### Positive Operator-Valued Measure (POVM)

From the third postulate of quantum mechan- ics, if we define

E_{m} ≡ M^{†}_{m}M_{m},
then

X m

E_{m} ^{=} 1 and p(m) = hψ|E_{m}|ψi.

The positive operators E_{m} are the POVM ele-
ments associated with the measurement, and
the set {E_{m}} is a POVM.

Since the POVM elements sum to identity,
they provide a partition of identity. In the
special case of projective measurements, the
E_{m}’s are orthogonal projections, and the di-
rect sum of the subspaces they project onto is
the Hilbert space of the measured system.

1

### Schmidt Decomposition

For any state |ψi_{AB} of a composite system AB,
there exists orthonormal states |ii_{A} and |ii_{B} for
systems A and B respectively such that

|ψi_{AB} = ^{X}

i

λ_{i}|ii_{A}|ii_{B},

where the λ_{i}’s are non-negative real numbers
satisfying ^{P}_{i} λ^{2}_{i} = 1.

So for any pure state |ψi_{AB}, the density oper-
ator for systems A and B are

ρ_{A} = tr_{B} (|ψi_{AB}) = ^{P}_{i} λ^{2}_{i} |ii_{AA}hi|

ρ_{B} = tr_{A}(|ψi_{AB}) = ^{P}_{i} λ^{2}_{i} |ii_{BB}hi|

They have the same eigenvalues.

2

The λ_{i}’s are the Schmidt coefficients, the
bases {|ii_{A}} and {|ii_{B}} are called the Schmidt
bases for A and B respectively, and the num-
ber of non-zero λ_{i}’s is called the Schmidt num-
ber. The Schmidt number provide a definition
and measure for entanglement.

Entanglement between A and B can be defined
as when the Schmidt number of |ψi_{AB} is larger
than one, since then the states are not sepa-
rable. The value of the Schmidt number also
reflect the degree of entanglement in that local
operations cannot change its value. If unitary
operation U is performed on A only, then the
new composite state is

|ψi_{AB} = ^{X}

i

λ_{i}(U|ii_{A})|ii_{B},

the Schmidt number remains constant.

### Purifications

Given a state ρ_{A} in system A, we can introduce
a system R and define a pure state |Ψi_{AR} of
the composite system AR such that

ρ_{A} = tr_{R} (|Ψi_{ARAR}hΨ|) .

That is, the mixed state ρ_{A} becomes the pure
state |Ψi_{ARAR}hΨ|. This is a purely mathemat-
ical procedure known as purification. System
R is called the reference system, and has no
physical reality.

Given an arbitrary state ρ_{A} of system A, ac-
cording to the basic properties of density ma-
trices, we can diagonalize it:

ρ_{A} = ^{X}

i

p_{i}|ii_{AA}hi|,

where the |ii_{A}’s are mutually orthogonal and
p_{i} ≥ 0, ^{P}_{i} p_{i} = 1.

3

Take system R to have the same state space
as A, and define an orthonormal basis {|ii_{R}},
then the pure state

|Ψi_{AR} = ^{X}

i

√p_{i}|ii_{A}|ii_{R}

is a purification for ρ_{A}.
Taking the partial trace

tr_{R} (|Ψi_{ARAR}hΨ|)

= tr_{R}

X

i

√p_{i}|ii_{A}|ii_{R}

X

j

(√

p_{j})^{∗}_{A}hj|_{R}hj|

= tr_{R}

X

ij

√p_{i}p_{j} (|ii_{A} ⊗ |ii_{R}) (_{A}hj| ⊗ _{R}hj|)

= ^{X}

ij

√p_{i}p_{j}|ii_{AA}hj|tr (|ii_{RR}hj|)

= ^{X}

ij

√p_{i}p_{j}|ii_{AA}hj|δ_{ij}

= ^{X}

i

p_{i}|ii_{AA}hi|

= ρ_{A}

The purification process defines a pure state by
its Schmidt decomposition, that is |ii_{A} and |ii_{R}
are the Schmidt bases for A and R respectively.

