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Introduction to Quantum Information, Certification link.

  1. Quantum Theory & Qubit
  2. Bipartite quantum systems
  3. Quantum computing

1. Quantum Theory & Qubit

  • State ( \(|\psi \rangle \in \mathcal{H}\) )
  • Dynamics (\(U^{+}U = UU^{+} = I\))
  • Measurement ( \(P(i) = \langle \psi | M_{i} | \psi\rangle\) )
  • Obserable (\(A = A^{+}\)).

1.1 Qubit State

  • Qubit state (of 2 level system): $ | \psi (\theta, \phi) \rangle = \cos \frac{\theta}{2} |0\rangle + e^{i\phi}\sin \frac{\theta}{2} |1\rangle $
  • Pauli matrices & Hadamard transformation (both unitary and hermitian)
    • Pauli X(Not gate) Y Z. \(X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\)
    • Hadamard : $ H |0\rangle = | + \rangle $ and $ H |1\rangle = | - \rangle $
  • Phase operation \(\begin{bmatrix} 1 & 0 \\ 0 & e^{i \alpha} \end{bmatrix}\)

Mixed state :

  • Pure state \(\rho = |\psi\rangle \langle \psi |\)
    • where $tr(\rho^{2}) = 1$
  • Mixed state (a combination of pure states) : \(\sum_{i = 0}^{n}q_{i}|\psi_{i}\rangle \langle \psi_{i} |\)
    • where $tr(\rho^{2}) < 1$
  • probability : \(P(i) = tr[M_{i}\rho]\)

1.2 Bloch Sphere

\[\begin{align} \rho(\theta, \phi) & = |\psi(\theta, \phi)\rangle \langle(\theta, \phi) \psi | \\ & = \begin{bmatrix} \cos^{2} \frac{\theta}{2} & e^{i\phi}\sin \frac{\theta}{2}\cos \frac{\theta}{2} \\ e^{-i\phi}\sin \frac{\theta}{2}\cos \frac{\theta}{2} & \sin^{2} \frac{\theta}{2} \end{bmatrix} \\ & = \frac{1}{2} (I + \hat{n} \vec{\sigma}) \end{align}\]

where n is the bloch vector (which lies in the bloch sphere). (to understand the sphere better, consider different cases of $\theta$ and $\phi$) Any point in the bloch sphere <==> a qubit state.

\[\begin{align} \hat{n} \vec{\sigma} & = \begin{bmatrix} n_{x} & n_{y} & n_{z} \end{bmatrix} \begin{bmatrix} X & Y & Z\end{bmatrix} \\ & = n_{x}X + n_{y}Y + n_{z}Z \\ & = (\sin\theta\cos\phi)X + (\sin\theta\sin\phi)Y + (\cos\theta)Z \end{align}\]

Mixture of states:

\[\begin{align} \rho &= q_{0}\rho_{0} + q_{1}\rho_{1} \\ & = \frac{1}{2} (I + (q_{0}\hat{n}_{0} + q_{1}\hat{n}_{1} ) \vec{\sigma}) \end{align}\] \[tr(\rho^{2}) < 1\]

1.3 Qubit Dynamics

\[e^{ixA} = \sum_{n=0}^{\inf} \frac{1}{n!}(ixA)^{n} = (\cos x) I + i(\sin x)A\]

Single-Qubit Dynamics : qubit transformation. (Rotation in the bloch sphere) Each transformation of qubit state, can be mapped to a combination of rotation operations.

\[\begin{align} & R_{x}(\theta) = exp[-i\frac{\theta}{2}X] \\ & R_{Y}(\theta) = exp[-i\frac{\theta}{2}Y] \\ & R_{Z}(\theta) = exp[-i\frac{\theta}{2}Z] \end{align}\]

1.4 Quantum state discrimination

(1) Discrimination: If the states are not orthogonal, perfect discrimination is impossible. Instead, the aim is to minimize error or maximize success probability in identifying the prepared state. (still possible to have wrong result)

\[\begin{align} P_{success} & = max_{M_{0}, M_{1}}(q_{0} \langle \psi_{0} | M_{0} | \psi_{0} \rangle + q_{1} \langle \psi{1} |M_{1} |\psi_{1}\rangle ) \\ & = \frac{1}{2} + \frac{1}{2} \| q_{0} |\psi_{0}\rangle \langle \psi_{0} | -q_{1} |\psi_{1}\rangle \langle \psi_{1} | \| \end{align}\]

Measurement Strategy: Two detectors are used to conclude the state based on detection events. The optimal measurement is defined by the relationship between the two states, often represented geometrically.

(2) Unambiguous Discrimination: This method introduces a third detector to eliminate ambiguity, ensuring that if a state is detected, the conclusion is always correct (for $M_{0}$ and $M_{1}$), although it may result in inconclusive outcomes (from $M_{2}$).

