Positive Operator-Valued Measure (POVM)
From the third postulate of quantum mechan- ics, if we define
Em ≡ M†mMm, then
X m
Em = 1 and p(m) = hψ|Em|ψi.
The positive operators Em are the POVM ele- ments associated with the measurement, and the set {Em} is a POVM.
Since the POVM elements sum to identity, they provide a partition of identity. In the special case of projective measurements, the Em’s are orthogonal projections, and the di- rect sum of the subspaces they project onto is the Hilbert space of the measured system.
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Schmidt Decomposition
For any state |ψiAB of a composite system AB, there exists orthonormal states |iiA and |iiB for systems A and B respectively such that
|ψiAB = X
i
λi|iiA|iiB,
where the λi’s are non-negative real numbers satisfying Pi λ2i = 1.
So for any pure state |ψiAB, the density oper- ator for systems A and B are
ρA = trB (|ψiAB) = Pi λ2i |iiAAhi|
ρB = trA(|ψiAB) = Pi λ2i |iiBBhi|
They have the same eigenvalues.
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The λi’s are the Schmidt coefficients, the bases {|iiA} and {|iiB} are called the Schmidt bases for A and B respectively, and the num- ber of non-zero λi’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 |ψiAB 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
|ψiAB = X
i
λi(U|iiA)|iiB,
the Schmidt number remains constant.
Purifications
Given a state ρA in system A, we can introduce a system R and define a pure state |ΨiAR of the composite system AR such that
ρA = trR (|ΨiARARhΨ|) .
That is, the mixed state ρA becomes the pure state |ΨiARARhΨ|. 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 ρA of system A, ac- cording to the basic properties of density ma- trices, we can diagonalize it:
ρA = X
i
pi|iiAAhi|,
where the |iiA’s are mutually orthogonal and pi ≥ 0, Pi pi = 1.
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Take system R to have the same state space as A, and define an orthonormal basis {|iiR}, then the pure state
|ΨiAR = X
i
√pi|iiA|iiR
is a purification for ρA. Taking the partial trace
trR (|ΨiARARhΨ|)
= trR
X
i
√pi|iiA|iiR
X
j
(√
pj)∗Ahj|Rhj|
= trR
X
ij
√pipj (|iiA ⊗ |iiR) (Ahj| ⊗ Rhj|)
= X
ij
√pipj|iiAAhj|tr (|iiRRhj|)
= X
ij
√pipj|iiAAhj|δij
= X
i
pi|iiAAhi|
= ρA
The purification process defines a pure state by its Schmidt decomposition, that is |iiA and |iiR are the Schmidt bases for A and R respectively.
The Schmidt coefficients are √ pi.
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
|Ψ0iAR = X
i
√pi|iiA ⊗ U|iiR,
trR |Ψ0iARARhΨ0|
= trR
X
i
√pi|iiA ⊗ U|iiR
X
j
(√
pj)∗Ahj| ⊗ Rhj|U†
= ρA.
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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 |xiA = a|0iA+ b|1iA, 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):
|xiA|yiB = (a|0iA + b|1iA) ⊗ (a|0iB + b|1iB),
|xiA|yiB = (a|0iA + b|1iA) ⊗ (b|0iB + a|1iB).
If they are entangled, then the composite states are:
a|0iA|0iB + b|1iA|1iB, a|0iA|1iB + b|1iA|0iB.
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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 PQ or PR by ran- dom, while particle 2 is measured for physical properties PS or PT by random. All measure- ment results are either 1 or −1. The measure- ment results are denoted Q, R, S, and T for PQ, PR, PS, and PT respectively.
Since
Q, R, S, T = ±1 we have
QS + RS + RT − QT
= (Q + R)S + (R − Q)T = ±2.
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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
!
= σ3, S = √1
2
−1 −1
−1 1
!
R = 0 1 1 0
!
= σ1, 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 = n1σ1 + n2σ2 + n3σ3 =
2 X m=1
λmPm, where
λ1 = 1, P1 = E(ˆn, +) = 12(1 + ˆn · ~σ), λ2 = −1, P2 = E(ˆn, −) = 12(1 − ˆn · ~σ),
and the probabilities for obtaining spin up and spin down are
p(1) = hψ|P1|ψi, p(2) = hψ|P2|ψi, respectively.
