Abstract
In contrast to abstract statistical analyses in the literature, we present a concrete physical diagrammatic model of entanglement characterization and measure with its underlying discrete phase-space physics. This paper serves as a pedagogical treatment of this complex subject of entanglement measures. We review the important inherent concurrence property of entangled qubits, as well as underscore its emergent qubit behavior. From the discrete phase space point of view, concurrence translates to translation symmetry of entangled binary systems in some quantitative measure of entanglement. Although the focus is on bipartite system, the technique is readily extendable to multi-partite system of qubits, as can easily be deduced from the physical inverter-chain link model. A diagrammatic analysis of the entanglement of formation for any multi-partite qubit system is given. We show that quantum mechanical joint distribution are entanglements.
Keywords
- discrete phase space
- inverter-chain link model
- Hadamard transform
- entanglement of formation
- concurrence
- ergodic distance
1. Introduction
Quantum entanglement has developed from a mere intellectual curiosity [1] of the fundamental structure of quantum mechanics1 to become an important and practical resource for quantum information processing in the evolving theory of quantum information and ultra-fast computing. Thus, the quantitative measure of entanglement has developed into one of the most active fields of theoretical and experimental research. Here we will try to shed more light on some of the important concepts in the quantification of quantum entanglement by using a concrete simple mechanical model of a bipartite system of qubits or chain of qubits. This treatment will be in contrast with mostly abstract and statistical treatment of entanglement measure in the literature.
We will focus on the so-called entanglement of formation and concurrence, two of the most important concepts to characterize entanglement resource. Here we consider a qubit as a two-state system. Moreover, we also consider an entangled qubit as effectively a two-state system, an
2. Bell basis deduced from inverter-chain link model
We sketch here the derivation of the Bell basis states from our physical model, as depicted in Figure 1. Clearly, the entanglement of two
Since the Hadamard transform is unitary, the inverse transform is well-defined, i.e., the transformation is bijective. We refer to
By virtue of this bijective relationship, any function of Wannier functions will have a corresponding function of Bloch functions. For example, a maximally
3. Entangled qubits as an emergent qubit
The virtue of our inverter-chain link model is that the emergent two-state property of entangled qubits is very transparent, since changing the state of one of the qubits also
Here, we employ the matrix representation of states and operators. We now represent
Then we have
proving the two eigenvalues, like a qubit property of the triplet system.
For the singlet the most straightforward manipulation is to recognize that the singlet system is an independent system with its own two states which is orthogonal to the triplet states. In this sense we can also represent the singlet states just like that of the triplet states, namely,
Note that in the ‘Bloch-state” space,
Thus, we also have for the singlet states
Following this notion, Eqs. (8) and (9) can easily be extended to all entangled qubits, bipartite or multi-partite qubit systems to yield an emergent qubit. It is far more simpler to analyze the inverter-chain link diagrams, i.e., employ diagrammatic analyses.
4. Translational/shift invariance of Bell basis
The translational invariance of maximally entangled Bell basis states demonstrates a unique characteristic of entangled qubits. This unique property has been used to detect or measure how much entanglement is present in arbitrary pure and mixed states. This is implemented in terms of the notion of concurrence, to be elaborated below.
The translational or shift operation (also referred to as the
Upon applying the translation (shift by
where addition obeys modular arithmetic (
4.1 Translational property of maximal superposition of Bell basis
Now let us consider the superposition of maximally entangled Bell basis states. We can easily see that a superposition of ‘plus’ entangled basis states with ‘minus’ entangled basis states yield
The thing to notice is that although the superposition is made up of two maximally entangled Bell basis, the results are not shift invariant (i.e., generation of another
Equation (15) is quite interesting because it holds on a complete expansion of a direct product of two qubit states. These relations can easily be deduced from the physical diagrammatic model, see Figure 1. For example, if we define the operation as a superposition such as,
We see that Eq. (17) is a direct product state of two qubits. We will see in what follows that this is an entangled state and corresponds to Eq. (15) of the
corresponds to Eq. (15) of the
However, the following combinations of ‘even’ and ‘odd’ entangled states result in
and
by virtue of the failure to have global sign factors. All these claims are justified through the concept of concurrence, an inherent property of entangled qubits, to be discussed in what follows. Moreover, this feature of failing to have global sign factor is also reflected in the failure to represent by our inverter-chain link diagrams.
