## What is Quantum Entanglement?

Quantum Entanglement … what a mysterious phenomenon. It happens at the subatomic scale and has no equivalent in the macroscopic world. It has historically been questioned and formalized by Schroedinger, Einstein, Podolski, and Rosen, and this since 1935 already [1, 2, 3].

It is frequently interpreted as a non-classical “link” between several quantum particles, considered as a correlation and a dependency between the states of several quantum systems, and this, independently from the distance (in space) between each of them. That is why the concept of entanglement invites us to consider the state of entangled particles as a unique system, without being able to explicitly express the state of each particle.

The nature and the physical reality of quantum entanglement are not yet known, and various theories were proposed to explain it or give an interpretation using modern physics [4, 5, 6, 7, 8, 9, 10, 11].

We can distinguish several types of entanglement, depending on which particle is entangled with the other(s), and at which amount. Being able to classify and detect the entanglement type of a state is a key competence in Quantum Computing. In fact, some classes of entanglement appear in quantum algorithms that show speedup, and other types of entanglement don’t appear or don’t show that much efficiency (when present).

In this context, we may want to evaluate what entanglement classes are we generating during the run of our algorithms, and with which amount of entanglement also (and the amount can sometimes be deduced from the entanglement class of the state). Some classes of entanglement, when all particles are entangled, are called genuine entanglement, and we may need these types when designing new quantum algorithms.

In this article, we will focus, from a mathematical point of view, on how we can classify different types of entanglement in some easy cases.

## Quantum Entanglement and separability

One of the most intuitive manners of defining entanglement, from a mathematical point of view, is to introduce the concept of separability. A quantum state is defined as separable, if we can separate each component, each particle, that forms the quantum system, and if we can consider the state of each particle independently from the state of the other particles of the system.

Mathematically, being able to separate particle is equivalent to being able to factorize the expression of the whole quantum system so it can be expressed as the product of vectors encoding the state of each particle. In fact, if we consider a quantum system |*ψ*⟩ composed of *n*-qubits, for which we know the state |*ψᵢ*⟩ of each *i*-th particle, then the state |*ψ*⟩ can always be written as :

Therefore, a state |*ψ*⟩ is said to be **separable **if it can always be written as the tensor product of its particles states. This implies that any state that cannot be written in this form is an **entangled **state.

*“One of the exciting things about an entanglement puzzle is there’s no end to it. Once you solve how to take it apart, you have to solve how to put it back together”*

*T. J. Miller.*

## Quantum Entanglement classification

In order to characterize quantum entanglement through several entanglement classes, we have to partition the Hilbert space into different equivalent classes, where the equivalence relation is defined with respect to the operations applied to quantum states [12]. Depending on the nature of the transformations (or operations) considered, the related classification (and the difficulty to establish them) can differ. We propose to recall the main ones that you can fin in the literature.

## LU transformations

One fundamental property of quantum entanglement is that it remains invariant under any local change of basis. More precisely, the entanglement class of a quantum state will stay the same after the application of *Local Unitary* (LU) transformations. These transformations preserve the orthonormal basis for each qubit.

A transformation U, acting on a quantum system with n particles is said to be LU if an and only if it can be defined as the product of n local unitary matrices, such that

Therefore, two states are considered as LU-equivalents if there exists an LU-transformation that can transform one state into another. These two states will then share the same physical properties, such as entanglement, because of the impossibility to create and destroy it using LU-transformations. Several studies were interested in how to determine necessary or sufficient conditions concerning the LU-equivalence of two arbitrary states, and the tools permitting to determine the normal forms related with each equivalence class under the action of the LU group [13, 14, 15, 16, 17].

## LOCC and SLOCC operations

Despite the local aspect of some transformations acting in a local way on particles, it is always possible to coordinate the quantum operations at distance, by using classical communication channels (such as phones).

This is how we define the group of *Local Operations and Classical Communication* (LOCC), representing to the closest the actions realized during experimentations: local unitary transformations, coordinations by classical communications and measurements of quantum systems. The LOCC group permit therefore to model in a larger way the actions on quantum systems, compared to the LU group. On another hand, in the case of pure state (which is the case of interest for us in this article), it is known that two states are LOCC-equivalents if they are LU-equivalents [18, 12]. Many studies about the classification of entanglement under the LOCC group have been developed [19, 20, 21, 22], and it is a difficult problem that still interests the scientific community [23].

It is from this previous concept that is defined the *Stochastic Local Operations with Classical Communication* (SLOCC). Introduced the first time by Bennet et al. in [21], and formalized later in [24], the SLOCC group is mathematically defined as the group of local invertible operations. When we consider the Hilbert space of general quantum state (defined as the tensor product of Hilbert spaces of each particle), an operator from the SLOCC group is defined as the cartesian product of invertible matrices, modeling an invertible and independent action on each subspace associated to each particle (local operators).

The action of the SLOCC group can be seen as similar to the LOCC’s one, except that the equivalence between two states is not necessarily deterministic. In fact, we introduce a stochastic aspect in the correspondence between two states, and the probability of success of the operation only has to be non-zero. Therefore, SLOCC operators are not equivalent to LU-transformations. The action of a SLOCC group element on a quantum state can increase or decrease the amount of entanglement (in the sense of a quantitative measure of entanglement), but cannot create entanglement (transform a separable state into an entangled one), and this, even in the presence of a stochastic process. Finally, the LOCC-equivalence between two states implies the SLOCC-equivalence between these same states, but the opposite is not necessarily true. Thus, a classification under the action of LOCC will be finer than the classification under SLOCC.