1.5 No-go theorems

  1. Non-orthogonal quantum states cannot be perfectly discriminated.
  2. Cloning of unknown quantum states is not possible.
    • Perfect Discrimination <==> Perfect Cloning

2. Bipartite quantum systems

2.1 Two-qubit entanglement

Two qubits forming a composite Hilbert space spanned by the basis states |00⟩, |01⟩, |10⟩, and |11⟩. A two-qubit state can be expressed as a linear combination of these basis states, with complex coefficients that must be normalized.

\[\begin{align} | \psi \rangle_{AB} &\in \mathcal{H}_{A} \otimes \mathcal{H}_{B} \\ &= span \{|00\rangle, |01\rangle, |10\rangle, |11\rangle \} \\ &= \alpha |00\rangle + \beta |01\rangle + \gamma |10\rangle + \delta |11\rangle, where \| |\psi \rangle \| = 1 \\ \end{align}\]

LOCC (Local Operations and Classical Communication): This framework allows two parties (Alice and Bob) to perform local operations and communicate classically to prepare quantum states.

graph LR
subgraph A["qubit A
Local Operation"]
end
subgraph B["qubit B
Local Operation"]
end
A <--"Classical Communication"--> B
  • Product states (which can be expressed as a product of individual qubit states). \(|\Psi \rangle_{AB} = |\psi \rangle_{A} \otimes | \psi'\rangle_{B}\)
\[\begin{align} |\Psi \rangle_{AB} \approx_{L.U} |\Psi'' \rangle_{AB} & = |\phi \rangle_{A} \otimes | \phi'\rangle_{B} \\ & = (U_{A}|\psi \rangle_{A}) \otimes (U_{B}| \psi'\rangle_{B}) \\ & = U_{A} \otimes U_{B} |\Psi \rangle_{AB} \end{align}\]
  • Entangled states (which cannot be expressed as a product of individual qubit states)

Schmidt Decomposition: This method is introduced as a way to determine if a two-qubit state is entangled. It simplifies the state into a form that reveals the entanglement properties through Schmidt coefficients.

\[\begin{align} |\psi \rangle & = \sum_{ij}C_{ij}|i\rangle |j\rangle \\ & C_{ij} =_{SVD} (UDV^{+})_{ij} \\ &= \sum_{k}\lambda_{k} (\sum_{i} U_{ik}|i\rangle) \otimes (\sum_{j} V_{jk}^{*}|j\rangle) \\ & = \sum_{k}\lambda_{k} |u_{k}\rangle \otimes |v_{k}\rangle \end{align}\]

2.2 Two-qubit

Universality: Any quantum circuit can be constructed using a combination of single qubit operations and two-qubit gates. This property allows for arbitrary transformations of quantum states.

Controlled Gates: The lecture explains the concept of controlled gates, such as the CNOT (Controlled NOT-X) gate, CPhase (Controlled Phase-Z) gate. The action on the second qubit depends on the state of the first qubit.

\[|0\rangle \langle 0| \otimes I + |1\rangle \langle 1| \otimes U\] \[|\phi^{+}\rangle = \frac{1}{\sqrt 2}(|00\rangle + |11\rangle) =U_{CNOT}(H\otimes I) |00\rangle\]

Entanglement: The lecture emphasizes the importance of entanglement in generating complex quantum states, which can be achieved through direct interactions between qubits or by using intermediate particles.

Two qubit measurement : joint measurement, can be formed with individual measurements.

Example for \(M_{\phi^{+}} = | \phi^{+} \rangle \langle \phi^{+} |\)

\[\begin{align} Prob(\phi^{+}) &= tr[\rho M_{\phi^{+}}] \\ &= tr[(H\otimes I )U^{+} \rho U (H\otimes)I |00\rangle \langle 00 | ] \end{align}\]

2.3 Quantum teleportation

Quantum Teleportation: It allows the transfer of quantum states without physically moving the particles.

One-Time Pad: A classical method where two parties share a secret bit (SA and SB) to securely transmit a message (M). The classical one-time pad can be translated into a quantum context, using quantum states and operations.

3. Quantum computing

Computation is the transformation of input into a meaningful output that solves a problem, utilizing finite resources (time, steps, or space).

  • Classical computing uses logic gates (like NAND and OR) to process bits.
  • Quantum computing transforms classical bits into qubits, allowing for more complex operations.
  • Two main quantum algorithms are discussed:
    • Grover’s Algorithm: Optimizes database searching, achieving a quadratic speedup by amplifying the probability of finding a target state.
    • Shor’s Algorithm: Focuses on prime number factorization, combining quantum and classical processes, with the quantum Fourier transformation being a key component.

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