Hilbert space

• Hilbert space, a complex vector space with an inner product. In Quantum mechanics, the term "Hilber space" is often reserved for an intinite-dimentional inner product space having the property that it is copmlete or closed. Here we will use it in finite-dimensional spaces.

• We use $Dirac$ notation $| v \rangle$ called $ket$. where $v$ is a symbol of a vector. You can analogous to $v$ or $\vec{v}$.

• The inner product of the vector $| v \rangle$ with $| w \rangle$ is written as $\langle v | w \rangle$. Think of this as a $\vec{v}\cdot\vec{w}$ in a familiar vector form.

Qubit

• Let's start with the 2-dimensional case. Every vector in the 2-dimensional Hilbert can be written as a linear combination of two vectors which form a basis for the space. The computational basis for quantum information are denoted as $|0\rangle$ and $|0\rangle$, and it is assumed that of them are normalized and that they are mutually orthogonal

  $$\langle 0 | 0 \rangle = 1 = \langle w | w \rangle, \ \langle 0 | 1 \rangle = 1 = \langle 1 | 0 \rangle.$$

• In quantum mechanics a 2-dimensional complex Hilbert space $H$ is used for describing the angular momentum or "spin of a spin-half particle (electron, proton, neutron, silver atom), which then provides a physical representation of a qubit.

• A polarization of a photon (particle of light) is also described by $d=2$ (2-dimensional).

• A state vector $|\psi\rangle$ says something about one component of the spin of the spin half particle. The usual convention is

  $$ |0\rangle = |z^{+}\rangle \leftrightarrow S_{z} = + 1/2, |1\rangle = |z^{-}\rangle \leftrightarrow S_{z} = - 1/2, $$ where $S_{z}$ is the $z$ component of angular momentum is measured in units of $\hbar$.

• The general rule is that $w$ is a direction in space correspnding to the angle $\theta$ and $\phi$ in polar coordinates,

  $$|0\rangle + e^{i\phi} \tan(\phi/2) |1\rangle \leftrightarrow S_{w} = +1/2, \ |0\rangle - e^{i\phi} \cot(\phi/2) |1\rangle \leftrightarrow S_{w} = -1/2, \ $$

  but see the comments below on normalization.

General $d$

• A collection of linearly independent vectors ${B_{j}}$ form a basis of $H$ provided any $| \psi \rangle$ in $H$ can be written as a linear combination

  $$| \psi \rangle = \sum_{j} c_{j} | B_{j} \rangle,$$

  where $c_{j}$ is a copmlex number.

• A orthonormal basis $\{|j\rangle \}$ for $j = 1, 2, \cdots, d$ with the property that

  $$\langle j | k \rangle = \delta_{jk}.$$

  The inner product of two basis vectors is 0 for $j \neq k$ if they are orthogonal, and 1 if they are normalized. $\delta_{jk}$ is also called as a Kronecker delta.

• If we write a vector as a combination of a orthonormal basis $\{|j\rangle \}$ for $j = 1, 2, \cdots, d$

  $$|v\rangle = \sum_{j} v_{j}|j\rangle, \ |w\rangle = \sum_{j} w_{j}|j\rangle$$

  where the coefficients $v_{j}$ and $w_{j}$ are given by

  $$v_{j} = \langle j | v \rangle, \ w_{j} = \langle j | w \rangle,$$

  the inner product $\langle v | w \rangle$ can be written as

  $$(|v\rangle^{\dagger})|w\rangle = \langle v | w \rangle = \sum_{j}v_{j}^{*}w_{j},$$

  which can be thought of as the product of a "bra" vector

  $$\langle v | = (|v\rangle)^{\dagger} = \sum_{j}^{*}w_{j}\langle j |$$

  with the "ket vector $| w \rangle$.

• It's convenient to think of $| w \rangle$ in a column vector

  $$| w \rangle \begin{pmatrix} w_{1} \\ w_{2} \\ \cdots \\ w_{d} \end{pmatrix} ,$$

  and $\langle v |$ by a row vector

  $$\langle v | = (v_{1}^{*},v_{2}^{*},\cdots, v_{d}^{*}).$$

  where d is a diemsnion.

