Introduction to Quantum Computation
量子計算導論
Topic 1 Fundamental Postulates
1
Fundamental postulates of quantum mechanics
• Postulate 1: state space
• Postulate 2: evolution of (closed) system
• Postulate 3 : measurement
• Postulate 4: composite system
2
Postulate 1: state space
3
Hilbert space
• Hilbert space is a real or complex vector space equipped with an inner product; meanwhile,
it is also complete with respect to the norm induced by the inner product.
• State space is a complex Hilbert space to which the states of a quantum system belong.
• State vector: a unit vector in the system’s state space.
→ Important constrain on the parameters of a state vector.
• Dirac ket-bra notation:
 ,
 = 
†
basis vector:
† (dagger): conjugate transpose
,
for qubit: computational basis
•
Dimension of the Hilbert space = Number of basis.
4
Hilbert space
• Arbitrary state vector (linear superposition):
•
Conjugate transpose:
• Porbability amplitude:
(complex numbers).
• Porbability amplitudes are NOT probabilities!
• Unity of state vector:
→ Important constrain on the parameters of a state vector.
5
Qubit state
• Arbitrary qubit state (linear superposition):
•
Conjugate transpose:
•
Porbability amplitude:
(complex numbers).
• Unity of state vector:
→ Important constrain on the parameters of a state vector.
6
Qubit state
• Arbitrary qubit state (linear superposition):
→ To determine a and b.
• Porbability amplitude:
→
→ There are 4 degrees of freedom.
•
Global phase, relative phase:
→ 3 degrees of freedom left.
• Unity of state vector:
neglect the global phase.
consider only the relative phase.
origin of quantum interference.
→
→ 2 degrees of freedom left.
7
Bloch sphere
• Bloch vector:
• The length is 1 for qubit state vectors (pure states):
• Bloch sphere representation is valid for qubit states exclusively; not available for higher
dimensional systems.
8
Inner product
• Inner product: a binary map 〈Φ|Ψ〉 brings an ordered pair of vectors to a complex
number satisfying the following properties:
1.
2.
3.
Positivity:〈Ψ|Ψ〉 > 0.
Linearity:〈Φ|(a1|Ψ1〉+a2|Ψ2〉) = a1〈Φ|Ψ1〉+ a2〈Φ|Ψ2〉.
Skew symmetry:〈Φ|Ψ〉* =〈Ψ|Φ〉.
• The norm is a real-valued function Ψ =
Ψ, Ψ ∈ ℝ.
1. It plays the role of the “length” of a vector.
2. The “distance” between two vectors: Ψ − Φ =
3. Triangle inequality: Ψ + Φ ≥ Ψ − Φ
•
Ψ − Φ, Ψ − Φ .
Orthonormality = orthogonality & normalization:
9
Inner product
Inner product of two qubit state:
•
Method 1 (Dirac ket-bra notation):
•
Method 2 (matrix multiplication):
10
Physical realization of qubit I: photon
A plane EM wave
Maxwell’s equations
Hamiltonian operator
2nd quantization
: Horizontal polarization
: Vertical polarization
11
Physical realization of qubit II: gate-defined quantum dots
: Spin up
: Spin down
Rev. Mod. Phys. 79, 1217 (2007).
12
Physical realization of qubit II: gate-defined quantum dots
Rev. Mod. Phys. 79, 1217 (2007).
13
Physical realization of qubit II: gate-defined quantum dots
S-T0 qubit
Rev. Mod. Phys. 79, 1217 (2007).
Chen et al., Nat. Commun. 10, 3794 (2019).
14
Physical realization of qubit III: diamond NV center
• Diamond NV- center: atomic structure
Spinless 12C (J = 0)
Bz
Spinful 13C (J = 1/2)
M.-C. Lin et al., J. Phys.: Condens. Matter 34, 505701 (2022)
Y.-H. Kuo and H.-B. Chen., Phys. Rev. Research 5, 043139 (2023)
• Primary advantage: extremely long quantum coherence time (~1 sec) at room temperature
Science 336, 1283 (2012)
• Existing applications:
quantum sensor
QI processor
quantum computer (commercialized)
Quantum Brilliance
Sci. Rep. 9, 6938 (2019)
Commun. Phys. 5, 102 (2022)
15
Physical realization of qubit III: diamond NV center
• Molecular orbitals of three NV centers (NV-, NV0, NV+)
New J. Phys. 13, 025019 (2011)
• Energy level
Effective qubit!
Phys. Rev. B 92, 115206 (2015)
M.-C. Lin et al., J. Phys.: Condens. Matter 34, 505701 (2022)
Y.-H. Kuo and H.-B. Chen., Phys. Rev. Research 5, 043139 (2023)
16
