Quantum Entangelments Prof. Leonard Susskind, Stanford University Álvaro Moreno Vallori
Contents
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Previou Previous s mathema mathematic tical al concept concepts s
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1. Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2. Dirac’s bra-ket notation: basic concepts . . . . . . . . . . . . . . . . . . . . . . . .
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3. Bra-ket notation and linear operators . . . . . . . . . . . . . . . . . . . . . . . . .
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States Sta tes and and observ observabl ables es
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1. Definitions and basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2. Eigenvectors and eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter Chapter 1
Previous mathematical concepts
1. Hilbert Hilbert spaces spaces
Definition 1.1. Let V be a vector space over a field F
there is a function (inner product) ·|· : V
× V −→ F for all vectors x,y,z ∈ V and all scalars α, β ∈ F : F : 1) x|y = y|x∗ . 2) αx + β y |z = αx|z + β y|z. 3) x|x ≥ 0, and x|x = 0 ⇐⇒ x = 0.
1
⊆ C. V is an inner product space if
that satisfies the following properties
1.2. Let Let V be an inner product space over a field K Definition 1.2.
C.
⊆
a function || · || : V
−→ R defined by the inner product as ||x|| = following properties for all vectors x, y ∈ V and for all scalar α ∈ K :
A norm on V is
x, x , that satisfies the
1) ||αx|| = |α| ||x||. 2) ||x + y|| 3) ||x||
≤ ||x|| + ||y||
≥ 0, and ||x|| = 0 ⇐⇒
x=0
Notice that then V is a metric space , where the distance is defined by the norm as d(x, y) =
||x
− y||. 1.3. Let Let M be a metric space. M is a complete complete space space if and only if every Definition 1.3.
Cauchy sequence converges with respect to this norm to an element in M .
1
We use here Dirac’s bra-ket notation x|y instead of the usual notation x, y .
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1.4. Let V be a vector space over a field F Definition 1.4.
⊆ C.
V is a Hilbert space space if and
only if V is an inner product space that is also a complete metric space with respect to the distance function induced by the inner product.
2. Dirac’s Dirac’s bra-ket bra-ket notation: notation: basic concepts concepts
Definition 2.1. Given a complex Hilbert space2 H, every vector in H is called a ket , and
is written as |ψ . If two vectors vectors correspond to the same quantum quantum state, they are written using
the same ket. A ket |ψ
∈ H can be viewed as a column vector written out in coordinates: |ψ =
2.3. Every Every ket ket |ψ Definition 2.3.
a1 a2 .. . an
: a1 , a2 , . . . , an
∈C
∈ H has a dual bra in H, written as ψ|, which is the
hermitian conjugate of |ψ , that is, a row vector containing the complex conjugate of the
elements of |ψ :
ψ| =
a1∗ a2∗ . . . an∗
2.3. The functi function on ψ| : H Definition 2.3.
−→ C defined by ψ|(|φ) = ψ|φ is a continuos
linear linear functional functional or linear form (i.e. a linear map (linear function) function) from a vector vector space to its field of scalars) where ψ|φ denotes the inner product of |φ with ψ|.
2
That is, a Hilbert space over the complex field C. During all this course, we will always always consider H as a finite-dimensional complex Hilbert space.
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3. Bra-ket Bra-ket notation and linear operators Definition 3.1. Let A be a matrix in M m×n (C), such that
A=
a11
a12
···
a1n
a21 .. .
a22 .. .
··· .. .
a2n .. .
am1 am2 · · ·
amn
then the hermitian conjugate of A is a matrix A†
†
A =
If A : H Note 3.2. If A
∈ M ×
n m (C)such
(a11)∗ (a21)∗ · · · (a12)∗ (a22)∗ · · ·
(am1 )∗
.
(am2 )∗ .. .
(a1n )∗ (a2n )∗ · · ·
(amn )∗
.. .
.. .
..
that
3
−→ H is a linear operator then A can be represented as a matrix . In
fact, a matrix is a linear operator and viceversa 4 .
3.3. If A If A is a matrix, we can apply A to a ket |ψ Note 3.3.
∈ H to obtain the ket A|ψ. Matric Matrices es can be also also viewe viewed d as acting acting on bras. bras. Composi Composing ng the bra φ| ∈ H with the matrix A results in the bra φ|A, defined as a linear functional φ|A : H −→ C by the rule (φ|A)(|ψ) = φ|(A|ψ). This expression is commonly written as φ|A|ψ. Definition 3.4. A given matrix A is a hermitian matrix 5if and only if satisfies A† = A.