Kets as physical properties

• In quantum mechanics, two vectors $|\psi\rangle$ and $c|\psi$, where $c$ is any nonzero complex number have exactly the same physical significance. For this reason, it is sometimes helpful to say that the physical state corresponds not to a particular vector in the Hilbert space, but the $ray$, or one-dimensional subspace, defined by the collection of all the complex multiples of a particular vector.

  • One can always choose $c$ in such way that the $| \psi \rangle$ corresponding to a particular physical situation is normalized, $\langle \psi | \psi \rangle = 1$ or $||\psi|| = 1$, where the norm $||\psi||$ of a state $|\psi\rangle$ is the positive square root of
  $$|| \psi||^{2} = \langle \psi | \psi \rangle,$$

  and is 0 if and only if $\lvert \psi \rangle$ is the unique zero vector, which will be written as 0 instead of $| \psi \rangle$.

  • Normalized vectors can always be multiplied by a phase factor, a complex number of the form $e^{i\phi}$ where $\phi$ is real, without changing the normalization or the physical interpretation.

• The state of a single qubit is always a linear combination of the basis vectors $|0\rangle$ and $|1\rangle$,

  $$|\psi\rangle = \alpha |0\rangle + \beta |1\rangle,$$

  where $\alpha$ and $\beta$ are complex numbers.

• Two nonzero vectors $|\psi \rangle$ and $|\phi\rangle$ which are orthogonal if $\langle \phi | \psi \rangle = 0$, represent distinct physical properties:

  • If one corresponds to a property, such as $S_{z} = + 1/2$, which is a correct description of a physical system at a particular time, then the other corresponds to a physical property which is not true for this system at this time. We call this mutaully exclusive.

• There are cases where $|\psi \rangle$ is neither a multiple of $| \phi \rangle$, nor is it orthogonal to $|\phi\rangle$. For example

  $$|0\rangle = |z^{+}\rangle \leftrightarrow S_{z} = +1/2 \ \text{and} \ \sqrt{2}|x^{+}\rangle = |0\rangle + |1\rangle \leftrightarrow S_{x} = +1/2$$

  These represent neither the same physical situation, nor do they represent distinct physical situations.

• A quantum system cannot simultaneously process two incompatible properties. For example, a spin-half particle cannot have both $S_{x} = +1/2$ and $S_{z} = +1/2$. There is nothing in the Hilbert space that could be used to represent such a combined property.

Operators

Definition

• Operators are linear maps of the Hilbert space $H$ onto itself. If $A$ is an operator, then for any $|\psi\rangle$ in H, $A|\psi\rangle$ is another element in $H$, and linearity means that

  $$A(b|\psi\rangle + c|\phi\rangle) = bA|\psi\rangle + cA|\phi\rangle$$

  for any pair $|\psi\rangle$ and $|\phi\rangle$, and any two complex number $b$ and $c$.

  • The product $AB$ of two operators $A$ and $B$ is defined by
  $$(AB)|\psi\rangle = A(B|\psi\rangle) = AB|\psi\rangle.$$

  In general $AB \neq BA$.

Dyads and completness

• The simplest operator is a dyad, $|x\rangle \langle w|$. This action is defined by

  $$(\color{red}{|x\rangle} \color{blue}{\langle w|})|\psi\rangle = \color{red}{|x\rangle} \color{blue}{\langle w|} \psi \rangle = (\color{blue}{\langle w|} \psi \rangle)\color{red}{|x\rangle}.$$

  Note

  The middle term, formed by removing the parentheses and replacing two vertivle bars $||$ with one bar $|$.

• The completness relation

  $$I = \sum_{j}|j\rangle \langle j|,$$

  where $I$ is the identity operator, $I|\psi\rangle = |\psi\rangle$ for any $|\psi\rangle$. The sum on the right is dyads $|j\rangle \langle j|$ corresponding to the elements $|j\rangle$ of an orthonormal basis.

