NoteNextra-origin/pages/Math401/Math401_T6.md

Math 401, Topic 6: Postulates of quantum theory and measurement operations

Section 1: Postulates of quantum theory

This part is a review of the quantum theory, I will keep the content brief.

If you are familiar with the linear algebra defined before, you can jump right into this section to keep your time as viewing those compact notations.

Pure states

Pure state and mixed state

A pure state is a state that is represented by a unit vector in \mathscr{H}^{\otimes N}.

A mixed state is a state that is represented by a density operator in \mathscr{H}^{\otimes N}. (convex combination of pure states)

if \rho_j=|\psi_j\rangle\langle\psi_j|, then \rho=\sum_{j=1}^N p_j\rho_j is a mixed state, where p_j\geq 0 and \sum_{j=1}^N p_j=1.

Coset space

Two non-zero vectors u,v\in \mathscr{H} are said to represent the same state if u=cv for some complex number c with |c|=1.

The set of states of a quantum system is called the coset space of \mathscr{H}, u\sim v if u=cv for some complex number c with |c|=1.

The coset space is called the projective space of \mathscr{H}, denoted by P(\mathscr{H})\colon=(\mathscr{H}\setminus\{0\})/\sim.

Any vector in the form e^{i\theta}|u\rangle for some u\in \mathscr{H} and \theta\in \mathbb{R} represents the same state as |u\rangle.

Example: the system of a qubit has a Hilbert space \mathbb{C}^2, the coset space is P(\mathbb{C}^2)\cong S^2 is the Bloch sphere.

Composite systems

Tensor product

The tensor product of two Hilbert spaces \mathscr{H}_1 and \mathscr{H}_2 is the Hilbert space \mathscr{H}_1\otimes\mathscr{H}_2 with the inner product \langle u_1\otimes u_2,v_1\otimes v_2\rangle=\langle u_1,v_1\rangle\langle u_2,v_2\rangle.

The tensor product of two vectors u_1\in \mathscr{H}_1 and u_2\in \mathscr{H}_2 is the vector u_1\otimes u_2\in \mathscr{H}_1\otimes\mathscr{H}_2.

Multipartite systems

For each part in a multipartite quantum system, each part is associated a Hilbert space \mathscr{H}_i. The total system is associated a Hilbert space \mathscr{H}=\mathscr{H}_1\otimes\mathscr{H}_2\otimes\cdots\otimes\mathscr{H}_n.

The state of the total system has the form u_1\otimes u_2\otimes\cdots\otimes u_n for some u_i\in \mathscr{H}_i.

Entanglement (talk later)

A state |\psi\rangle is entangled if it cannot be expressed as a product state v_1\otimes v_2 for any single-qubit states |v_1\rangle and |v_2\rangle. In other words, an entangled state is non-separable.

Example: the Bell state |\psi^+\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle) is entangled.

Assume it can be written as |\psi\rangle=|\psi_1\rangle\otimes|\psi_2\rangle where |\psi_1\rangle=a|0\rangle+b|1\rangle and |\psi_2\rangle=c|0\rangle+d|1\rangle. Then:

|\psi\rangle=a|00\rangle+b|01\rangle+c|10\rangle+d|11\rangle

Setting this equal to |\psi^+\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle) gives:

ac|00\rangle+ad|01\rangle+bc|10\rangle+bd|11\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)

This requires:

ac=bd=\frac{1}{2} ad=bc=0

This is a contradiction, so |\psi^+\rangle is entangled.

Mixed states and density operators

Density operator

A density operator is a Hermitian, positive semi-definite operator with trace 1.

The density operator of a pure state |\psi\rangle is \rho=|\psi\rangle\langle\psi|.

The density operator of a mixed state is given by the unit vector u_1,u_2,\cdots,u_n in \mathscr{H} with the probability p_1,p_2,\cdots,p_n, p_i\geq 0 such that \sum_{i=1}^n p_i=1.

The density operator is \rho=\sum_{i=1}^n p_i|u_i\rangle\langle u_i|.

Trace 1 proposition

Density operator on the finite dimensional Hilbert space \mathscr{H} are positive operators having trace equal to 1.

