# 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

$p : H_1 \times H_2 \to H_1 \otimes H_2$

which sends a pair of states $(\psi_1, \psi_2)$ 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 $|\Psi^{(A)}\rangle$ and $|\Psi^{(B)}\rangle$ respectively. The combined state of the system may be described by a single state vector $|\Psi^{(AB)}\rangle=|\Psi^{(A)}\rangle \otimes |\Psi^{(B)}\rangle$.

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

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

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

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

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. The entangled states discussed above are, in fact, pure states rather than mixed states because they cannot be broken down further. However, there is also a notion of entanglement for mixed states.

## Properties

### LOCC and SLOCC

The following refers to (Coecke-Kissinger).

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 $|\Psi\rangle,|\Phi\rangle \in \bigotimes H_i$ 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 $U : H_i \rightarrow H_i$ 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 $|\Psi\rangle,|\Phi\rangle \in \bigotimes H_i$ 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 $L : H_i \rightarrow H_i$ 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 $|\Psi\rangle \in H_A \otimes H_B$ and Alice wishes to perform some operation $L$. Alice prepares an ancilla qubit $|0\rangle \in \mathbb{C}^2$ and performs a unitary operation

$U : \mathbb{C}^2 \otimes H_1 \rightarrow \mathbb{C}^2 \otimes H_1$

on her qubit as well as her part of the state $|\Psi\rangle$. She then measures the ancilla qubit. If she gets an outcome of $|0\rangle$, she has performed some operation $L : H_A \rightarrow H_A$ and if she gets outcome $|1\rangle$ she has performed $L' : H_A \rightarrow H_A$. The probability of Alice successfully performing $L$ is then the probability of getting the outcome of $|0\rangle$ 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 $\mathbb{C}^2 \otimes \mathbb{C}^2$, 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 |$\Psi$> $\in \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ 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:

$|W\rangle = |100\rangle + |010\rangle + |001\rangle \qquad\qquad |GHZ\rangle = |000\rangle + |111\rangle$

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

## References

Introduction:

Exposition of entanglement as a phenomenon of non-Cartesian monoidal categories:

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

• W. Dür, G. Vidal, J. I. Cirac, Three qubits can be entangled in two inequivalent ways, Phys. Rev. A. 62, 062314

Discussion in quantum optics:

A connection to algebraic geometry is proposed in

• Frédéric Holweck, Jean-Gabriel Luque, Jean-Yves Thibon, Geometric descriptions of entangled states by auxiliary varieties, J. Math. Phys. 53, 102203 (2012); doi

The following work included the consideration of identical particles into the study of quantum entanglement. In this case, the usage of partial trace may not be suitable and instead subsystems are described in terms of subalgebras. The work is in operator algebraic framework, based on usage of GNS construction and related to the consideration of von Neumann entropy.

• A. P. Balachandran, T. R. Govindarajan, Amilcar R. de Queiroz, A. F. Reyes-Lega, Entanglement and particle identity: a unifying approach, Phys. Rev. Lett. 110, 080503 (2013) arxiv/1303.0688; Algebraic approach to entanglement and entropy, arxiv/1301.1300; Entanglement, particle identity and the GNS construction: a unifying approach arxiv/1205.2882 (earlier, longer version, overlapping with 1303.0688)

Use of homological algebra for quantifying entanglement:

category: physics