Algebraic and group structure for bipartite three dimensional anisotropic Ising model on a non-local basis

Algebraic and group structure for bipartite three dimensional anisotropic Ising model on a non-local basis

Francisco Delgado fdelgado@itesm.mx Escuela Nacional de Posgrado en Ciencias e Ingeniería, Tecnológico de Monterrey, México. Departamento de Física y Matemáticas, Tecnológico de Monterrey, Campus Estado de México, Atizapán, Estado de México, CP. 52926, México.
July 22, 2019
Abstract

Entanglement is considered as a basic physical resource for modern quantum applications in Quantum Information and Quantum Computation theories. Interactions able to generate and sustain entanglement are subject to deep research in order to have understanding and control on it, based on specific physical systems. Atoms, ions or quantum dots are considered a key piece in quantum applications because is a basic piece of developments towards a scalable spin-based quantum computer through universal and basic quantum operations. Ising model is a type of interaction which generates and modifies entanglement properties of quantum systems based on matter. In this work, a general anisotropic three dimensional Ising model including an inhomogeneous magnetic field is analyzed to obtain their evolution and then, their algebraic properties which are controlled through a set of physical parameters. Evolution denote remarkable group properties when is analyzed in a non local basis, in particular those related with entanglement. These properties give a fruitful arena for further quantum applications and their control.

pacs:
03.67.-a; 03.67.Bg; 03.65.Ud; 03.65.Ge; 03.65.Fd; 03.65.Aa; 02.20.Uw

I Introduction

Quantum entanglement is one of the most interesting properties of Quantum Mechanics which was noted since early times of theory vneumann1 (); schrod1 (); schrod2 (); einstein1 (); schrod3 (). Nowadays, this property has been exploited by quantum applications as central aspect to improve information processing in terms of capacity and speed jozsa1 (); jozsa2 (); bennet1 (). Thus, Quantum Information studies entanglement as an important aspect to codify and manage information in several quantum applications developed since seminal proposals in Quantum Computation feyn1 (); deutsch1 (); steane1 (), Quantum Cryptography bennet2 (); ekert1 () and discoveries about superdense coding bennet3 () and teleportation bennet4 (). A complete entanglement map of road will not be constructed until its quantification and behavior could be understood since a general mathematical theory and a deep knowledge about quantum interactions which generates it. It last means, Hamiltonian models which are able to generate entanglement, which are actually studied in order to understand how this quantum feature is generated on several physical systems. For magnetic systems, Ising model ising1 (); brush1 () in statistical physics and Heisenberg model baxter1 () in quantum mechanics are Hamiltonian models derived from interaction between spin systems when they include a magnetic field, it works as a driven element in Hamiltonian. Nielsen nielsen2 () was the first reporting studies of entanglement in magnetic systems based on a two spin systems using that model including an external magnetic field.

Magnetic driven Ising interaction is well known by developing an evolution depending on local parameters. Still its simplicity, for only two particles it exhibit four energy levels introducing a non periodical behavior in terms of Rabi frequencies phenomenon and their control kiri1 (); meng1 (). Several simplified models has been analyzed in order to understand quantum behavior of these kind of systems when they approach to different concrete systems as quantum dots or electronic gases. Still, research around of control and entanglement in bipartite qubits terzis1 () and lattices stelma1 (); novo1 () is fundamental because these simple systems let the possibility to control quantum states of a single or a couple of electron spins at time, standing at the heart of developments towards a scalable spin-based quantum computer.

Control being depicted, in combination with controlled exchange between neighboring spins, would let obtain universal quantum operations recher1 (); saraga1 (); kopp1 () in agreement with DiVincenzo criteria vincenzo1 () in terms of reliability of state preparation and identification of well identified qubits. Thus, the aim of this paper is analyze algebraic properties of a bipartite system with a general three dimensional anisotropic Ising interaction including an inhomogeneous magnetic field strength in a fixed direction. One of the central aspects is that analysis of dynamics is conducted on a non-local basis in terms of classical Bell states, which lets to discover outstanding algebraic aspects of this interaction around entanglement and a regular group structure, obtaining possible direct applications for quantum control and quantum computer processing.

Ii Anisotropic Ising model in three dimensions

Different models of Ising interaction (XX, XY, XYZ depending on focus given by each author) has been considered in order to reproduce calculations related with bipartite and tripartite systems berman1 (); wang2 (); aless1 ()). Similar models which requires interaction with radiation are modeled in terms of Jaynes-Cummings and Jaynes-Cummings-Hubbard Hamiltonians meek1 (); rai1 (); ange1 (). As example, in quantum control, different versions of Ising interaction have been considered in terms of homogeneity of magnetic field, dimensions and directions involved branczyk1 (); xi1 (); delgado1 (); delgado2 (). Thus, restrictions in dimensions, number of particles and strength of external fields in these models are due for simplicity, geometry of lattices and other properties of physical systems involved wang2 (); kamta1 (); sun1 (); zhou1 (); gunlycke1 ().