Pure state lemma

A state is pure if and only if Tr(\rho^2)=1.

For any mixed state \rho, Tr(\rho^2)<1.

[Proof ignored here]

Unitary freedom in the ensemble for density operators theorem

Let v_1,v_2,\cdots,v_l and w_1,w_2,\cdots,w_l be two collections of vectors in the finite dimensional Hilbert space \mathscr{H}, the vectors being arbitrary (can be zero) except for the requirement that they define the same density operator \rho.

\sum_{i=1}^l |v_i\rangle\langle v_i|=\sum_{i=1}^l |w_i\rangle\langle w_i|

Then there exists a unitary matrix U=(\mu_{ij})_{1\leq i,j\leq l} such that:

v_i=\sum_{j=1}^l \mu_{ij}w_j

The converse is also true.

If \rho is a density operator on \mathscr{H} given by: \sum_{i=1}^l |w_i\rangle\langle w_i| and vector v_i is given by: v_i=\sum_{j=1}^l \mu_{ij}w_j, then \rho_1=\sum_{i=1}^l |v_i\rangle\langle v_i| is the density operator of the subsystem \mathscr{H}_1.

[Proof ignored here]

Density operator of subsystems

Schmidt Decomposition theorem

Let |u\rangle\in \mathscr{H}_1\otimes\mathscr{H}_2 be a unit vector (pure state), then there exists orthonormal bases |v_i\rangle of \mathscr{H}_1 and |w_j\rangle of \mathscr{H}_2 and \{\lambda_k\},k\leq r, where r is the Schmidt rank of |u\rangle, such that:

|u\rangle=\sum_{k=1}^r \lambda_k|v_k\rangle\otimes|w_k\rangle

where \lambda_k are non-negative real numbers. such that \sum_{k=1}^r \lambda_k^2=1.

[Proof ignored here]

Remark: non-zero vector u\in \mathscr{H}_1\otimes\mathscr{H}_2 decomposes as a tensor product u=u_1\otimes u_2 if and only if the Schmidt rank of u is 1. A state that cannot be decomposed as a tensor product is called entangled.

Reduced density operator

In \mathscr{H}_1\otimes\mathscr{H}_2, the reduced density operator of the subsystem \mathscr{H}_1 is:

\rho_1=\operatorname{Tr}_2(\rho)=\sum_{k=1}^r \lambda_k^2|v_k\rangle\langle v_k|

where \rho is the density operator in \mathscr{H}_1\otimes\mathscr{H}_2.

Example:

Let \rho=\frac{1}{2}(|01\rangle+|10\rangle)\in \mathbb{C}^2\otimes\mathbb{C}^2,

Expand the expression of \rho in the basis of \mathbb{C}^2\otimes\mathbb{C}^2:

\rho=\frac{1}{2}(|01\rangle\langle 01|+|01\rangle\langle 10|+|10\rangle\langle 01|+|10\rangle\langle 10|)

then the reduced density operator of the subsystem \mathbb{C}^2 in first qubit is:

\begin{aligned}
\rho_1&=\operatorname{Tr}_2(\rho)\\
&=\frac{1}{2}(\langle 1|1\rangle|0\rangle\langle 0|+\langle 1|0\rangle|0\rangle\langle 1|+\langle 0|1\rangle|1\rangle\langle 0|+\langle 0|0\rangle|1\rangle\langle 1|)\\
&=\frac{1}{2}(|0\rangle\langle 0|+|1\rangle\langle 1|)\\
&=\frac{1}{2}I
\end{aligned}

State purification

Every mixed state can be derived as the reduction of a pure state on an enlarged Hilbert space.

State purification theorem

Let \rho be a mixed state in a finite dimensional Hilbert space \mathscr{H}, then there exists a unit vector |w\rangle\in \mathscr{H}\otimes\mathscr{H} such that:

\rho=\operatorname{Tr}_2(|w\rangle\langle w|)

Hint of proof:

Let u_1,u_2,\cdots,u_d be an orthonormal basis of \mathscr{H}, \sum_{i=1}^d p_i=1, p_i\geq 0, then:

\rho=\sum_{i=1}^d p_i|u_i\rangle\langle u_i|

Let w=\sum_{i=1}^d \sqrt{p_i}u_i\otimes u_i.