Physical realization of qubit IV: superconducting circuit
Superconductivity
• Superconductivity is a phenomenon observed in certain materials where
electrical resistance drops abruptly to zero and magnetic fields are
expelled from the material at temperature lower than a critical one.
• Superconductivity was discovered in 1911 by Dutch physicist Heike
Kamerlingh Onnes.
• He was devoted to the properties of pure metals at very low temperatures.
• On April 8 1911, he foud that at 4.2 K the resistance in a solid mercury
wire immersed in liquid helium suddenly vanished.
• He won the 1913 Nobel Prize in Physics.
• Meissner effect is the expulsion of a magnetic field from a superconductor during its
transition to the superconducting state.
• This phenomenon is discovered in 1933 by German physicists Walther Meiβner and
Robert Ochsenfeld.
• LK-99?
17
Physical realization of qubit IV: superconducting circuit
BCS theory
• The BCS theory is the first microscopic theory of superconductivity since its discovery
in 1911.
• It was proposed by Bardeen, Cooper and Schrieffer in 1957; they won the Nobel Prize in
Physics for this theory in 1972.
Cooper pair
18
Physical realization of qubit IV: superconducting circuit
Josephson junction
• The Josephson effect is a phenomenon describing the supercurrent, in the
absence of external voltage, penetrating a barrier sandwiched between
two superconductors placed in proximity.
• It was predicted mathematically in 1962 by the British physicist Brian
Josephson.
• He won the Nobel Prize in Physics for his prediction in 1973.
• Electrical symbol:
Nature 474, 589 (2011)
𝜕 𝜓𝐿 𝑒 𝑖𝜙𝐿
𝑒𝑉
𝑖ℏ
=
𝐾
𝜕𝑡 𝜓𝑅 𝑒 𝑖𝜙𝑅
𝜓𝐿 𝑒 𝑖𝜙𝐿
𝐾
−𝑒𝑉 𝜓𝑅 𝑒 𝑖𝜙𝑅
Supercurrent: I(t) ∝ sin(𝜙𝐿 − 𝜙𝑅 )
19
Physical realization of qubit IV: superconducting circuit
The first Josephson charge qubit
box size: 700 × 50 × 15 nm Al strip
Superconductor Insulator
Cooper pair box Josephson junction
Nature 398, 786 (1999)
Superconductor
Reservoir
Vg
• Hamiltonian
20
Physical realization of qubit IV: superconducting circuit
• Three primary superconducting qubit archetypes.
Nature 474, 589 (2011)
𝐸𝐽
≪1
𝐸𝐶
𝐸𝐽
~101
𝐸𝐶
𝐸𝐽
~106
𝐸𝐶
• There are many other hybridizations of these primary archetypes.
Transmon, Fluxonium, Xmon, Gatemon, Unimon
21
Physical realization of qubit IV: superconducting circuit
Transmon qubit
Devices designed by IBMQ
npj QI 3, 2 (2017)
[Advantage] An increase of the ratio EJ /EC leads to an exponential
decrease of the charge dispersion and thus a qubit transition frequency
that is extremely stable with respect to charge noise.
22
Physical realization of qubit V: quantum transport in QDs
T. Fujisawa and S. Tarucha,
Superlatt. Microstr. 21, 247 (1997).
L
ΓL
Δ
R
ΓR
Phys. Rev. Lett. 105, 176801 (2010).
23
Physical realization of qubit VI: photosynthetic complexes
• Photosystem II (PSII) and reaction center (RC) of Synechococcus Elongatus
SCIENCE 303, 1831 (2004).
• Simplified energy struture
Chen et al., Phys. Rev. E 94, 052101 (2016).
24
Postulate 2: evolution
25
Photon polarization and half-wave plate
HWP @ 22.5˚
26
Photon polarization and half-wave plate
HWP @ 22.5˚
27
Photon polarization and half-wave plate
HWP @ 22.5˚
28
Photon polarization and half-wave plate
HWP @ 22.5˚
29
Photon polarization and half-wave plate
=
=
+
30
Photon polarization and half-wave plate
HWP @ 22.5˚
31
Photon polarization and half-wave plate
HWP @ 22.5˚
32
Photon polarization and half-wave plate
HWP @ 22.5˚
33
Photon polarization and half-wave plate
HWP @ 22.5˚
34
Mathematical description
HWP @ 22.5˚
35
Mathematical description
HWP @ 22.5˚
36
Mathematical description
HWP @ 22.5˚
37
Mathematical description
HWP @ 22.5˚
38
Mathematical description
HWP @ 22.5˚
39
Mathematical description
HWP @ 22.5˚
40
Unitary evolution
HWP @ 22.5˚
• The evolution of a closed quantum system is described by a unitary operator.
• Unitarity:
(Identity operator)
Example: half-wave plate (HWP)
• HWP @ 22.5˚:
• Unitarity:
=
=
1 1 1 


