Reading Notes of: Quantum Computation and Quantum Information
2.1 Linear Algebra
vector space (e.g. \(\mathbb{C}^n\))
a vector subspace of a vector space \(V\) is a subset \(W\) of \(V\) such that \(W\) is also a vector space
a basis of a vector space
the dimension of a vector space is the number of elements in the basis
a linear operator between vector spaces \(V\) and \(W\)
a matrix representation of the operator \(A:V\rightarrow W\) \[ A|v_j\rangle=\sum_iA_{ij}|w_i\rangle \] To make the connection between matrices and linear operators we must specify a set of input and output basis states for the input and output vector spaces of the linear operator.
inner product : 1. linear 2. commutation 3. (v,v)>=0
inner product space : a vector space equipped with an inner product
a Hilbert space is exactly the same thing as an inner product space
orthogonal
norm
unit vector
orthonormal
orthonormal basis (produced by Gram-Schmidt procedure from a basis)
a matrix representation is with respect to orthonormal input and output bases
a vector has a matrix representation with respect to an orthonormal basis
the inner product of two vectors is equal to the vector inner product between two matrix representations of those vectors
define outer product \(|w\rangle\langle v|\) as a linear operator from \(V\) to \(W\), making use of the inner product
completeness relation: \(\sum_i|i\rangle\langle i|=I\)
representation of any linear operator in the outer product notation: \[ A=\sum_{ij}|w_i\rangle\langle w_i|A|v_j\rangle\langle v_j| =\sum_{ij}\langle w_i|A|v_j\rangle |w_i\rangle\langle v_j|=\sum_{ij}A_{ij}|w_i\rangle\langle v_j| \] Cauchy-Schwarz inequality : \(|\langle v|w\rangle|^2\leq\langle v|v\rangle\langle w|w\rangle\)
eigenvector, eigenvalue of a linear operator
The characteristic function depends only upon the operator \(A\), and not on the specific matrix representation used for \(A\).
eigenspace corresponding an eigenvalue \(v\) is a vector subspace of \(V\)
diagonal representation \(A=\sum_i\lambda_i|i\rangle\langle i|\) , where \(|i\rangle\) form an orthonormal set of eigenvectors for \(A\).
an eigenspace is called degenerate when it's more than one dimensional
the adjoint / Hermitian conjugate of a linear operator on a Hilbert space
a Hermitian / self-adjoint operator
the projector onto the subspace \(W\) is \(P\equiv\sum_{i=1}^k|i\rangle\langle i|\). \(W\) is a \(k\)-dimensional vector subspace of the \(d\)-dimensional vector space \(V\). \(|1\rangle,\dots,|k\rangle\) is an orthonormal basis for \(W\). The definition is independent of the orthonormal basis used for \(W\).
The orthogonal complement of \(P\) is the operator \(Q\equiv I-P\).
normal : \(AA^\dagger=A^\dagger A\)
normal \(\Longleftrightarrow\) diagonalizable
Hermitian \(\Longleftrightarrow\) diagonalizable and having real eigenvalues
projector \(\Longleftrightarrow\) diagonalizable and having 0/1 eigenvalues
unitary \(\Longleftrightarrow\) diagonalizable and having modulus-1 eigenvalues
unitary : \(U^\dagger U=I\)
unitary operators also satisfies \(UU^\dagger=I\), therefore normal and diagonalizable. unitary operators preserve inner products between vectors.
outer product representation of unitary operators : \(U=\sum_i|w_i\rangle\langle v_i|\), where \(|v_i\rangle\) is any orthonormal set and \(|w_i\rangle\equiv U|v_i\rangle\) is therefore also an orthonormal set.
All eigenvalues of a unitary matrix have modulus 1.
positive operator : for any \(|v\rangle\), \((|v\rangle,A|v\rangle)\) is real, non-negative
A positive operator is necessarily Hermitian. (Any \(A\) can be written as \(A=B+iC\), where \(B\) and \(C\) are Hermitian)
tensor product of two Hilbert spaces
Kronecker product of two matrices
operator functions \(f(A)\equiv\sum f(\lambda)|\lambda\rangle\langle\lambda|\)
trace \(tr(A)=\sum A_{ii}=\sum \langle i|A|i\rangle\)
trace is invariant under the unitary similarity transformation
thus the trace of an operator is any trace of its matrix representation
commutator \([A,B]=AB-BA\)
anticommutator \(\{A,B\}=AB+BA\)
\([A,B]=0\) if and only if \(A\) and \(B\) are simultaneously diagonalizable
Polar decomposition \(A=UJ=KU\), where \(U\) is unitary, \(J\) and \(K\) are positive operators.