The Schmidt coefficients are √
p_{i}.

So the pure state produced by purification us- ing the same reference system R is unique up to a local unitary transformation. That is, for

|Ψ^{0}i_{AR} = ^{X}

i

√p_{i}|ii_{A} ⊗ U|ii_{R},

tr_{R} ^{}|Ψ^{0}i_{ARAR}hΨ^{0}|^{}

= tr_{R}

X

i

√p_{i}|ii_{A} ⊗ U|ii_{R}

X

j

(√

p_{j})^{∗}_{A}hj| ⊗ _{R}hj|U^{†}

= ρ_{A}.

4

### Relations Between the Physical States of Two Systems

For systems A and B, if their states are classi- cally correlated, then their state vectors have some algebraic relation to each other; if they are entangled, then the relation between their state vectors becomes inseparable.

If system A is in the state |xi_{A} = a|0i_{A}+ b|1i_{A},
then the states of the composite system when
there are some classical corrrelations with sys-
tem B are (for equality and reverse in the Bloch
sphere):

|xi_{A}|yi_{B} = (a|0i_{A} + b|1i_{A}) ⊗ (a|0i_{B} + b|1i_{B}),

|xi_{A}|yi_{B} = (a|0i_{A} + b|1i_{A}) ⊗ (b|0i_{B} + a|1i_{B}).

If they are entangled, then the composite states are:

a|0i_{A}|0i_{B} + b|1i_{A}|1i_{B},
a|0i_{A}|1i_{B} + b|1i_{A}|0i_{B}.

5

There is no unitary transformation on the com- posite system AB that produces classical cor- relations between the states of A and B for arbitrary states of A. The best one can do is to produce an entangled state. This is the basis of the no-cloning theorem.

The purpose of cloning is to produce an inde- pendent copy of data. Entanglement causes the copy to be dependent on future values of the source data, this defeats the purpose of cloning, yet it can be seen as the ultimate copy mechanism.

### Bell Inequality

Two particles are prepared, particle 1 is mea-
sured for physical properties P_{Q} or P_{R} by ran-
dom, while particle 2 is measured for physical
properties P_{S} or P_{T} by random. All measure-
ment results are either 1 or −1. The measure-
ment results are denoted Q, R, S, and T for
P_{Q}, P_{R}, P_{S}, and P_{T} respectively.

Since

Q, R, S, T = ±1 we have

QS + RS + RT − QT

= (Q + R)S + (R − Q)T = ±2.

6

Let p(q, r, s, t) denote the probability that be- fore measurement the particles have value

Q = q, R = r, S = s, T = t, then

hQS + RS + RT − QT i

= ^{X}

qrst

p(q, r, s, t)(qs + rs + rt − qt)

≤ ^{X}

qrst

2p(q, r, s, t)

= 2.

So we have

hQSi + hRSi + hRT i − hQT i ≤ 2.

This is an instance of Bell inequality.

Now suppose the two particles sent are in the qubit state

√1

2(|0i|1i − |1i|0i),

and the measurements are projective measure- ments where

Q = 1 0

0 −1

!

= σ_{3}, S = ^{√}^{1}

2

−1 −1

−1 1

!

R = 0 1 1 0

!

= σ_{1}, T = ^{√}^{1}

2

1 −1

−1 −1

!

Simple calculation shows that hQSi = 1

√2, hRSi = 1

√2, hRT i = 1

√2, hQT i = − 1

√2, thus

hQSi + hRSi + hRT i − hQT i = 2√ 2.

The Bell inequality is violated.