2 1 −1
1 1 1 


2 1 −1
• State transformation:
• Inverse transformation:
=
=

1 1 1  1 1 1  1 0 



=

2 1 −1 2 1 −1 0 1 

1 1 1  1  1 1 1 1  1  0

  =
 =
 +
 
2 1 −1 0 
2 1
2 0
2 1 

1 1 1  1 1 1 


 = 
2 1 −1 2 1 0 
41
Equivalent notations
Matrix representaiton
1  1 1
0 =
1
2 
 
0  1  1 
1  =
 −1
2
 
 

1 1 1 
=


2 1 −1
Ket-bra notation
=
=

42
Generator of a unitary operator
• The set of all unitary operators form a unitary group U(n).
• The unitary group U(n) can be generated by the u(n) Lie algebra, which consist of
Hermitian operators.
• Every unitary operator can be expressed in the exponential form
•
•
• Given
Hermitian operator
Generator of the unitary operator.
→ Fully determine
•
Solving the eigenvalue problem will be helpful.
• Diagonalization
• Very important physical implications!
(proof?)
43
Bloch sphere
Qubit unitary operations
• Rotation about x-axis at angle α.
α
44
Bloch sphere
Qubit unitary operations
α
α
α
45
Bloch sphere
Qubit unitary operations
• Rotation about y-axis at angle α.
α
46
Bloch sphere
Qubit unitary operations
• Rotation about z-axis at angle α.
α
47
Unitarity: further insights
• The evolution of a closed quantum system is described by a unitary operator.
• Unitarity:
(Identity operator)
• The magnitude of the state vector should be conserved.
HWP @ 22.5˚
=
++ = H
H = H ˆI H = H H = 1
48
Unitarity: further insights
• The evolution of a closed quantum system is described by a unitary operator.
• Unitarity:
(Identity operator)
=
=
• Both {|H﹥, |V﹥} and {|+﹥, |-﹥} form orthonormal basis.
H H = V V =1
+ + = - - =1
H V = V H =0
+ - = - + =0
• Unitary transformation is a basis transform; i.e., rotation.
• Information of a state vector is conserved under unitary transform.
49
Unitarity: further insights
• The evolution of a closed quantum system is described by a unitary operator.
• Unitarity:
(Identity operator)
HWP @ 22.5˚
• Unitary transformation is a basis transform; i.e., rotation.
• Information of a state vector is conserved under unitary transform.
50
Postulate 2’: time-evolution
51
Two types of evolution
• Input and output. Ignorance of time.
HWP @ 22.5˚
• Initial time and final time. Continuous time variable.
t1
t2
52
Time-evolution
• The time-evolution of a closed quantum system is described by the Schrödinger equation.
: reduced Plank’s constant.
: Hamiltonian operator of the closed system satisfying the Hermitian condition
• [Theorem] The eigenvalues of an Hermitian operator are all real.
• If one knows the Hamiltonian operator of a closed quantum system, then he can fully
understand its dynamics.
Schrödinger equation:
Formal solution:
Time evolution operator:
• Solving the Schrödinger equation is equivalent to finding the eigenvalues of the
Hamiltonian operator.
53
Homework I
Pauli matrices:
1. Verify their Hermiticity
, and unitarity
2. Find their eigenvectors and eigenvalues.
3. According to the above, show that
and verify its unitarity, where
and
4. Apply the unitary operator in 3. to
,
, and
And express the results in the form of
54
Postulate 3: measurement
55
Two kinds of quantum measurement
1. Projective measurement.
2. Positive operator-valued measurement (POVM).
56
Polarizing Beam Splitter (PBS)
• Horizontal polarization
PBS
• Vertical polarization
PBS
57
Photon counting statistic (ideal cases)
PBS
Input state:
58
Photon counting statistic (ideal cases)
PBS
Input state:
Outcome: H
59
Photon counting statistic (ideal cases)
PBS
Input state:
Outcome: H
100 counts / 100 rounds: p(H) = 1 = |1|2
※ Counting statistics is relevant after
running the experiment many times.
60
Photon counting statistic (ideal cases)
PBS
Input state:
61
Photon counting statistic (ideal cases)
PBS
Input state:
Outcome: V
62
Photon counting statistic (ideal cases)
PBS
Input state:
Outcome: V
100 counts / 100 rounds: p(V) = 1 = |1|2
63
Photon counting statistic (ideal cases)
PBS
Input state:
64
Photon counting statistic (ideal cases)
PBS
Input state:
Outcome: H
65
Photon counting statistic (ideal cases)
PBS
Input state:
Outcome: V
66
Photon counting statistic (ideal cases)
PBS
Input state:
Outcome: H
1
1
50 counts / 100 rounds: p(H) = =
2
2
2
PBS
Input state:
Outcome: V
1
1
50 counts / 100 rounds: p(V) = =
2
2
2
67
Photon counting statistic (ideal cases)
PBS
Input state:
Measurement outcome: H
Post-measurement state:
Probability (after many rounds): p(H) = |a|2
Two outcomes
(one in each single round)
Measurement outcome: V
Post-measurement state:
Probability (after many rounds): p(V) = |b|2
68
Mathematical description (projective measurement)
About the observable
•
A projective measurement is described by an observable.
•
Observable:
•
Eigenvalue equation (spectral decomposition of the observable):
→ Hermitian operator → Real eignevalues
,
•
Eigenvalue mj: possible measurement outcome.
•
Eigenvector |Xj〉: post-measurement state corresponding to mj.
→ forms an orthonormal basis
69
Mathematical description (projective measurement)
About the measurement operators
•
Measurement operators:
,
→being projection operators and Hermitian operators
•
Condition 1:
→ summation of possibilities is 1.
→ completeness equation
•
Condition 2:
→ eigenvalue equation
70
Mathematical description (projective measurement)
Before measurement
unknown state
•
After measurement