  • Theus, we can rewrite a state $|\psi\rangle$ $$ \psi = (\sum_{j} \color{red}{|j\rangle}\color{blue}{\langle j|})|\psi\rangle = \sum \color{red}{|j\rangle}\color{blue}{\langle j|}\psi\rangle = \sum_{j}\color{blue}{\langle j |}\psi\rangle \color{red}{|j\rangle}. $$

Matrices

• Given an operator $A$ and a bisis $\{ |\beta_{j} \rangle\}$ , which need not to be orthonormal, the matrix associated with $A$ is the square array of numbers $A_{jk}$ defined by

  $$A|\beta_{k}\rangle = \sum{j}|\beta_{j}\rangle A_{jk} = \sum_{j}A_{jk}|\beta_{j}\rangle.$$

• For an orthonormal basis, we can rewrite the completness relation

  $$A|k\rangle = I \cdot A|k\rangle = (\sum_{j}\color{red}{|j\rangle} \langle j|)A|k\rangle = \sum_{j}\color{red}{|j\rangle} \langle j| A|k\rangle = \sum_{j}\langle j|A|k\rangle\color{red}{|j\rangle}.$$

  In $\langle j|A|k\rangle$, the inner product of $|j\rangle$ with $A|k\rangle$ is just the $A_{jk}$ above. Thus, $\langle j|A|k\rangle$, can be written as $\langle \psi|A|\omega\rangle$, is referred to as a "matrix element" when using Dirac notation.

• In a similar way the expression

  $$A = I \cdot A \cdot I = \sum_{j}|j\rangle \langle j| \cdot A \cdot \sum_{k}|k\rangle \langle k| = \sum{jk}\langle j|A|k\rangle\cdot|j\rangle\langle k|$$

  allows us to express operator $A$ as a sum of dyads, with coefficients given by its matrix elements.

• When $A$ refers to a qubit the usual wawy of writing the matrix in the standard basis is

  $$\begin{pmatrix} \langle 0|A|0\rangle & \langle 0|A|1\rangle \\ \langle 1|A|0\rangle & \langle 1|A|1\rangle \end{pmatrix}$$

• We can also write the matrix element of the product of two operators in terms of the individual matrix elements:

  $$\langle j|AB|k\rangle = \langle j|A \cdot I \cdot B|k\rangle = \sum_{m}\langle j|A|m\rangle\langle m|b|k\rangle.$$

  If we write this equation in subscripts form,

  $$(AB)_{jk} = \sum_{m}A_{jm}B_{mk}.$$

Dagger or adjoint

• Here are some dagger $^\dagger$ properties:

  $$\begin{array}{c} (|\psi\rangle)^{\dagger} = \langle\psi|,\\ (\langle \psi|)^{\dagger} = |\psi \rangle,\\ (a|\psi\rangle+b|\psi\rangle)^{\dagger} = a^{*}\langle\psi|+b^{*}\langle\psi|,\\ (|\psi\rangle \langle \omega|)^{\dagger} = |\omega\rangle \langle \psi|,\\ \langle j|A^{\dagger}|k\rangle = (\langle k|A|j\rangle)^{*},\\ (aA+bB)^{\dagger} = a^{*}A + b^{*}B,\\ (AB)^{\dagger} = B^{\dagger}A^{\dagger}, \end{array}$$

  where $a$ and $b$ are complex numbers; $a^{*}$ and $b^{*}$ are its complex conjugate. The operator $A^{\dagger}$ is called the adjoint of the operator $A$. The matrix of $A^{\dagger}$ is the complex conjugate of the transpose of the matrix of $A$.

Normal operators

• A normal operator $A$ on a Hilbert space in one that commutes with its adjoint, that is, $AA^{\dagger} = A^{\dagger}A$.

