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Product States

Consider two quantum systems, $A$ and $B$, with state vectors $\mid {\Psi }^{(A)}⟩$ and $\mid {\Psi }^{(B)}⟩$ respectively. The combined state of the system may be described by a single state vector $\mid {\Psi }^{(\mathrm{AB})}⟩=\mid {\Psi }^{(A)}⟩\otimes \mid {\Psi }^{(B)}⟩$.

Example

As an example, suppose that in the basis $\{\mid 0⟩,\mid 1⟩\}$, $\mid {\Psi }^{(A)}⟩=\frac{1}{\sqrt{2}}(\mid 0⟩+\mid 1⟩)$. This can be interpreted as system $A$ being in state $\mid 0⟩$ with probability 1/2 and state $\mid 1⟩$ with probability 1/2. Suppose further that $\mid {\Psi }^{(B)}⟩=\mid 0⟩$. Then we have

$\mid {\Psi }^{(\mathrm{AB})}⟩=\mid {\Psi }^{(A)}⟩\otimes \mid {\Psi }^{(B)}⟩=\frac{1}{\sqrt{2}}(\mid 0⟩+\mid 1⟩)\otimes \mid 0⟩=\frac{1}{\sqrt{2}}(\mid 00⟩+\mid 10⟩)$.

Such a state is said to be a product state because it is “factorable” or equivalently separable, i.e. it can be formed from some combination of individual states in the basis.

Entangled States

Compare the above example to the state

$\mid {\Psi }^{(\mathrm{AB})}⟩=\frac{1}{\sqrt{2}}(\mid 00⟩+\mid 11⟩)$.

This state is not a product state since it cannot be formed from any combination of individual states in the given basis. Such a state is known as an entangled state because it is said to be non-factorable or non-separable. Entangled states are, in fact, pure states rather than mixed states because they cannot be broken down further.

The formation of entangled states requires an external action that, mathematically, takes the form of some type of unitary operator acting on a product state. Physically this usually entails interacting systems $A$ and $B$ in some way, e.g. one method for entangling photons is producing them from the same source.

LOCC and SLOCC

Often if multi-party state?s can be inter-converted via local operations, they are considered to be the same. This can be made formal by the following definition.

Such a protocol is reversible, so since protocols compose, this generates an equivalence relation. While this removes a good deal of redundancy from the study of entanglement, it is often useful to use an even more course-grained relation.

An example of a local stochastic operation is as follows. Suppose Alice and Bob share a state $\mid \Psi ⟩\in H_A\otimes H_B$ and Alice wishes to perform some operation $L$. Alice prepares an ancilla qubit $\mid 0⟩\in {ℂ}^2$ and performs a unitary operation

on her qubit as well as her part of the state $\mid \Psi ⟩$. She then measures the ancilla qubit. If she gets an outcome of $\mid 0⟩$, she has performed some operation $L:H_A\to H_A$ and if she gets outcome $\mid 0⟩$ she has performed $L\prime :H_A\to H_A$. The probability of Alice successfully performing $L$ is then the probability of getting the outcome of $\mid 0⟩$ when she performed her measurement.

Its easy to show using the Schur decomposition that there are only two SLOCC-equivalence classes in ${ℂ}^2\otimes {ℂ}^2$, namely the product state class and the Bell state class. Perhaps more surprising is the following result to to Dur, Vidal, and Cirac. [2]

By genuine, they mean a state that is not a product of smaller states. The two states are defined as:

Each of these states yields the structure of a commutative Frobenius algebra. $\mid \mathrm{GHZ}⟩$ yields a special CFA and $\mid W⟩$ yields an “anti-special” CFA. This structure serves to uniquely identity these states (up to SLOCC) in ${ℂ}^2$. [1]

A category-theoretic model for the creation of entangled states

Let’s start with two qubits, $\mid s⟩$ and $\mid t⟩$, in the initial independent states $\mid s⟩=\frac{1}{\sqrt{2}}(\mid 0⟩+\mid 1⟩)$ and $\mid t⟩=\mid 1⟩$. The combined state (which is not entangled) may be written

Now consider the unitary operator, $U=\mid 00⟩⟨00\mid +\mid 01⟩⟨01\mid +\mid 11⟩⟨10\mid +\mid 10⟩⟨11\mid$ (known as the CNOT operator) acting on the non-entangled state $\mid {\psi }_0⟩$:

The state $\mid {\psi }_1⟩$ is entangled. The entanglement was accomplished via the unitary operator $U$.

One potentially general model of entanglement via category theory would be the following.

Consider the state $\mid {\psi }_0⟩$. Since it is a factorable state (as described above), we can interpret it as being representable as an ordered pair, i.e. a Cartesian product of two elements of some set. The state $\mid {\psi }_1⟩$, however, cannot be represented in this manner. Thus we could model the process of entangling two particles as a bit like an arrow (morphism) out of a product state, i.e.

where $S$ and $T$ represent the states $\mid s⟩$ and $\mid t⟩$ respectively and $P$ represents the state $\mid {\psi }_1⟩$.

References

1. Bob Coecke, Aleks Kissinger. The compositional structure of multipartite quantum entanglement. arXiv:1002.2540v1 quant-ph
2. W. Dür, G. Vidal, and J. I. Cirac. Three qubits can be entangled in two inequivalent ways. PRA. Vol 62, 062314