•
measurement outcome: mj
•
measurement operator:
•
probability:
•
post-measurement state:
measure the observable
Average (or expectation) value of the projective measurement:
71
Issue I: collapse of quantum state after measurement
Before measurement
Input state:
Post-measurements states:
PBS
•
•
•
•
After measurement
PBS
PBS
Initial states and post-measurement states are in general different.
Initial states are projected into one of the eigenstates of the observable.
Measurement equipment will destroy the initial quantum states.
Quantum nondemolition measurement (not introduced here).
72
Issue II: a philosophical question-reality
•
•
Does the value of an observable pre-exist before measurement?
A classical measurement can, in principle, keep the system intact. It is merely used to reveal
the pre-existing information.
Albert Einstein:
Is the moon there when we are not looking?
Niels Bohr:
God does no play dice.
Stop telling God what to do!
Einstein insisted on classical mechanics, even though he
interpreted the photoelectric effect in terms of discretized
photon energy, which led to his Nobel Prize in 1921.
•
遇事不決，量子力學
73
Issue III: (tomographically) incomplete information
PBS
 =
1
1
H +
V
2
2
Measurement outcome: H
Post-measurement state:
1
Probability (after many rounds): p(H) =
2
Measurement outcome: V
Post-measurement state:
1
Probability (after many rounds): p(V) =
2
PBS
1
1 i 4
 =
H +
e V
2
2
Measurement outcome: H
Post-measurement state:
1
Probability (after many rounds): p(H) =
2
Measurement outcome: V
Post-measurement state:
1
Probability (after many rounds): p(V) =
2
PBS
1
1 i 2
 =
H +
e V
2
2
•
•
Same statistics!
Does these initial states are the same?
Measurement outcome: H
Post-measurement state:
1
Probability (after many rounds): p(H) =
2
Measurement outcome: V
Post-measurement state:
1
Probability (after many rounds): p(V) =
2
74
Issue III: (tomographically) incomplete information
PBS
 =
1
1
H +
V
2
2
Measurement outcome: H
Post-measurement state:
1
Probability (after many rounds): p(H) =
2
Measurement outcome: V
Post-measurement state:
1
Probability (after many rounds): p(V) =
2
PBS
1
1 i 4
 =
H +
e V
2
2
Measurement outcome: H
Post-measurement state:
1
Probability (after many rounds): p(H) =
2
Measurement outcome: V
Post-measurement state:
1
Probability (after many rounds): p(V) =
2
PBS
1
1 i 2
 =
H +
e V
2
2
•
•
•
Measurement outcome: H
Post-measurement state:
1
Probability (after many rounds): p(H) =
2
Measurement outcome: V
Post-measurement state:
1
p(V)
=
Probability
(after
many
rounds):
Different initial states! Particularly the relative phase!
2
Single observable is insufficient for fully characterizing a quantum state.
75
Quantum states tomography (will be introduced latter).
Issue IV: distinguishability of quantum states
Guess what state?
•
•
For classical world, distinct states of an object are (in principle) distinguishable.
This is not always the case in quantum mechanics.
•
For orthogonal states:
Alice’s side
Pick up one state in each round from:
Bob’s side
Perform the measure defined by the operators:
•
Bob can successfully distinguish orthogonal states.
•
For non-orthogonal states:
Alice’s side
Pick up one state in each round from:
Bob’s side
No measurement observable enables Bob to
1
1
1
3
distinguish the states deterministically in each
1 = H ,  2 =
H +
V , 3 = H +
V
2
2
2
2
round.
76
• Non-orthogonal states can’t be reliably distinguished by single observable!
Issue V: non-commutative observables
•
Measurement on z-direction:
measurement
Measurement outcome: +1
Post-measurement state:
Probability (after many rounds): p(+1) = 1
measurement
•
•
•
Measurement outcome: -1
Post-measurement state:
Probability (after many rounds): p(-1) = 1
Spin up and down states are the eigenstates of Pauli-z operator.
Pauli-z operator is diagonal in this basis with eigenvalues ±1.
z-direction measurement always gets definite outcome with p = 1.
77
Issue V: non-commutative observables
•
Measurement on x-direction:
measurement
Measurement outcome: +1
Post-measurement state:
1
Probability (after many rounds): p(+1) =
2
Measurement outcome: -1
Post-measurement state:
1
Probability (after many rounds): p(-1) =
2
measurement
Measurement outcome: +1
Post-measurement state:
1
Probability (after many rounds): p(+1) =
2
•
•
•
Measurement outcome: -1
Post-measurement state:
1
Probability (after many rounds): p(-1) =
2
Spin up and down states are not the eigenstates of Pauli-x operator.
Each measurement has two possible outcomes with p = 1/2.
The outcome of x-direction measurement becomes unpredictable!
[Equivalent statements]
• Two observables cannot be deterministically measured simultaneously.
• Two observables cannot be diagonalized simultaneously.
• Two observables do not commute to each other.
78
Issue V: non-commutative observables
•
ˆ ˆ := AB
ˆ ˆ - BA
ˆˆ
Commutator: [A,B]
•
•
•
Commutative operators
Non-commutative operators
ˆ ˆ = 0  AB
ˆ ˆ = BA
ˆˆ
[A,B]
ˆ ˆ  0  AB
ˆ ˆ  BA
ˆˆ
[A,B]
•
•
•
Simultaneously diagonalizable.
Common eigenvectors.
Can be deterministically measured
simultaneously.
Not simultaneously diagonalizable.
Not all eigenvectors are in common.
Cannot be deterministically measured
simultaneously.
Example: Pauli matrices
0 1 
ˆ x = 