### Projective Measurements of the Qubit

In the qubit system, a projective measurement of the spin on axis ˆn (on the Bloch sphere) is represented by

M = ˆn · ~σ = n_{1}σ_{1} + n_{2}σ_{2} + n_{3}σ_{3} =

2 X m=1

λ_{m}P_{m}^{,}
where

λ_{1} = 1, P_{1} ^{=} E^{(ˆ}n, +) = ^{1}_{2}(1 ^{+ ˆ}n · ~σ),
λ_{2} = −1, P_{2} ^{=} E^{(ˆ}n, −) = ^{1}_{2}(1 − ˆn · ~σ),

and the probabilities for obtaining spin up and spin down are

p(1) = hψ|P_{1}|ψi,
p(2) = hψ|P_{2}|ψi,
respectively.

7

### Another Bell Inequality

The Bell state |ψ^{−}i = ^{√}^{1}

2(|0i|1i − |1i|0i) satis- fies

((ˆn · ~σ) ⊗ 1 ^{+} 1 ⊗ (ˆn · ~σ))|ψ^{−}i = 0.

This means that if we measure the first qubit along ˆn and the second along −ˆn, their results will always be anticorrelated:

h(ˆn · ~σ) ⊗ 1i

= hψ^{−}|((ˆn · ~σ) ⊗ 1)|ψ^{−}i

= −hψ^{−}|(1 ⊗ (ˆn · ~σ))|ψ^{−}i

= hψ^{−}|(1 ⊗ (−ˆn · ~σ))|ψ^{−}i

= h1 ⊗ (−ˆn · ~σ)i

8

If we measure the spin of the first qubit along ˆn, and the spin of the second qubit along ˆm (The measurement (ˆn · ~σ) ⊗ ( ˆm · ~σ)), then the probability of obtaining the same or different results for the two qubits is

hψ^{−}| (E^{(ˆ}^{n, +) ⊗} E^{( ˆ}^{m, +)) |ψ}^{−}i

= hψ^{−}| (E^{(ˆ}n, −) ⊗ E^{( ˆ}m, −)) |ψ^{−}i

= 1

4(1 − cos θ)

hψ^{−}| (E^{(ˆ}n, +) ⊗ E^{( ˆ}m, −)) |ψ^{−}i

= hψ^{−}| (E^{(ˆ}^{n, −) ⊗} E^{( ˆ}^{m, +)) |ψ}^{−}i

= 1

4(1 + cos θ)
So we have probabilities ^{1}

2(1− cos θ) of obtain-
ing the same result and ^{1}_{2}(1 + cos θ) of obtain-
ing different results.

Consider measuring along three co-plane axes
60^{◦} apart, measuring the first qubit along ˆn_{1}
with result r_{1} and the second along −ˆn_{2} with
result r_{2} we can conclude that if we could
somehow measure the first qubit along ˆn_{1} and
ˆn_{2} we would obtain the results r_{1} and r_{2}. The
sum of the probabilities that the same result is
obtained for any two axes is

P_{same}(ˆn_{1}, ˆn_{2}) + P_{same}(ˆn_{2}, ˆn_{3}) + P_{same}(ˆn_{1}, ˆn_{3})

= 3 · ^{1}_{2}(1 − cos 60^{◦}) = ^{3}_{4}.

Since there are only two possible results, the Bell inequality in this case is

P_{same}(ˆn_{1}, ˆn_{2})+P_{same}(ˆn_{2}, ˆn_{3})+P_{same}(ˆn_{1}, ˆn_{3}) ≥ 1,
which is violated in this particular instance.

Two assumptions made in the proof of the Bell inequality need to be reconsidered:

1. Realism. The physical properties have def- inite values that exist independent of ob- servation.

2. Locality. The measurement of the two par- ticles does not influence each other’s re- sult.

That is, the world may not be locally realistic.

The concept of entanglement is the key to un- derstanding non-locality.

9

### The Collapse of the State Vector

In quantum measurement, we let the measured system Q interact with the measurement de- vice A, the state of the composite system be- comes

X n

a_{n}|ψ_{n}i_{Q}|ϕ_{n}i_{A}.