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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
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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(ˆn, +) ⊗ E( ˆm, +)) |ψ−i
= hψ−| (E(ˆn, −) ⊗ E( ˆm, −)) |ψ−i
= 1
4(1 − cos θ)
hψ−| (E(ˆn, +) ⊗ E( ˆm, −)) |ψ−i
= hψ−| (E(ˆn, −) ⊗ E( ˆm, +)) |ψ−i
= 1
4(1 + cos θ) So we have probabilities 1
2(1− cos θ) of obtain- ing the same result and 12(1 + cos θ) of obtain- ing different results.
Consider measuring along three co-plane axes 60◦ apart, measuring the first qubit along ˆn1 with result r1 and the second along −ˆn2 with result r2 we can conclude that if we could somehow measure the first qubit along ˆn1 and ˆn2 we would obtain the results r1 and r2. The sum of the probabilities that the same result is obtained for any two axes is
Psame(ˆn1, ˆn2) + Psame(ˆn2, ˆn3) + Psame(ˆn1, ˆn3)
= 3 · 12(1 − cos 60◦) = 34.
Since there are only two possible results, the Bell inequality in this case is
Psame(ˆn1, ˆn2)+Psame(ˆn2, ˆn3)+Psame(ˆn1, ˆn3) ≥ 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.
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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
an|ψniQ|ϕniA.
The {|ϕniA}’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
|ψniQ|ϕniA for a particular value of n, or
X n
|an|2 |ψniQ|ϕniA Qhψn|Ahϕn| , the state has collapsed (become mixed).
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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 (σ1).
2. Phase flip error: |0i → |0i
|1i → −|1i (σ3).
3. Both errors: |0i → +i|1i
|1i → −i|0i (σ2).
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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, σ1 (error 1), σ2 (error 3), and σ3 (error 2) respectively.
For initial state |ψiA|0iE = (a|0iA+ b|1iA)|0iE, and probability of error p with each kind of error equally possible, then the state of system AE evolves as
|ψiA|0iE
→ p1 − p|ψiA|0iE +
rp
3 (σ1|ψiA|1iE +σ2|ψiA|2iE + σ3|ψiA|3iE) .
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 |ψiA =
|0iA, which has spin pointing in the (0, 0, 1) di- rection, the initial density matrix is 12(1 + σ3), and the evolved density matrix is
1 − p + 3p 0 0 23p
!
= 1 2
1 + (1 − 4p 3 )σ3
,
which is a mixed state with Bloch ball repre- sentation (0, 0, 1 − 4p3 ). With enough evolution cycles the state would become 121, which is a completely random state. In other words, when the qubit has depolarized from the state
|0iAAh0| to 121, 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
|0iA|0iE → p1 − p|0iA|0iE + √
p|0iA|1iE
|1iA|0iE → p1 − p|1iA|0iE + √
p|1iA|2iE The environment occasionally gets changed to states |1iE and |2iE when qubit A is in |0iA and |1iA respectively. The basis {|0iA, |1iA} 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.
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For the initial state |ψiA|0iE = (a|0iA+b|1iA)|0iE of system AE, evolution produces the new state
a√
1 − p|0iA|0iE + a√
p|0iA|1iE + b√
1 − p|1iA|0iE + b√
p|1iA|2iE The initial density matrix of A is
ρA = |a|2 ab∗ a∗b |b|2
!
.
We obtain the state of A by partial trace ρ0A = |a|2|0iAAh0| + ab∗(1 − p)|0iAAh1|
+a∗b(1 − p)|1iAAh0| + |b|2|1iAAh1|
= |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
ρ00A = |a|2 0 0 |b|2
!
= |a|2|0iAAh0| + |b|2|1iAAh1|.
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:
|0iA|AliveiCat,
After interaction they are entangled:
√1
2|0iA|AliveiCat + 1
√2|1iA|DeadiCat,
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((|0iA|AliveiCat)(Ah0|CathAlive|) +(|1iA|DeadiCat)(Ah1|CathDead|)) 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.
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