Observables

The observables in the quantum theory are self-adjoint operators on the Hilbert space \mathscr{H}, denoted by A\in \mathscr{O}

In finite dimensional Hilbert space, A can be written as \sum_{\lambda\in \operatorname{sp}{(A)}}\lambda P_\lambda, where P_\lambda is the projection operator onto the eigenspace of A corresponding to the eigenvalue \lambda. P_\lambda=P_\lambda^2=P_\lambda^*.

Effects and Busch's theorem for effect operators

Below is a section on Topic 4, about Gleason's theorem and definition of states, and Born's rule for describing the states using density operators.

Definition of states (non-commutative (quantum) probability theory)

Do a double check on this section, this notation is slightly different from the one in Topic 4.

A state on (\mathscr{B}(\mathscr{H}),\mathscr{P}) is a map \mu:\mathscr{P}\to[0,1] such that:

1. 0\leq \mu(E)\leq 1 for all E\in \mathscr{P}(\mathscr{H}).
2. \mu(I_{\mathscr{H}})=1.
3. If E_1,E_2,\cdots,E_n are pairwise disjoint orthogonal projections, whose sum is also in \mathscr{P}(\mathscr{H}) then \mu(E_1\lor E_2\lor\cdots\lor E_n)=\sum_{i=1}^n\mu(E_i).

Where projections are disjoint if P_iP_j=P_jP_i=O.

Definition of density operator (non-commutative (quantum) probability theory)

A density operator \rho on the finite-dimensional Hilbert space \mathscr{H} is:

1. self-adjoint (A^*=A, that is \langle Ax,y\rangle=\langle x,Ay\rangle for all x,y\in\mathscr{H})
2. positive semi-definite (all eigenvalues are non-negative)
3. \operatorname{Tr}(\rho)=1.

If (|\psi_1\rangle,|\psi_2\rangle,\cdots,|\psi_n\rangle) is an orthonormal basis of \mathscr{H} consisting of eigenvectors of \rho, for the eigenvalue p_1,p_2,\cdots,p_n, then p_j\geq 0 and \sum_{j=1}^n p_j=1.

We can write \rho as

\rho=\sum_{j=1}^n p_j|\psi_j\rangle\langle\psi_j|

(under basis |\psi_j\rangle, it is a diagonal matrix with p_j on the diagonal)

Every basis of \mathscr{H} can be decomposed to these forms.

Theorem: Born's rule

Let \rho be a density operator on \mathscr{H}. then

\mu(P)\coloneqq\operatorname{Tr}(\rho P)=\sum_{j=1}^n p_j\langle\psi_j|P|\psi_j\rangle

Defines a probability measure on the space \mathscr{P}.

[Proof ignored here]

Theorem: Gleason's theorem (very important)

Let \mathscr{H} be a Hilbert space over \mathbb{C} or \mathbb{R} of dimension n\geq 3. Let \mu be a state on the space \mathscr{P}(\mathscr{H}) of projections on \mathscr{H}. Then there exists a unique density operator \rho such that

\mu(P)=\operatorname{Tr}(\rho P)

for all P\in\mathscr{P}(\mathscr{H}). \mathscr{P}(\mathscr{H}) is the space of all orthogonal projections on \mathscr{H}.

[Proof ignored here]

Extending the experimental procedure in quantum physics, many of the outcome probabilities are expectation of effects instead of projections. (POVMs)

Definition of effect

An effect is a positive (self-adjoint) operator E on \mathscr{H} such that 0\leq E\leq I.

The set of effects on \mathscr{H} is denoted by \mathscr{E}(\mathscr{H}).

An operator E is said to be the extreme point of the convex set \mathscr{E}(\mathscr{H}) if it cannot be written as a convex combination of two other effects.