The {|ϕ_{n}i_{A}}’s represent values that could be
read out on the measurement device. The
classical nature of the measurement device im-
plies that after measurement, the state (read-
ings) of the measurement device takes on a
well defined value. The state of the compos-
ite system would then be

|ψ_{n}i_{Q}|ϕ_{n}i_{A}
for a particular value of n, or

X n

|a_{n}|^{2} ^{}|ψ_{n}i_{Q}|ϕ_{n}i_{A}^{ }_{Q}hψ_{n}|_{A}hϕ_{n}|^{} ,
the state has collapsed (become mixed).

10

The Copenhagen interpretation of measure- ment is based on the state collapse of a quan- tum system due to interaction with a macro- scopic, classical, measurement device. But interaction with a classical system cannot be part of a consistent quantum theory.

Von Neumann separates the measurement pro- cess into two stages. The first stage (“Von Neumann measurement”) describes the inter- action between Q and A that gives rise to the entangled composite state. The second stage (“observation”) is the state collapse af- ter which a definite read out on the measure- ment device is obtained.

In our formulation the realization of general measurements as projective measurements is combined with the first stage of quantum mea- surement according to von Neumann. (See The Measurement Process I.)

### Views on the Collapse of the State Vector

During the collapse, a completely known state vector seems to have evolved into one of sev- eral possible outcome states (a mixed state when the result is not known). Yet such a pro- cess is not describable with unitary evolution, giving rise to the third postulate of quantum mechanics and two distinct ways for a system to evolve; one deterministic and linear, and the other probabilistic.

Von Neumann divided the measurement pro- cess in an attempt to resolve the paradox, yet the observation (interaction with a classical system) must occur at some stage, the para- dox still exists. Von Neumann brings the hu- man consciousness into the picture.

The Everett-DeWitt interpretation suggests that the observation stage never takes place, only

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one of each possibility is available for one (con- scious or mechanical) observer, each observer records a possible version of reality. Thus this unprovable proposition states that the universe’s state branches at each quantum event. The universe observes itself ?

Environment decoherence is also used to ex- plain the paradox. But while irreversibility is accounted for by the interpretation, the col- lapse is still not explained, thus the cat is still a paradox.

### Decoherence I: Depolarization

Decoherence is the process in which a pure (coherent) state becomes mixed due to in- teraction with another system (environment).

That is, it becomes entangled with another system.

In the depolarization of a qubit system, such an entanglement is manifested in the form of qubit errors:

1. Bit flip error: |0i → |1i

|1i → |0i (σ_{1}).

2. Phase flip error: |0i → |0i

|1i → −|1i (σ_{3}).

3. Both errors: |0i → +i|1i

|1i → −i|0i (σ_{2}).

12

Suppose the qubit system A becomes entan-
gled with the environment E, since there are
four situations of interest (no error and three
kinds of errors), we can set the state space of E
to four dimensions with basis {|0i, |1i, |2i, |3i},
representing no error, σ_{1} (error 1), σ_{2} (error
3), and σ_{3} (error 2) respectively.

For initial state |ψi_{A}|0i_{E} = (a|0i_{A}+ b|1i_{A})|0i_{E},
and probability of error p with each kind of
error equally possible, then the state of system
AE evolves as

|ψi_{A}|0i_{E}

→ ^{p}1 − p|ψi_{A}|0i_{E} +

rp

3 (σ_{1}|ψi_{A}|1i_{E}
+σ_{2}|ψi_{A}|2i_{E} + σ_{3}|ψi_{A}|3i_{E}) .

The state of system A becomes mixed (entan- gled with system E). Measuring E puts A in a definite state and any errors can be correct by consulting the measured value.

Without loss of generality, we can assume |ψi_{A} =

|0i_{A}, which has spin pointing in the (0, 0, 1) di-
rection, the initial density matrix is ^{1}_{2}(1 + σ_{3}),
and the evolved density matrix is

1 − p + _{3}^{p} 0
0 2_{3}^{p}

!