1 0 
•
Commutation relations:
0 −i 
ˆ y = 

i 0 
[ˆ x , ˆ y ] = 2iˆ z
[ˆ y , ˆ z ] = ?
1 0 
ˆ z = 

0 −1
[ˆ z , ˆ x ] = ?
(Check this!)
79
Heisenberg uncertainty principle
•
From the forementioned issues, we can deduced that, for a quantum state |Ψ〉 and two noncommutative observables Ô 1 and Ô 2, the fluctuations (standard deviation) of the measurement
statistics are constrained. It is impossible to make them arbitrarily precise simultaneously!
•
Expectation values:
•
Fluctuations (standard deviation):
•
Heisenberg uncertainty principle:
80
Homework II
Pauli matrices:
1. Given an observable
,
where
, and
Find its eigenvectors and eigenvalues.
2. Given states
,
, and
What is the possible measurement outcomes and the
corresponding probabilities?
81
Postulate 4: composite system
82
Hilbert space
• Arbitrary state vector (linear superposition):
• Porbability amplitude:
• Unity of state vector:
Example: single photon state
: Horizontal polarization
: Vertical polarization
83
2-photon state
•
Two photons:
photon 1
photon 2
•
State vector: direct product
•
Column vector: tensor product
 1   1 
1     
1  1    0    0 
|H1〉⊗|H2〉=      = 
=