• A normal operators can be diagonalized using an orthonormal basis, that is

  $$A = \sum_{j} \alpha_{j} |\alpha_{j}\rangle \langle a_{j}|$$

  where the basis vectors $|a_{j}\rangle$ are eigenvectors of $A$ and the complex numbers $\alpha_{j}$ are its eigenvalues. Equivalently, the matrix of $A$ in this basis is diagonal

  $$\langle a_{j} | A | a_{k} \rangle = \alpha_{j}\delta_{jk}.$$

Hermitian operators

• A hermitian or self-adjoint operator $A$ is defined by the property that $A = A^{\dagger}$, so it is a normal operator. It is the qunatum analog of a real (not complex) number. Its eigenvalues $\lambda_{j}$ are real numbers.
  • The term "Hermitian" and "self-adjoint" mean the same thing for a finite-dimensional Hilber space.
  • Hermitian operators in quantum mechanics are used to represent physical variables, quantities such as energy, momentum, angular momentum, position, etc. The operator representing the energy is the Hamiltonian $H$.
  • For example, the operator $S_{z} = \frac{1}{2}(|z^{+}\rangle\langle z^{+}| - |z^{-}\rangle\langle z^{-}|)$ is the $z$ component of angular momentum of a spin-half particle.
• There is no always has a well-defined value for a physical system in a particulate in quantum mechanics. Let's say a quantum state $|\psi\rangle$, the physical variable corresponding to the operator $A$ has a well-defined value if and only if $| \psi\rangle$ is an eigenvector of $A$, $A|\psi\rangle = \alpha|\psi\rangle$, where $\alpha$m must be a real number since $A^{\dagger} = A$.
  • The eigenstates of $S_{z}$ for a spin-half particle are $|z^{+}\rangle$ and $|z^{-}\rangle$, with eigenvalues of +1/2 and -1/2, respectively.
  • If $|\psi\rangle$ is not an eigenstate of $A$, then in this state the physical quantity $A$ is undefined , or meaningless in the sense that quantum theory can assign it no meaning.
  • The state $|x^{+}\rangle$ is an eigenstate of $S_{x}$ but not of $S_{z}$. hence in this state $S_{x}$ has well-defined value (1/2), where as $S_{z}$ is undefined.

Projectors

• A projectors, short for orthogonal projection operator, is a Hermitian operator $P = P^{\dagger}$ which is idempotent in the sense that $P^{2} = P$. It is a Hermitian operator all or whose eigenvalues are either 0 or 1. Therefore, there is always a basis in which its matrix is diagonal $\langle a_{j}|A|a_{k}\rangle = \alpha_{j} \delta_{jk}$, with only 0 or 1 on the diagonal. Such a matrix always represents a projector.

  • There is a one-to-one correspondence between a projector $P$ and the subspace $P'$ of the Hilbert space that it projects onto. $P'$ consists of all the kets $|\psi\rangle$ such that $P|\psi\rangle = |\psi \rangle$. That is, it is the eigenspace consisting of eigenvectors a projector.
  • Projector operator "projects" a vector in a "perpendicular" manner onto a subsapce.
  • The dyad $|\psi\rangle \langle \psi|$ for a normalized state $|\psi\rangle$. If $|\psi\rangle$ is not normalized (and not zero), the corresponding projector is
  $$P = \frac{|\psi\rangle \langle \psi|}{\langle \psi|\psi\rangle}$$
  • If $|\psi\rangle$ and $|\phi\rangle$ are two normalized states orthogonal to each other, i.e., the inner product is 0. Then the sum $|\psi\rangle\langle\psi|+|\phi\rangle\langle\phi|$ of the corresponding dyads is also a projector.

• The physical significance of projectors is that they represent physical properties of a quantum system that can be either true or false.

  • The property $P$ corresponding to a projector $P$ is trueuf the physical state $|\psi\rangle$ of the system is an eigenstate of $P$ with eigenvalue of 1, and false if it is an eigenstate with eigenvalue of 0. If $|\psi\rangle$ is not an eigenstate of $P$, then the corresponding property is undefined (meaningless) for this state.