= 1 2

1 ^{+ (1} − 4p
3 )σ_{3}

,

which is a mixed state with Bloch ball repre-
sentation (0, 0, 1 − ^{4p}_{3} ). With enough evolution
cycles the state would become ^{1}_{2}1^{, which is}
a completely random state. In other words,
when the qubit has depolarized from the state

|0i_{AA}h0| to ^{1}_{2}1, information is completely lost
to the environment.

### Decoherence II: Phase-damping

In phase-damping decoherence, the state of the system is not changed after entanglement with the environment, yet the relative phase information is lost, resulting in a mixed state.

For a qubit system A and environment E, phase- damping occurs as the unitary transformation

|0i_{A}|0i_{E} → ^{p}1 − p|0i_{A}|0i_{E} + √

p|0i_{A}|1i_{E}

|1i_{A}|0i_{E} → ^{p}1 − p|1i_{A}|0i_{E} + √

p|1i_{A}|2i_{E}
The environment occasionally gets changed to
states |1i_{E} and |2i_{E} when qubit A is in |0i_{A}
and |1i_{A} respectively. The basis {|0i_{A}, |1i_{A}}
is preferred by the environment in this case in
that they are the only states that does not
change due to environment interaction, any
other state will be changed in the process.

13

For the initial state |ψi_{A}|0i_{E} = (a|0i_{A}+b|1i_{A})|0i_{E}
of system AE, evolution produces the new state

a√

1 − p|0i_{A}|0i_{E} + a√

p|0i_{A}|1i_{E}
+ b√

1 − p|1i_{A}|0i_{E} + b√

p|1i_{A}|2i_{E}
The initial density matrix of A is

ρ_{A} = |a|^{2} ab^{∗}
a^{∗}b |b|^{2}

!

.

We obtain the state of A by partial trace
ρ^{0}_{A} = |a|^{2}|0i_{AA}h0| + ab^{∗}(1 − p)|0i_{AA}h1|

+a^{∗}b(1 − p)|1i_{AA}h0| + |b|^{2}|1i_{AA}h1|

= |a|^{2} ab^{∗}(1 − p)
a^{∗}b(1 − p) |b|^{2}

!

.

The off-diagonal elements would decrease af- ter evolution, with enough evolution cycles the density matrix would become

ρ^{00}_{A} = |a|^{2} 0
0 |b|^{2}

!

= |a|^{2}|0i_{AA}h0| + |b|^{2}|1i_{AA}h1|.

Phase information is lost and we are left with classical probabilities of the preferred basis states.

Phase-damping can be used to explain the for- mulation of the unnatural Schr¨odinger cat state, but it still remains to explain why only one of the outcome is preceived.

Initially the cat and the atom is not entangled:

|0i_{A}|Alivei_{Cat},

After interaction they are entangled:

√1

2|0i_{A}|Alivei_{Cat} + 1

√2|1i_{A}|Deadi_{Cat},

The environment prefers either a live cat or a dead cat, but not any combination of these two, so the whole Atom-Cat system phase- damps to the density matrix state

1

2((|0i_{A}|Alivei_{Cat})(_{A}h0|_{Cat}hAlive|)
+(|1i_{A}|Deadi_{Cat})(_{A}h1|_{Cat}hDead|))
The cat has probability ^{1}

2 to be dead or alive, but what does that mean?

### N. J. Cerf and C. Adami’s Views on the Quantum Measurement Process

The collapse of the physical state in the quan- tum measurement process is an illusion brought about by the observation of part of a com- posite system that is quantum entangled and thus inseparable. Rather than collapsing, the state of a measured system becomes entangled with the state of the measurement device. The fact that no state collapse or quantum jump occured is implicit in the quantum eraser phe- nomenon.

Due to the absence of state collapse, the uni- tary description of quantum measurement is reversible. But in a general measurement situ- ation where the measured system is entangled with a macroscopic system, it is practically im- possible to track all the atoms involved, thus it is practically irreversible.

14