5. Bipartite system
Let the Hilbert spaces of a bipartite system consisting of
Let the density matrix for the whole system be denoted by
The entanglement entropy,
Upon multiplying by the Boltzmann constant,
which is the Boltzmann thermodynamic entropy, based on ergodic theorem.
A qubit is simply a quantum bit whose number of distinct eigenstates is 2. We denote these eigenstates as
Here, the first bit refers to subsystem
then
An operator whose square is equal to itself must have an eigenvalue equal to unity. Let us write for pure state of the two qubits as,
We have,
Thus, indeed,
We refer to
Now clearly
so that
6. Entanglement of formation of multi-partite qubit systems
The entanglement entropy of subsystem
where exponent base 2, here 1, is the number of qubits that is entangled with system
If each subsystem
Any combination or superposition of both ‘even’ triplet and singlet states comprised a maximally entangled state, i.e.,
Thus, from the states given above, we have for example the maximally entangled state,
and similarly, we can also form another entangled state,
where the first two bits belongs to subsystem
Therefore
In general, the following maximally entangled state corresponds to a chain of entangled basis states, namely,
of a four qubit system.
The density matrix operator for the whole 4-qubit system can be written as,
The reduced density matrix operator for subsystem
To determine the eigenvalues for
So we have
and the eigenvalues of
The exponent 2 correspond to the number of qubits that is entangled with subsystem
Now of course, for this bipartite system,
which simply means a complete matching of configurations of each system
The following diagrams represent the entangled tripartite system of qubits (Figure 3).
The entangled basis states are as follows,
A chain or superposition of any choice of entangled
are all, respectively, maximally entangled states with emergent two-state or qubit properties.
It is important to point out that any entangled qubits, irrespective of their number, identically behave like a single qubit, i.e., behaving exactly like a two-state system. Thus, the natural order of disentangling is as follows: First, one entangles one qubit from the remaining two entangled qubits. The entanglement of formation is equivalent to one qubit. Next, one disentangled the remaining entangled two qubits. This further give an entanglement of formation of one qubit. Thus, the total entanglement of formation is
6.1 Monogamy inequality of entanglement of formation
The reasoning we have given above yields exact equality of the monogamy, usually given as
The exact equality comes about by construction, since on both sides of Figure 4, two 2-qubit entangled states are being untangled to obtain the entanglement of formation of this multi-partite system. This comes about through the knowledge that any multi-partite entangled qubits behave as an emergent qubit.
Similarly, for an entangled 4-partite system, the entanglement of formation is determined schematically by the diagrams of Figure 5, where the unentangling operations to be done on the left side are itemized on the unentangling operations of
This sort of diagrammatic analysis of the entropy of entanglement of formation can straightforwardly be employed to all multi-partite qubit systems by a simple counting argument as is done here. For example, for 18 multi-partite entangled qubits in Figure 6, we have the entanglement of formation schematically depicted by the equality where there is 17 number of monogamy in the right-hand side yielding 17 qubits of entropy of entanglement of formation.