Singular value decomposition \(A=UDV\), where \(U\) and \(V\) are unitary, \(D\) is a diagonal matrix with non-negative entries.
2.2 The postulates of quantum mechanics
Postulate 1: state space
Postulate 2: evolution (of a closed system)
Hadamard gate \(H=\frac{1}{\sqrt 2}(|0\rangle+|1\rangle)\langle 0|+\frac{1}{\sqrt 2}(|0\rangle-|1\rangle)\langle 1|\)
\(H\) is Hermitian and unitary.
Postulate 3: quantum measurement
the measurement of a qubit in the computational basis
distinguishability of quantum states
projective measurements
- observable \(M\), a Hermitian
- \(\langle M\rangle\)
- \(\Delta(M)=\sqrt{\langle M^2\rangle-\langle M\rangle}\)
the Heisenberg uncertainty principle : \[ \Delta(C)\Delta(D)\geq \frac{|\langle\psi|[C,D]|\psi\rangle|}{2} \] measurement of spin along the \(\vec{v}\) axis: an observable \(\vec{v}\cdot\vec{\sigma}=v_1\sigma_1+v_2\sigma_2+v_3\sigma_3\)
POVM \(\{E_m\}\), \(\sum_m E_m=I\), positive operator
one sided error of distinguishing two (not necessarily orthogonal) states
${M_m^}M_m=E_m$ unitary \(U_m\) s.t. \(M_m=U_m\sqrt{E_m}\)
equivalence up to the global phase factor
Postulate 4: the state space of a composite system
a linear operator on a subspace which preserves inner product can be extended to the entire space as a unitary operator
unitary dynamics + projective measurements + ability to introduce ancillary systems = Postulate 3
superdense coding
Bell states: 00+11, 00-11, 01+10, 01-10
for all bell states, \(\langle\psi|E\otimes I|\psi\rangle=\frac{1}{2}(\langle 0|E|0\rangle+\langle 1|E|1\rangle)\), thus undistinguishable by only the first qubit.
density operator \(\rho\):
- positive operator
- \(tr(\rho)=1\)
\(tr(\rho^2)\leq 1\) with equality iff \(\rho\) is a pure state \[ \rho=\sum_i p_i|\psi_i\rangle\langle\psi_i|=\sum_j q_j|φ_j\rangle\langleφ_j|\Longleftrightarrow \exists U\ \sqrt {p_i}|\psi_i\rangle=\sum_j u_{ij}\sqrt{q_j}|φ_j\rangle \] an arbitrary density matrix for a qubit may be written as \[ \rho=\frac{I+\vec{r}\cdot\vec{\sigma}}{2} \] partial trace \(\rho^A\equiv tr_B(\rho^{AB})\) defined by \(tr_B(\rho\otimes\sigma)=\rho\ \mathrm{tr}(\sigma)\)
the reduced density operator for Bell states is \(\frac{I}{2}\)
why partial trace? the unique function that satisfies \(tr(M\rho^A)=tr((M\otimes I)\rho^{AB})\), the correct measurement statistics
Schmidt decomposition: \[ |\psi\rangle=\sum_i\lambda_i|i_A\rangle|i_B\rangle \]
\(|\psi\rangle\) is any pure state of \(AB\)
\(|i_A\rangle\) are orthonormal states for \(A\), \(|i_B\rangle\) are orthonormal states for \(B\)
\(\lambda_i\) are non-negative real numbers satisfying \(\sum_i\lambda_i^2=1\)
\(\rho^A=\sum_i\lambda_i^2|i_A\rangle\langle i_A|\), \(\rho^B=\sum_i\lambda_i^2|i_B\rangle\langle i_B|\), the eigenvalues of \(\rho^A\) and \(\rho^B\) are identical
a symmetry between component systems of a pure state
Schmidt bases, Schmidt number
Schmidt number quantifies the "amount" of entanglement, preserved under unitary transformation on component system. \(|\psi\rangle\) is a product state if and only if it has Schmidt number 1.
purification:
suppose \(\rho^A=\sum_i p_i|i^A\rangle\langle i^A|\)
define \(|AR\rangle\equiv \sum_i\sqrt{p_i}|i^A\rangle|i^R\rangle\)
then \(tr_R(|AR\rangle\langle AR|)=\rho^A\)
for a pure state phi of AB, calculate the decomposition of rho^A and rho^B, then the Schmidt decomposition of phi is derived? \[ (\rho^A,\rho^B)\text{ uniquely determine } \rho^{AB}? \]