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What is a Tensor Product \otimes ?

The tensor product of two vectors is defined as:

\begin{pmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{pmatrix} \otimes \begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_m \end{pmatrix} = \begin{pmatrix} a_1 \begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_m \end{pmatrix} \\ a_2 \begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_m \end{pmatrix} \\ \vdots \\ a_n \begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_m \end{pmatrix} \end{pmatrix} = \begin{pmatrix} a_1 b_1 \\ a_1 b_2 \\ \vdots \\ a_1 b_m \\ a_2 b_1 \\ a_2 b_2 \\ \vdots \\ a_2 b_m \\ \vdots \\ a_n b_1 \\ a_n b_2 \\ \vdots \\ a_n b_m \end{pmatrix}

The tensor product is a mathematical operation that combines two quantum states into a single state describing a composite system. If you have two systems, A and B, with states \lvert 0 \rangle and \lvert 0 \rangle, their combined state is written as: $$ \lvert a \rangle \otimes \lvert b \rangle $$ For a single qubit has a state of dimension 2 (\lvert 0\rangle,\lvert 1\rangle), for two qubis together forma a state space of dimension 2 \times 2 = 4 \ (\lvert 00 \rangle, \lvert 01 \rangle, \lvert 10 \rangle, \lvert 11 \rangle).

  1. Linearity: $$ (a\lvert u \rangle + b\lvert v \rangle) \otimes \lvert w \rangle = a(\lvert u \rangle \otimes \lvert w \rangle)+ b(\lvert v \rangle \otimes \lvert w \rangle)
    $$

  2. Dimensionality: If \lvert u \rangle is in a d_{1}-dimensional space and \lvert v \rangle is in a d_{2}-dimensional space, the the combination state, $$ \lvert d_{1} \rangle \otimes \lvert d_{2} \rangle $$ is in a d_{1} \times d_{2}-dimensional space.

  3. Associativity $$ (\lvert u \rangle \otimes \lvert v \rangle) \otimes \lvert w \rangle \approxeq \lvert u \rangle \otimes (\lvert v \rangle\otimes \lvert w \rangle) $$

  4. Independence: The tensor prodict assumes that the two systems are independent and their joint state is simply the product of their individual states. For example: $$ \lvert 0 \rangle \otimes \lvert 1 \rangle = \lvert 01 \rangle $$

Here are some properties of the tensor product operation: 5. Basis States If \lvert u \rangle and \lvert v \rangle are basis states, their tensor product \lvert 0 \rangle \otimes \lvert 0 \rangle forms a basis for the composite system. For example, the basis states \lvert 0 \rangle, \lvert 1 \rangle for a sinagle qubi combine to form the two-qubi basis \{ \lvert 00 \rangle, \lvert 01 \rangle, \lvert 10 \rangle, \lvert 11 \rangle \}.

What \lvert 00 \rangle = \lvert 0 \rangle \otimes \lvert 0 \rangle implies

This representation emphasizes:

  1. Individuality of Qubits: Each qubit has its own state space, but when combined, they form a composite system.
  2. Non-Entangled States: The tensor product \lvert 0 \rangle \otimes \lvert 0 \rangle imples a product state, meaning two qubits are independent and not entangled.

Non-cloing in quantum mechanics:

The expression \lvert \psi \rangle \otimes \lvert \psi \rangle represents the tensor product of a quantum state \lvert \psi \rangle with itself. In the context of cloning, this means having two identical copies of the quantum state \lvert \psi \rangle.

Understanding Tensor Products

In quantum mechanics: 1. The tensor product combines two quantum systems into a single composite system. 2. If two systems are in identical states \lvert \psi \rangle, the tensor product \lvert \psi \rangle \otimes \lvert \psi \rangle describes a system where both subsystems are in the same state \lvert \psi \rangle.

Why \lvert \psi \rangle \otimes \lvert \psi \rangle Represents Cloning

  1. Single Qubit:

    • A single qubit in the state \lvert \psi \rangle has amplitudes \alpha and \beta for the basis states \lvert 0 \rangle and \lvert 1 \rangle, respectively: $$ \lvert \psi \rangle = \alpha \lvert 0 \rangle + \beta \lvert 1 \rangle $$

  2. Cloning:

    • To clone \lvert \psi \rangle, we want to produce a second qubit in the exact same state, so the final system contains two qubits, both in state \lvert \psi \rangle.

  3. Mathematical Representation:

    • If the first qubit is \lvert \psi \rangle and the second (blank) qubit starts in \lvert 0 \rangle, then after cloning, the system would be described as: $$ \lvert \psi \rangle \otimes \lvert \psi \rangle = (\alpha \lvert 0 \rangle + \beta \lvert 1 \rangle) \otimes (\alpha \lvert 0 \rangle + \beta \lvert 1 \rangle) $$

  4. Expansion:

    • Expanding the tensor product: $$ \lvert \psi \rangle \otimes \lvert \psi \rangle = \alpha^2 \lvert 00 \rangle + \alpha \beta \lvert 01 \rangle + \beta \alpha \lvert 10 \rangle + \beta^2 \lvert 11 \rangle $$
  5. This describes a two-qubit system where both qubits are in the same state \lvert \psi \rangle.

Why Cloning is Different from Classical Duplication

In classical systems: 1. Information can be copied exactly without any loss or disturbance.

In quantum systems: 2. A state like \lvert \psi \rangle = \alpha \lvert 0 \rangle + \beta \lvert 1 \rangle cannot be perfectly duplicated into \lvert \psi \rangle \otimes \lvert \psi \rangle because of the no-cloning theorem.

Summary

  1. \lvert \psi \rangle \otimes \lvert \psi \rangle represents a quantum system where two qubits are in the same state \lvert \psi \rangle, which corresponds to a cloned system.
  2. Achieving this for an arbitrary \lvert \psi \rangle is impossible because cloning would violate the linearity of quantum mechanics, as shown by the no-cloning theorem.

Reference:

  1. Tensor product