# Contents

## Idea

While in classical mechanics a (pure) state is an element of an object in a cartesian monoidal category, in contrast in quantum mechanics a pure state is an element of an object in a non-cartesian monoidal category (say of Hilbert spaces). As a result, in quantum mechanics a state of a compound physical system may not come from a pair of states of the two subsystems, but instead be a nontrivial sum – a superposition – of such. These non-classical combinations of states of subsystems are called entangled states.

## Definition

In quantum mechanics a state of a physical system is represented by a vector in some (Hilbert-)vector space $H$. If the system is the composite of two subsystems with state spaces ${H}_{1}$ and ${H}_{2}$, respectively, then the state space of the total system is the tensor product $H={H}_{1}\otimes {H}_{2}$. The universal property of the tensor product gives a linear map

$p:{H}_{1}×{H}_{2}\to {H}_{1}\otimes {H}_{2}$p : H_1 \times H_2 \to H_1 \otimes H_2

which sends a pair of states $\left({\psi }_{1},{\psi }_{2}\right)$ to their tensor product ${\psi }_{1}\otimes {\psi }_{2}$. States in the image of $p$ are called product states or separable states. An entangled state is a state which is not a product state.

## Examples

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

As an example, suppose that in the basis $\left\{\mid 0⟩,\mid 1⟩\right\}$, $\mid {\Psi }^{\left(A\right)}⟩=\frac{1}{\sqrt{2}}\left(\mid 0⟩+\mid 1⟩\right)$. 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 $\mid {\Psi }^{\left(B\right)}⟩=\mid 0⟩$. Then we have

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

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.

Compare the above example to the state

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

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.

## Properties

### LOCC and SLOCC

The following refers to (Coecke-Kissnger).

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.

###### Definition

Two states $\mid \Psi ⟩,\mid \Phi ⟩\in ⨂{H}_{i}$ are said to be equivalent up to local operations with classical communication (LOCC) if they can be inter-converted by a protocol involving any number of steps where (i) one party applies a local unitary operation $U:{H}_{i}\to {H}_{i}$ or (ii) one party sends some classical information to another.

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.

###### Definition

Two states $\mid \Psi ⟩,\mid \Phi ⟩\in ⨂{H}_{i}$ are said to be equivalent up to stochastic LOCC (SLOCC) if they can be inter-converted with some non-zero probability a protocol involving any number of steps where (i) one party applies a an arbitrary local operation $L:{H}_{i}\to {H}_{i}$ or (ii) one party sends some classical information to another.

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

(1)$U:{ℂ}^{2}\otimes {H}_{1}\to {ℂ}^{2}\otimes {H}_{1}$U : \mathbb{C}^2 \otimes H_1 \rightarrow \mathbb{C}^2 \otimes H_1

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 $\mid 1⟩$ 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.

###### Theorem

Two states are SLOCC-equivalent iff they can be inter-converted by applying arbitrary invertible local operations (ILOs).

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]

###### Theorem

Any genuine tripartite state |$\Psi$> $\in {ℂ}^{2}\otimes {ℂ}^{2}\otimes {ℂ}^{2}$ is SLOCC-equivalent to either |W> or |GHZ>;.

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

(2)$\mid W⟩=\mid 100⟩+\mid 010⟩+\mid 001⟩\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\mid \mathrm{GHZ}⟩=\mid 000⟩+\mid 111⟩$|W\rangle = |100\rangle + |010\rangle + |001\rangle \qquad\qquad |GHZ\rangle = |000\rangle + |111\rangle

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]

## References

An introduction is in

A discussion in quantum mechanics in terms of dagger-compact categories is in