That is, If E is an extreme point, then E=\lambda E_1+(1-\lambda)E_2 for some 0\leq \lambda\leq 1 and E_1,E_2\in \mathscr{E}(\mathscr{H}) implies E=E_1=E_2.

Proposition: Effect operator lemma

The set of orthogonal projections on \mathscr{H}, \mathscr{P}(\mathscr{H}), is the set of extreme points of \mathscr{E}(\mathscr{H}).

Theorem: Generalized measures on effects

Let \mathscr{H} be a finite-dimensional Hilbert space. Then any generalized probability measure

\mu:E\in \mathscr{E}(\mathscr{H})\to \mu(E)\in[0,1]

with the properties (same as the definition of states):

1. 0\leq \mu(E)\leq 1 for all E\in \mathscr{E}(\mathscr{H}).
2. \mu(I_{\mathscr{H}})=1.
3. If E_1,E_2,\cdots,E_n are pairwise disjoint orthogonal effects, whose sum is also in \mathscr{E}(\mathscr{H}) then \mu(E_1\lor E_2\lor\cdots\lor E_n)=\sum_{i=1}^n\mu(E_i).

is the form:

\mu(E)=\operatorname{Tr}(\rho E)

for some density operator \rho on \mathscr{H}.

[Proof ignored here]

If \mu is a positive linear functional on the space of self-adjoint operators on the finite dimensional Hilbert space \mathscr{H}.

Then, there exists a density operator \rho on \mathscr{H} such that \mu(E)=\operatorname{Tr}(\rho E).

Measurements

A measurement (observation) of a system prepared in a given state produces an outcome x, x is a physical event that is a subset of the set of all possible outcomes.

To each x\in X, we associate a measurement operator M_x on \mathscr{H}.

Given the initial state (pure state, unit vector) u, the probability of measurement outcome x is given by:

p(x)=\|M_xu\|^2

After the measurement, the state of the system is given by:

v=\frac{M_xu}{\|M_xu\|}

Note that to make sense of this definition, the collection of measurement operators \{M_x\} must satisfy the completeness requirement:

1=\sum_{x\in X} p(x)=\sum_{x\in X}\|M_xu\|^2=\sum_{x\in X}\langle M_xu,M_xu\rangle=\langle u,(\sum_{x\in X}M_x^*M_x)u\rangle

So \sum_{x\in X}M_x^*M_x=I.

An example of measurement is the projective measurements (von Neumann measurements).

It is given by the set of orthogonal projections M_x on \mathscr{H} with the property:

1. M_x=M_x^*
2. M_xM_y=\delta_{xy}M_x for all x,y\in X
3. \sum_{x\in X}M_x=I

Composition of measurements

Given two complete collections of measurement operators \{M_x\} and \{N_y\} on \mathscr{H}_1 and \mathscr{H}_2 respectively, the composition of the two measurements is given by the collection of measurement operators \{M_xN_y\} on \mathscr{H}_1\otimes\mathscr{H}_2.

Proposition of indistinguishability

Suppose that we have two system u_1,u_2\in \mathscr{H}_1, the two states are distinguishable if and only if they are orthogonal.

Ways to distinguish the two states:

1. set X=\{0,1,2\} and M_i=|u_i\rangle\langle u_i|, M_0=I-M_1-M_2
2. then \{M_0,M_1,M_2\} is a complete collection of measurement operators on \mathscr{H}.
3. suppose the prepared state is u_1, then p(1)=\|M_1u_1\|^2=\|u_1\|^2=1, p(2)=\|M_2u_1\|^2=0, p(0)=\|M_0u_1\|^2=0.

If they are not orthogonal, then there are no choice of measurement operators to distinguish the two states.

[Proof ignored here]

intuitively, if the two states are not orthogonal, then for any measurement there exists non-zero probability of getting the same outcome for both states.

Quantum operations and CPTP maps

Conditional operations

Section 2: Quantum entanglement

Bell states and the EPR phenomenon

Von Neumann entropy and maximally entangled states

Section 3: Information transmission by quantum systems

Transmission of classical information

Making use of entanglement and local operations

Superdense coding [very important]

Section 4: Quantum automorphisms and dynamics

Section ignored.