If A is a matrix and |ψ , |φ Proposition 3.5. If A 1) ψ|A|φ = φ|A† |ψ ∗
∈ H, then:
2) If A is hermitian, ψ|A|ψ = ψ|A|ψ∗ , and therefore ψ|A|ψ ∈ R. 3
During this course, we will just assume any matrix is a linear operator A : H −→ H. This is only true when H is finite dimensional dimensional,, otherwise otherwise a linear linear operator operator cannot cannot be represente represented d as a matrix. 5 More generally generally,, if we have just a linear linear operator, operator, this definition definition is equivalen equivalentt by saying the operator operator is self-adjoint. 4
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Chapter Chapter 2
States and observables
1. Definitions Definitions and basic concepts
Definition 1.1. Given a quantum system, an observable is a property of the system state
than can be determined by some sequence of physical operations. Definition 1.2. A quantum state is set of observables that fully describes a quantum sys-
tem. Quantum Quantum states can be either pure or mixed. A pure quantum state is one that can be obtained obtained in a measuremen measurement. t. A mixed quantum quantum state is the state of a quantum quantum system system while it is not observed. Any given given physical physical system is identified identified with some Hilbert space, space, such such that Axiom 1.3. Any each non-zero ket in the Hilbert space corresponds to a pure quantum state 1 . In other words, each each pure state is a ray in the Hilbert space. In adittion, adittion, two two kets that diff er er only by a nonzero complex scalar α correspond to the same state: |ψ
≈ α|ψ.
Axiom 1.4. If H is describing a quantum system, any given observable is identified with
a hermitian matrix A. Given a hermitian hermitian matrix A and given |ψ Definition 1.5. Given
∈ H, the expression ψ|A|ψ gives the expect expectation ation value or average average value of the observable A in a the state |ψ . In this this case, we can just write the expression as A . ψ
1
Notice that if a quantum state in the given system is defined by n obersvables, then the Hilbert space has dimension n.
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1.6. If {|e1 , |e2 , . . . |en } is a basis of a H, then any ket |ψ Definition 1.6
expressed as a linear combination of the basis kets:
∈ H can be
n
/ |ψ =
∃c , c , . . . , c ∈ C 1
n
2
c1 |ei
i=1
In physical terms, this is described by saying that |ψ has been expressed as a quantum super-
position of the states |ei .
If H descri describes bes a physic physical al system system which which can be in the pure states states
Proposition 1.7.
|φ1 , |φ2 . . . |φm
∈ H, then the general mixed state of the system while is not observed is
defined by a the ket
|ψ = c1 |φ1 + c2 |φ2 + . . . + cm |φm
∈ C. In this linear combination, for all i ∈ {1, 2, . . . , m} c c∗ represent the probability that the state of the system is |φ. In adittion, adittion, the ket should should be normalize normalized, d, i.e.
where c1 , c2 , . . . cn
i i
the sum of the probabilities should be one: n
ci c∗i = 1
i=1
Moreover, |ψ can be represented by a column vector |c containing the coe fficients ci in a m-
dimensional complex Hilbert space, where each dimension is identified with each pure state. In this case the fact that |c is a normalized vector can be written c|c = 1.
Examples 1.8.
1) If we bring an electron into a vertical magnetic field, the spin will orientate either in the direction of the field or in the opposite direction, so the state of the electron, when measured, will be either |up or |down . In this case case the kets kets are elemen elements of a 2-dimensional complex
Hilbert space H. The spin of the electron, electron, while not measured, measured, may be b e any normalized normalized ket
|ψ = c1 |up + c2 |down , where c1 , c2
T
c1 c2
∈ C. According to the proposition 1.7., the column vector
represents the state |ψ in a 2-dimension -dimensional al complex complex Hilbert space. space. For example, example,
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Álvaro Moreno Vallori - Quantum Entangelments
the vector
1
1
√ , √ 2 2
T
being up or down, since
Previous mathematical mathemat ical concepts concepts
represent a state in which the electron has the same probability of
√1
2
2
= 12 .
2) If we have have an electr electron on in a Neon Neon athom, athom, its state, state, when when measur measured ed,, is described described by its quantum numbers. The possible pure states are:
1 0 0 1 2
,
1 0 0
−
1 2
,
2 0 0 1 2
,
2 0 0 1 2
2 1 0 1 2
2 1
,
0
−
1 2
−
2 1
,
1
1 2
−
2
2
1
,
1
−
−
1 2
1
,
1
1 2
−
2 1
,
−
1 1 2
In this case the kets are elements of a 4-dimensional complex Hilbert space H. The state of the electron, while not measured, may be any normalized ket |ψ = c1 |φ1 + c2|φ2 + . . . + c10|φ10 ,
where c1 , c2 , . . . c10
∈ C and φ , φ , .. . , φ 1
2
proposition proposition 1.7., the column vector
10
are the possible possible pure states. states. Accor Accordin dingg to the
c1 c2 . . . c10
dimensional complex Hilbert space.
T
represents the state |ψ in a 10 10--
2. Eigenvectors Eigenvectors and eigenvalues
2.1. Given Given a matrix matrix A, a non-zero vector |a is defined to be an eigenvector Definition 2.1.
of A if and only if satisfies the equation A|a = λa |a for some λ
∈ C.
In this situat situation ion,, the
scalar λ is called an eigenvalue of A corresponding to the eigenvector |a .
