ).
The rule Prρ (Π) = tr(Πρ) is called Born’s rule.
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Measurements
Schmidt decomposition
Marginals
State purification
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P OSITIVE OPERATOR - VALUED MEASURES
O R SIMPLY POVM S
Can we have more than two outcomes for a qubit measurement?
No! We can only have two (nonzero) orthogonal vectors in 2-d.
A “generalized” measurement, or positive operator-valued measure is:
A collection of positive operators {Ej }m
j=1 (i.e. Ej ≥ 0),
Pm
that have sum j=1 Ej = 1.
We define the POVM probability rule to mimic Born’s rule:
Prρ {Ej } = tr(Ej ρ).
A common example is where the outcomes might be {0, 1, ⊥}.
Here {0, 1} could be a bit or a no/yes response, and ⊥ is “fail.”
Problem 4.6 deals with POVMs {E0 , E1 , E⊥ } having a fail option.
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Preamble
Measurements
Schmidt decomposition
Marginals
State purification
Postscript
O UTLINE
1
Measurements
2
Schmidt decomposition
3
Marginals
4
State purification
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Preamble
Measurements
Schmidt decomposition
Marginals
State purification
Postscript
B IPARTITE SYSTEMS
Some of the most interesting science comes from multipartite systems.
We describe bipartite systems via two actors Alice and Bob.
Today: Alice will be “our local system” and Bob “an external system.”
Later: Alice will send (quantum) information, and Bob receives it.
If Alice and Bob are completely independent:
their joint probabilities would be
Pr{A = a and B = b} = Pr{A = a} · Pr{B = b}.
in the quantum case this can be achieve by ρA ⊗ ρB .
In particular, bipartite systems are composed as HA ⊗ HB .
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T HE S CHMIDT DECOMPOSITION
Theorem (Schmidt decomposition)
(A)
Let |ψi ∈ HA ⊗ HB . Then there exists orthonormal basis {|φj i} and
(B)
{|φµ i} of HA and HB and positive values {λj } so that
r
X
p
(A)
(B)
|ψi =
λj |φj i ⊗ |φj i.
j=1
r ≤ min{dim HA , dim HB } is the “Schmidt number” (or “rank”) of |ψi.
p
The “Schmidt coefficients” are the λj .
The Schmidt decomposition is basically the singular value decomposition.
P
(A)
(B)
1
Take any bases of HA and HB and write |ψi = jµ αjµ |χj i ⊗ |χµ i.
2
Use the singular value decomposition: α = UDV where
p
p
D = diag( λ1 , . . . , λr , 0, . . . ).
3
(A)
(B)
Form {|φj i} and {|φµ i} using U † and V as change of basis.
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Measurements
Schmidt decomposition
Marginals
State purification
Postscript
O UTLINE
1
Measurements
2
Schmidt decomposition
3
Marginals
4
State purification
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Measurements
Schmidt decomposition
Marginals
State purification
Postscript
N OTION OF MARGINALS IN PROBABILITY
For a bipartite system a marginal (probability or distribution) refers to:
probabilities associated with one part of the system,
without reference to the other part of the system.
This is different that conditional probability where the outcome in the other
part is known and accounted for.
For classical probability the formula reads:
X
X
p(a) = Pr{A = a} =
Pr{A = a and B = b} =
p(a, b).
b
b
In the quantum case:
the joint probability is give by a density ρ on HA ⊗ HB ,
Alice’s marginal probability should be given by a density ρ0 on HA .
In the above classical formula, we “sum over Bob’s outcomes.”
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PARTIAL TRACE
In the quantum setting, “summing over Bob’s state” is called “partial trace.”
We can formalize it via the scenario:
Alice wants to make a measurement, but Bob’s not so interested.
So measurement projections will be of the form Π ⊗ 1B .
The resulting probabilities (w.r.t. Alice) are tr((Π ⊗ 1B )ρ).