0
0
1
  0 
   
0     
 0   0 
 0   0 
0     
1
0
0
0
|V1〉⊗|V2〉=      =     =  
0 
0 
1  1 
1     
 1   1 
84
2-photon Hilbert space
•
4 bases:
•
According to the postulate 1, an arbitrary 2-photon state is a linear superposition of
the four bases :
85
Entanglement: a first glimpse
•
Consider a special state:
•
How can you separate the 2-photon state?
•
In fact, not separable!
•
The two photons are entangled together!
?
N. Bohr
•
E. Schrödinger
It turns out that entanglement becomes a very important resource in the quantum
information theory!
86
2-photon evolution
•
Local operation
HWP @ 22.5˚
HWP @ 22.5˚
•
Tensor product
87
2-photon evolution
•
Ket-bra notation
? (Check this!)
•
Matrix multiplication
? (Check this!)
88
2-photon measurement
PBS
Input state: |H1〉⊗|H2〉
Outcome: HH
100 counts / 100 rounds: p(HH) = 1 = |1|2
PBS
89
2-photon measurement
PBS
Input state: |V1〉⊗|V2〉
Outcome: VV
100 counts / 100 rounds: p(VV) = 1 = |1|2
PBS
90
2-photon measurement
PBS
Input state: a|H1〉⊗|H2〉+b|H1〉⊗|V2〉
+c|V1〉⊗|H2〉+d|V1〉⊗|V2〉
• 4 bases, 4 possible outcomes
Measurement outcome: HH
Post-measurement state: |H1〉⊗|H2〉
Probability (after many rounds): p(HH) = |a|2
PBS
Measurement outcome: HV
Post-measurement state: |H1〉⊗|V2〉
Probability (after many rounds): p(HV) = |b|2
Measurement outcome: VH
Post-measurement state: |V1〉⊗|H2〉
Probability (after many rounds): p(VH) = |c|2
Measurement outcome: VV
Post-measurement state: |V1〉⊗|V2〉
Probability (after many rounds): p(VV) = |d|2
91
Entanglement: correlations in the measurement statistics
PBS
Input state:
• 4 bases, 4 possible outcomes
Measurement outcome: HH
Post-measurement state: |H1〉⊗|H2〉
1
Probability (after many rounds): p(HH) =
2
PBS
Measurement outcome: HV
Post-measurement state: |H1〉⊗|V2〉
Probability (after many rounds): p(HV) = 0
Measurement outcomes
Measurement
outcome: VH
are correlated!
Post-measurement state: |V1〉⊗|H2〉
Probability (after many rounds): p(VH) = 0
Measurement outcome: VV
Post-measurement state: |V1〉⊗|V2〉
1
Probability (after many rounds): p(VV) =
92
2
Entanglement: correlations in the measurement statistics
Charlie distributes an entangled photon pair:
Alices’s side
Bob’s side
Measure the photon polarization in this round.
Then keep silent.
Measure the photon polarization in this round.
Measurement outcome: H
Post-measurement state: |H2〉
Bob cannot say anything on the probability.
Bob can still deduce that the measurement
outcome of Alices’s side should be H
according to the knowledge of entanglement!
•
•
•
Steering Alice’s photon? EPR paradox?
Very important resource in the quantum information theory.
More details will be introduced latter.
93
Partial measurement
PBS
Input state:
What if?
• Only perform measurement on the first
photon.
• The second photon is gone.
•
•
This will give rise to the loss of
quantum coherence.
This can be understood after the
introduction of density matrices (next
topic).
94
Partial measurement
PBS
Input state:
What if?
• Only perform measurement on the first
photon.
• The second photon is kept for quantum
state tomography.
?
•
•
This will lead to Bob’s assemblage.
This is the central issue of one of the
quantum correlations: steering.
95