7. Joint distributions as entanglements
In this section, we will explicitly show the discrete phase-space physics of qubit entanglements. We will demonstrate that a
Consider the generalized Fourier transformation between two Hilbert spaces,
where the
where
7.1 Two-point joint distributions
We have a joint distribution in
Therefore, for a qubit or a two-state system, we write,
Consider the
We will denote the joint distribution in
7.2 Three-point joint distributions
Similarly, for three-fold joint distribution in
Again, in matrix form, we express the above equation as
From Eq. (68), we can define 8 joint distributions in
In Eqs. (69)–(76), we made use of the following relations:
A crucial observation is that only even
8. Chiral degrees of freedom and entangled qubits
Observe that the pseudo-spin variables in semiconductor Bloch equation is defined by the following expressions,
This maps to
We refer to the basis
This is a very important observation which will aid in understanding the
9. The concurrence concept and emergent qubit
By virtue of our diagrammatic construction, coupled with discrete phase space Hadamard transformation, in deriving the entangled basis states, the concurrence concept defined by Wooters [2, 3], as well as the two-state (qubit) properties of entangled multi-partite qubits, naturally coincides with the mathematical description of our physical model of qubit entanglement. In other words, the essence of the quantum description of an entangled qubits in Figure 1 consists of the superposition of product ‘site states’ and their corresponding translated ‘site states’ of all qubits. This means a superposition of the two states of the entangled two qubits of Figure 1. One observe that irrespective of how many qubits are entangled, the resulting entangled state is an
The concept of concurrence is basically contained by construction of our physical model of qubit entanglement. It is defined by
where
10. A natural measure of entanglement
The two properties of any entangled qubits, namely, concurrence and its
Equation (90) basically say that if there is complete concurrence, i.e.,
affirming that entangled qubits behave as a two-state system or as an
10.1 Entropic distance in entanglement measure
From the above developments, one can introduce an entanglement entropic distance by the formula
where
An example where the
where
Then
11. Mixed states and pure states
In discussing mixed and pure states, one makes use of density-matrix operators. This is the domain of abstract statistical treatment usually found in the literature, perhaps following the statistical tradition of Bell’s theorem. To elucidate the basic physics, we will here avoid abstract statistical treatment and only discuss specific situations and examples.
11.1 Mixed states and mixed entanglements
Consider the maximally mixed state of a triplet system,
We obtain the mixed entanglements given by,
Similarly, consider the maximally mixed state of a singlet system,
We obtain the mixed entanglements given by
11.2 Mixed state from pure state (entangled state)
Consider the example of a bipartite of two qubit system. Consider a pure entangled state,
Then the density matrix is,
where the first qubit belongs to party
so that Eq. (103) is a pure state. Again, we have, by tracing the party
Now clearly
so that
12. Concluding remarks
The inverter-chain rigid-coupling mechanical model of entanglement link has been demonstrated to faithfully implement the discrete phase space viewpoint [4]. The crucial observation that arise from this inverter-chain link model is that any multi-partite qubit entangled system has the property of an
The natural measure of entanglement is based on entropy of entanglement formation, concurrence, and entropic distance from maximally entangled reference.
References
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Hill S, Wooters WK. Entanglement of a pair of quantum bits. Physical Review Letters. 1977; 78 :5022 - 3.
Wooters WK. Entanglement of a pair of quantum bits. Physical Review Letters. 1998; 80 :2245 - 4.
Buot FA, Elnar AR, Maglasang G, Galon CM. A mechanical implementation and diagrammatic calculation of entangled basis states. arXiv:2112.10291 [cond-mat.mes-hall]. 2021 - 5.
Kim JS. Entanglement of formation and monogamy of multi-party quantum entanglement. Scientific Reports. 2021; 11 :2364. DOI: 10.1038/s41598-021-82052-3 - 6.
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Notes
- Note that although Bell’s theorem asserts the nonlocality of quantum mechanics, the EPR inquiry is still not resolved, i.e., what is still left unanswered is the mysterious ‘link’ between qubits corresponding to our “see-saw” or mechanical inverter-chain link. (Note: Bell’s inequality theorem is widely discussed in the literature and websites, we prefer not to cite specific reference).
- The use of the term “triplet” is actually a misnomer here since the entangled system is not free to assume a singlet or zero spin state. Thus, this term is used here only as a label.
- The idea that entanglement is due to conservation of momentum does not hold for triplet entanglement since its two states give opposing spin-angular momentum. However, one may interpret that quantum superposition of the two-opposing angular momentum states conserves the overall zero net spin-angular momentum, as supported by their eigenvalues similar to a single qubit. On the other hand, for singlet entanglement, the zero angular momentum is conserve in both two states. This seemingly apparent physical difference of the triplet and singlet entanglements underscores the importance of resolving the “mysterious link” in the EPR inquiry, [1] in order to further advance theoretical physics.
- We maintained the Bloch function/Wannier function analogy for convenience, and to stress the wide-ranging impact of the discrete phase-space physics.
- Concurrency is one of the most important characterization of entangled qubits, to be discussed in more details below.
- This is also deduced when concurrence C=1 [3].
- What we mean by party A and B is in the general sense since any entangled number of qubits behave as an emergent qubit.
- If the parties A and B are entangled states, then what we have obtain are also mixed entanglement through the inverse Hadamard transformation.