By Gleason’s theorem, there exists a density ρ0 with
tr(Πρ0 ) = tr((Π ⊗ 1B )ρ).
This ρ0 is the partial trace (over Bob’s subsystem).
We denote this as ρ0 = trB (ρ).
Warning: we write trB but this is an operator on HA !
It is the quantum analogue of Alice’s marginal probability.
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Schmidt decomposition
Marginals
State purification
Postscript
C OMPUTING THE PARTIAL TRACE
(A)
(B)
To compute: take orthonormal basis {|φj i} and {|φµ i} of HA and HB .
tr((Π ⊗ 1B ) ⊗ ρ) =
X
=
X
(A)
(hφj
(A)
(B)
| ⊗ hφµ |)((Π ⊗ 1B )ρ)(|φj
(B)
i ⊗ |φµ i)
jµ
(A)
hφj
(B)
(A)
(B)
| Π · hφµ |ρ|φµ i |φj i
jµ


=
X
(A)
hφj | Π
·
X
(B)
(A)
(B)
hφµ |ρ|φµ i |φj i
µ
j
And so we see why this is called the “partial” trace:
X
(B)
trB (ρ) =
hφ(B)
µ |ρ|φµ i.
µ
A specific example: ρ = 12 (|00i + |11i)(h00| + h11|) Then
1
2
1
=
2
trB (ρ) =
PARTIAL T RACE AND S TATE P URIFICATION
h0(B) |ρ|0(B) i + h1(B) |ρ|1(B) i
1
|0(A) ih0(A) | + |1(A) ih1(A) | = 1A .
2
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Schmidt decomposition
Marginals
State purification
Postscript
O UTLINE
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Measurements
2
Schmidt decomposition
3
Marginals
4
State purification
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Measurements
Schmidt decomposition
Marginals
State purification
Postscript
M ARGINALS OF PURE STATES
Theorem
Let |ψi ∈ HA ⊗ HB be a pure state. Then the marginals trA (|ψihψ|) and
trB (|ψihψ|) have the same nonzero eigenvalues.
P p
(A)
(B)
Proof. Let |ψi = j λj |φj i ⊗ |φj i be the Schmidt decomposition.
Then:
Xp
(A)
(A)
(B)
(B)
λj λk |φj ihφk | ⊗ |φj ihφk |.
|ψihψ| =
jk
(B)
Using the orthonormal basis {|φµ i} to compute the partial trace:
trB (|ψihψ|) =
X
(B)
(B)
hφµ |(|ψihψ|)|φµ i
µ
=
Xp
X (B) (B)
(B) (B)
(A)
(A)
λj λk |φj ihφk |
hφµ |φj ihφk |φµ i
=
X
µ
jk
(A)
(A)
λj |φj ihφj |.
j
This form is diagonal, so λj are the nonzero eigenvalues. Same for trA .
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State purification
Postscript
S TATE PURIFICATION
A fascinating result is that this theorem can be reversed:
If Alice has mixed state ρ, she can realize it as ρ = trB (|ψihψ|).
I.e. including external (Bob’s) information, |ψi ∈ HA ⊗ HB is pure.
Here’s how it works:
1
2
3
4
Use the spectral theorem to write ρ =
Pr
j=1
λj |φj ihφj |.
{|χj i}rj=1
orthonormal for a Hilbert space HB = span{|χj i}.
P √
Define |ψi = j λj |φj i ⊗ |χj i ∈ HA ⊗ HB .
Declare
The previous computation shows ρ = trB (|ψihψ|).
In words, given an ensemble {(|φj i, λj )}rj=1 :
each species gets associated to a new quantum state (via the |χj i),
and attaching this information to the prepared |φj i purifies ρ.
√
There’s some question about using coefficient λj , but the math sorts it out.
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Schmidt decomposition
Marginals
State purification
Postscript
N EXT TIME ...
Superoperators.
Kraus operators.
Examples of some quantum channels.
Stinespring dilation theorem.
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