In the next section, we focus on the classification of entanglement under the action of SLOCC for the 2-qubit and 3-qubit cases.

## The 2-qubit case: Bell state

As a first example, let us study first the 2-qubit case. A general 2-qubit state is composed of two particles, each particle having two possible basis states. For each particle, the two possible states are denoted |0⟩ and |1⟩. This means that the quantum state can potentially be in four different basis states, denoted by |00⟩, |01⟩, |10⟩ and |11⟩. Since any quantum state can always be expressed as the superposition of its basis states, therefore any 2-qubit state |*ψ*⟩ can be written as:

, with a normalization condition on the complex coefficients. The two-qubit state is separable if and only if it can be written in the following form:

If we develop the tensor product of the previous expression, we retrieve the following state:

By identification, if we take the expression of a generic 2-qubit state, in order to be separable it has to satisfy the following conditions:

This implies that:

This is equivalent to writing:

Therefore, any state that verifies this algebraic equation is separable. On another hand, any state that doesn’t verify this equation is entangled. This equation permits, then, to distinguish between separable and entangled states. This algebraic equation defines a geometrical object called the Segre variety (that will be discussed in another medium article). Any state that is outside this variety will be entangled. This equation is invariant under SLOCC transformation, which means that the separable states form an orbit under the action of SLOCC group.

If we represent the 2-qubit state as a matrix (instead of a vector),

then the equation defining the separable state can be rewritten as:

This result can be generalized to higher dimensions, and we will say that any quantum state is separable if it annihilates all the possible 2x2 minors.

In the 2-qubit case, the other equivalence class under SLOCC is the entangled states. The first example of entangled state was proposed by Einstein, Podolski and Rosen, and it is named either EPR state or Bell state, and it is expressed as:

One can easily verify that it is impossible to write this state as a tensor product of two qubits and that the determinant of the associated matrix is not equal to 0.

From the expression of the state, one can see that if we measure the first qubit and retrieve the state |0⟩, then the whole system will be projected in the state |00⟩, and thus the second qubit must be also in the state |0⟩. We see with this example how by acting on a particle, we can instantaneously affect the second without manipulating it, and this is the magic of entanglement.

In the next section, we add a particle to the quantum system and go to the 3-qubit case, and see how we can classify entanglement between these particles.

## The 3-qubit case: GHZ and W states

The 3-qubit case is the first to illustrate the existence of non-equivalent entangled states (|*GHZ*⟩ and |W⟩ states, as explained later), as it was published for the first time by Dür et al [24]. We can enumerate six orbits under the action of SLOCC for 3-qubit systems, which are represented in the table below. One can establish geometrical interpretations of this classification [25, 26], and we will discuss it another time in a different medium article.

The first orbit regroups all separable states, i.e. all 3-qubit states that can be expressed as the tensor product of 3 qubits. The simplest representant is the state |000⟩, which is the tensor product of three qubits, each one in the state |0⟩ (we use here simplification in notations to avoid writing all tensor products).

The biseparable states correspond to quantum systems that can be decomposed into two independent subsystems: the first is composed of only one particle, and the second is composed of two entangled particles. Therefore, biseparable state, in the context of 3-qubits, can be written as the tensor product between a qubit and a Bell state. Depending on which qubit is independent, we can define 3 orbits of biseparable states. It is still an entangled state, but not from a global perspective since only two particles are entangled in the three-particle system.

The |W⟩ state is the first type of entangled state for pure 3-qubit states. One representant of the class can be defined as the sum of basis states that have a Hamming distance of 1 (in their binary notation). Geometrically, it is a special state because it has (as a tensor) a rank equal to 3, compared to other entangled states that are of rank 2, and belong to the tangential variety. The interesting thing about this state (and all SLOCC-equivalent states), is that it cannot be generated by Grover’s algorithm. Besides, the tangential seem also to be absent during most of the possible runs of Shor’s algorithm. This may have a link with the role of entanglement classes in the speedup of quantum algorithms, and it is discussed in [27]**.**

The last entanglement class is one of the|*GHZ*⟩ states. The |*GHZ*⟩ state is defined like the Bell state, i.e. the sum of two basis states that are complementary in their binary notations. It is the generic state of the Hilbert space for 3-qubits. This means that if you take a random tensor in the Hilbert space, with a high probability you will retrieve a state SLOCC equivalent to |*GHZ*⟩. It is the second entanglement type where all particles are entangled, and it is non-equivalent to the class-related with the |W⟩ state.

## At ColibrITD

Quantum Entanglement is a fundamental resource in quantum computing. At ColibrITD, we believe that quantum entanglement is necessary to reach quantum advantage when we design and run quantum algorithms and circuits.

We work on leveraging quantum superposition and quantum entanglement in order to define states that can help solve faster and more precisely problems that industry or scientific community are facing. We believe that being able to quantify and classify entanglement during such computations is key in the context of Quantum Computation and Quantum Information Processing.

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[27]: PhD thesis : « Étude de l’Intrication dans les Algorithmes Quantiques : Approche Géométrique et Outils Dérivés », Hamza JAFFALI, UBFC, UTBM, 2020 https://tel.archives-ouvertes.fr/tel-03162835