Entangled quantum states

• If we have available more than one (physical) qubit, we may be able to entangle them. The tensor product of the Hilbert spaces for the individual qubits is the appropriate model for these entangled systems.

• For example, if we have two qubits with bases $\{\vert{0}\rangle_1,\vert{1}\rangle_1\}$ and $\{ \vert{0}\rangle_2,\vert{1}\rangle_2\}$ respectively, the tensor product space has the basis
  
  $$\{\vert{0}\rangle_1\otimes\vert{0}\rangle_2,\vert{0}\rangle_1... ...s\vert{0}\rangle_2,\vert{1}\rangle_1\otimes\vert{1}\rangle_2\}.$$
  
  
  We can (conveniently) denote this basis as
  
  $$\{\vert{00}\rangle,\vert{01}\rangle,\vert{10}\rangle,\vert{11}\rangle\}.$$
  
  

• More generally, if we have n qubits to which we can apply common measurements, we will be working in the 2ⁿ-dimensional Hilbert space with basis
  
  $$\{\vert{00\ldots00}\rangle,\vert{00\ldots01}\rangle,\ldots,\vert{11\ldots10}\rangle,\vert{11\ldots11}\rangle\}$$
  
  

• A typical quantum state for an n-qubit system is
  
  $$\sum_{i = 0}^{2^n-1}a_i\vert{i}\rangle$$
  
  
  where $a_i\in\mathbb{C}$, $\sum\vert a_i\vert^2=1$, and $\{\vert{i}\rangle\}$ is the basis, with (in our notation) i written as an n-bit binary number.

• A classical (macroscopic) physical object broken into pieces can be described and measured as separate components. An n-particle quantum system cannot always be described in terms of the states of its component pieces. For instance, the state
  $\vert{00}\rangle+\vert{11}\rangle$ cannot be decomposed into separate states of each of the two qubits in the form
  
  $$(a_1\vert{0}\rangle + b_1\vert{1}\rangle)\otimes(a_2\vert{0}\rangle + b_2\vert{1}\rangle).$$
  
  
  This is because
  
  $$(a_1\vert{0}\rangle + b_1\vert{1}\rangle)\otimes(a_2\vert{0}\rangle + b_2\vert{1}\rangle) =$$
  
  
  
  
  $$a_1a_2\vert{00}\rangle + a_1b_2\vert{01}\rangle + b_1a_2\vert{10}\rangle + b_1b_2\vert{11}\rangle$$
  
  
  and a₁b₂ = 0 implies that either a₁a₂ = 0 or b₁b₂ = 0. States which cannot be decomposed in this way are called entangled states. These are states that don't have classical counterparts, and for which our intuition is likely to fail.

• Particles are entangled if a measurement of one affects a measurement of the other. For example, the state $\frac{1}{\sqrt{2}}(\vert{00}\rangle+\vert{11}\rangle)$ is entangled since the probability of measuring the first bit as $\vert{0}\rangle$ is 1/2 if the second bit has not been measured. However, if the second bit has been measured, the probability that the first bit is measured as $\vert{0}\rangle$ is either 1 or 0, depending on whether the second bit was measured as $\vert{0}\rangle$ or $\vert{1}\rangle$, respectively. On the other hand, the state $\frac{1}{\sqrt{2}}(\vert{00}\rangle+\vert{01}\rangle)$ is not entangled. Since $\frac{1}{\sqrt{2}}(\vert{00}\rangle+\vert{01}\rangle) = \vert{0}\rangle\otimes\frac{1}{\sqrt{2}}(\vert{0}\rangle+\vert{1}\rangle)$, any measurement of the first bit will yield $\vert{0}\rangle$ regardless of measurements of the second bit. Similarly, the second bit has a fifty-fifty chance of being measured as $\vert{0}\rangle$ regardless of measurements of the first bit. Note that entanglement in terms of particle measurement dependence is equivalent to the definition of entangled states as states that cannot be written as a tensor product of individual states.