If A is a hermitian matrix and λ is an eigenvalue of A, then λ Proposition 2.2. If A
∈ R.
.3. If A is an observable, the values of A that can be measured by an Proposition 2.3. experiment are the eigenvalues of A. In adittion, an eigenvector |a of A represents a state in
which the observable A has a probability 1 to be λa .
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Examples 2.4.
1) The vector since
In particular,
2) The vector since
In particular,
1 0
1
is an eigenvector of any matrix
m11 m12 0
1
m22
0
0
0 1
0
m11
0
0
m21 m22
1
2
0
2
and S y
=
σ3
2
2
, where
σ2
where
σ3
0
m22
, with eigenvalue m11,
1 0
= m22
1
m11
0
m21 m22
, with eigenvalue m22,
0 1
is an eigenvector of any diagonal matrix, with eigenvalue m22.
1
since
=
is an eigenvector of any matrix
2
1
0
0
−1
− −
the z direction2 . S z has two eigenvectors,
S z
= m11
m11 m12
is an eigenvector of any diagonal matrix, with eigenvalue m11.
3) The spin projection operator S z =
2
1 0
1 0
0
1
1
0
1
0 0
−
and
=
=
1
aff ects ects a measurement of the spin in 0 1
1
2
0
0
2
1
, with eigenvalues
2
and
is the third Pauli matrix. There are also two more spin operators, which are S x
σ1
=
»
0
1
1
0
–
and
σ2
=
»
0
−i
−i
0
turns out that S x , S y and S z have the same eigenvalues.
8
–
− 2 ,
=
2
σ1
are the first and the second Pauli matrice matrices. s. It
Álvaro Moreno Vallori - Quantum Entangelments
2.5. If A If A Proposition 2.5.
Previous mathematical mathemat ical concepts concepts
∈ M (C) is a hermitian matrix, then A has n egeinvectors, each n
one with its own eigenvalue. 2.6. Let A be a hermitian matrix and let |a and |b be two eigenvalues of A. Theorem 2.6.
If the eigenvalues λa and λb are diff erent, erent, then the vectors |a and |b are orthogonal.
Proof.
We begin from the following equations: A|a = λa |a
A|b = λ |b b
Using the known rules of inner product in bra-ket notation: A|a = λa |a
=⇒ b|A|a = λ b|a =⇒ b|A|a = λ b|a =⇒ a|A|b = λ a|b b|A|a = λ∗b|a A|b = λ |b λ ∈R b
a
a
b
b
⇒ b|A|a = λ b|a =⇒ λ b|a = λ b|a =⇒ b|A|a = λ b|a Therefore |a ⊥ |b. a
=
a
b
b
(λa
b
− λ )b|a = 0 =⇒ b|a = 0 = λ λ b
a
b
Theorem 2.6. Let A be an observable and let |a be a certain normalized eigenvector of
A. If the system is prepared prepared in any arbitrary state described described as a normalized normalized vector vector |b , the
probability of measuring λa is a|b a|b ∗ = a|b b|a .
2.7. Let Let |b = Example 2.7.
1
be a vector which describes the state of the spin of an 0 electron, so that the electron is pointing upwards. |b is an eigenvector of the observable S z , so
we know precisely that when we measure S z we will obtain 1. The probability of measure the eigenvalue 1 of S x is given by a|ba|b∗ , where a is the eigenvector to the eigenvalue 1:
1
√
2
1
√
2
1 0
1
√
2
1
√
2
1 0
9
∗
1 = 2
√1
2
1
of S x associated
√
2
1 ∗ 1 1 1 = = 2 2 2 2
√ √
√ √
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Notice that in order to measure the observable S x we should create a horizontal magnetic field, so the spin of the electron will stop ponting upwards and will change to point in the direction of the field. The fact that the spin is in a vertical vertical direction implies implies that the electron has the same probability of emiting a photon (acting as if it was prepared in the opposite direction of the field) as emiting none (acting as if it was prepared on the direction of the field), so that is why the probability of measure the eigenvalue 1 of S x is 12 . 2.8. The operator operator associate associated d with with spin spin in an arbitr arbitrary ary directio direction n given given by a Definition 2.8. unitary vector n ˆ=
n1 n2 n3
is the sum of the three spin operators for the three spatial
directions each one multiplied by the component of n ˆ in each of the axis:
n1S x + n2 S y + n3 S z = n1
=
2
0
n1
n1
0
+
2
0
−n i
n2 i
0
2
0 1 1 0
+
+ n2
2
n3
0
0
−n
−
0 i
i
0
=
3
+ n3
1
0
2
0
−1
n3
n1
2
n1 + n2 i
=
−n i 2
n3
Proposition 2.9. The square of the operator associated with spin in an arbitrary direction
(including the directions of the axis, with the operators S x , S y and S z ) is 2 I , where I is the identity identity matrix. That means the eigenvalues of these matrices are always
10
1 0
and
0 1
.