• Two quantum properties represented by projectors $P$ and $Q$ are said to be compatible if $PQ = QP$ (if $P$ and $Q$ commute). Incompatible oterwise.

  • When $P$ and $Q$ are compatible, the product $PQ$ is itself a projector, and represents the property "$P$ AND $Q$". That is, the property that the system has both properties $P$ and $Q$ at the same time.
  • The projectors $|z^{+}\rangle\langle z^{+}|$ and $|x^{+}\rangle\langle x^{+}|$ do not commute, and so $S_{z} = +1/2$ and $S_{x}=+1/2$ are examples of incompatible properties.

Positive operators

• The positive operators is useful when dealing with probabilities. They are defined as that for every ket $|\psi\rangle$

  $$\langle \psi |A|\psi\rangle \geq 0.$$

  or

  $$A = \sum_{j} \alpha_{j} |\alpha_{j}\rangle \langle a_{j}|,$$

  where $\alpha_{j} \geq 0$. Both representations should be normalized.

Unitary operators

• A unitary operator $U$ has the property that

  $$U^{\dagger}U = I = UU^{\dagger}.$$

  • Since $U$ commutes with its adjoints, it is a normal operator and can diagonalized using an orthonormal basis. Aboove equation also implies that all the eigenvalues of $U$ are complex numbers of magnitude 1. That is, they lie on the unit circuile in the complex plane.

  • In a finite-dimentional Hilbert space, with $U$ mapping the space into itself, thus, we only have to check one of the above equation to see if $U$ is unitary.

  • In quantum mechnics, unitary operators are used to change from one orthonormal basis to another, to represent symmetries, such as rotational symmetry, and to describe some aspects of the dynamics or time development of a quantum system.

Bloch sphere

• Any physical state $\psi$ of a qubit (ray or normalized vector in the two-dimensional Hilbert space) can be associated in this way with a direction $w = (w_{x}, w_{y}, w_{z})$ in space for which $S_{w} = 1/2$, i.e., the $w$ component of angular momentum is positive. There is therefore a one-to-one coorespondence bewteen directions, or the correspondence points on the unit shpere, with rays of a two-dimensional Hilber space. This is also known as the ***Bloch sphere representation *** of qubit states.

  • We often write a state
  $$\begin{array}{c} \cos{(\theta/2)}|0\rangle + e^{i\phi}\sin{(\theta/2)}|1\rangle \leftrightarrow S_{w} = +1/2, \\ \sin{(\theta/2)}|0\rangle - e^{i\phi}\cos{(\theta/2)}|1\rangle \leftrightarrow S_{w} = -1/2, \end{array}$$

  where the ket are normalized.

  • Note that it is the surface of the Bloch sphere - vector $w$ of unit length - that correspond to different rays.

• Two states of a qubit are orthogonal, physically distinct ot distinguishable, if they are antipodes, two points at opposite ends of a diagonal. For example, $|0\rangle$ $|1\rangle$ are the north and south pole of the Bloch sphere.

  • Any orthonormal basis of a qubit is associated with a pair or antipodes of that Bloch sphere.

• A linear operator maps a ray onto a ray, or onto a zero vector. A linear operator on a qubit maps the Bloch sphere onto itself, or in the case of a noninvertible operator, onto a single point on the sphere.

• Of particular importance are rotations of 180${^\circ}$ about the $x$, $y$, and $z$ axes, obtained using the unitary operators $X = \sigma_{x}$, $Y = \sigma_{y}$, and $Z = \sigma_{z}$, respectively. In the standard basis the corresponding matrices are the Pauli matrices:

$$X = \begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix} , \ Y = \begin{pmatrix} 0 & -i\\ i & 0 \end{pmatrix} , \ Z = \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix} .$$

Composite systems and tensor products

Definition

Product and entangled states

Operators on tensor products

Example of two qubits

Multiple systems. Identical particles.

Reference

1. Hilbert Space Quantum Mechanics