In this work, we focus on the following Hamiltonian for the bipartite anisotropic Ising model ising1 (); nielsen2 () including an inhomogeneous magnetic field restricted to the -direction ( corresponding with respectively):

(1)

which attempts to generalize most of several models considered in the cited works before. By diagonalizing and finding the corresponding eigenvalues, which are independent of :

(2)

where and are defined as follows: if is a cyclic permutation of which will be simplified by using symbol, being equivalent to the pair of indexes :

(3)

ii.1 Reduced notation and definitions

One more suitable selection of reduced parameters will establish an appropriate notation which we will follow in the whole of work (in order to make finite some parameters and to reduce extent of some expressions):

(4)

Note that subscripts are settled for these variables in relation with their internal operations in (4). It is known that when anisotropic Ising evolution matrix is expressed in the computational basis, it has in general a complex full form (with full 4 4 entries complex and different of zero), except if magnetic field is in direction delgado1 (); delgado2 (), which denotes the privileged basis selected. For this reason, in the next section, the analysis will be based on Bell state basis to develop a different structure in it. This selection suggests to change the notation by using for lower and upper scripts. Nevertheless, when these labels appear in mathematical expressions, it will be convenient recover them to express operations, so they will be assumed as respectively when they are forming part of algebraic expressions. Reader should be alert about it. In the same sense, capital scripts will be reserved for referred to the computational basis; greek scripts will reserved for or ; latin scripts will reserved spatial coordinates or ; and (between parenthesis) as scripts for energy levels . will be used sometimes to emphasize number multiplication between terms in scripts and avoid confusions. For example, in this notation we will write the standard Bell states as:

(5)

ii.2 Eigenvectors and Evolution operator

In last terms, correspond with respectively and they could written in the current notation as:

the corresponding eigenvectors for each direction are:

where is the custom Kronecker delta.

An arbitrary bipartite state is written in computational basis or in Bell basis respectively as:

(8)

then it is possible demonstrate by direct calculation that concurrence and Schmidt coefficients wott1 (), in terms of coefficients , are in the Bell basis:

(10)

so, for eigenstates , coefficients are simply:

(11)

then, their entropy of entanglement becomes:

(12)

which is maximal only if (symmetric or antisymmetric fields), in agreement with delgado1 ().

Iii Form and structure of evolution operator

Using last expressions for eigenvalues and eigenvectors, and introducing the following convenient definitions related with energy levels:

(13)

and:

(14)

then, if evolution operator is written in Bell basis as:

(15)

so, for those elements different from zero, its explicit form becomes :

(16)

which gives close forms for evolution operators in each case when they are expressed in the non-local basis of Bell states. It express, in some sense, more explicitly the evolution of entanglement. Reader should note that scripts in variables defined in current section are related with energy labels more than internal operations (as in ) as before. Awareness about this detail in notation will avoid later misconceptions.

iii.1 Sector structure in the Evolution operator

Last expressions could be appreciated better in matrix form:

(17)

clearly have a sector structure (sectors are not consecutive to form matrix blocks, instead, each non zero entry is a vertex of square sectors embed in the whole matrix). Note that is included by set simply . By the properties of and , sectors are unitary with as determinant. Because is unitary, inverses are obtained just by take (nevertheless those structure, it is required prove if it can be obtained as for some other physical parameters for the same system). In addition, as the sum of eigenvalues is zero, then , which is an important aspect of our evolution operator.

Referring only to their structure, special unitary matrices in formed by unitary sectors in as is depicted in (17) (clearly with unitary and reciprocal determinants), they form groups. Thus, it is easy show that each one, , are subgroups of (identity, inverses and multiplication are included in each and product of two elements in the set remains in it):

(18)
(19)

where . In addition, we state the symbol to each set of matrices able to be generated by in (17). In them, the general structure for their sectors in is:

(21)

where denotes the associated spatial coordinate of magnetic field, is an ordering label for sector as it appears in the rows of the evolution matrix, corresponding with , the labels for its rows in each matrix of (17) (by example, are the labels for the second sector, , in it means . Note particularly that determinant for each sector, , are reciprocal because . Note that as was previously stated, but not all elements of is a for any parameters (because the form of in entries and respectively, which not a general complex number because their arguments are limited to integer or half integer factors of ). This implies that is not necessarily a subgroup of , which open opportunities to extend their coverage in with two o more pulses.

For a further discussion, it is notable to write the generic sector in exponential form in terms of Pauli matrices:

(22)

where , is the identity matrix and is the matrix sector with lacking of its numeric exponential factor, which is defined by further convenience. That structure shows that each kind of interaction (with external magnetic field in ) exclusively transforms Bell states in specific pairs as a operation plus a phase term in . Then, each sector for a given is responsible to combine linearly two Bell states and under a operation, with:

(23)

where is the sum module 2. Thus, it is possible associate, by eliminating the sector phase and instantaneous exponential factor in component (it is equivalent to use a certain rotating frame to whole system), a Bloch sphere between and in which, each part of in (8) corresponding with each sector:

evolves ’locally’ driven by in sector . Note that this evolution does not change probabilities between parts in each sector, but introduces relative phases because and the complex mixing generated by . Clearly are elements of a Lie group with parameters , which will be important further.

Figure 1: Schematic representation of states related between each interaction in (17). Each dotted line should be understood as linear combination of states on vertex. Each line represents an space equivalent to a Bloch sphere for related states and . On this sphere, evolution of linear combination could be represented as lines depicted in the picture on the right driven by an operator in .

Figure 1 shows Bell states related with several interactions or in (17). Each dotted line is a linear combination of states in their vertex (III.1). Given a initial state of this type and depending on interaction being considered (), evolution, for this part of quantum state, develops on a Bloch sphere as a trajectory, depicted in scheme on right-down, which shows its evolution trajectory for some pair and related by an specific interaction in their respective Bloch sphere.

By combining several adequate interactions, could be possible switch any Bell state into another (Figure 1) if sectors adopt the following combined forms for (referred in the following as diagonalization and antidiagonalization cases respectively):

(26)

Thus, by combining these two types of sector forms, we can achieve evolution loops mielnik1 (); fernandez1 (); delgado3 (); delgado4 () and exchange operations delgado2 () in for Bell states, which will let obtain several control effects. Note that last expressions give only the matrix form for one sector , avoiding a confusion between operators in computational basis and Bell basis by the use of Pauli matrices in last expression, which are used only to set a desired form in that matrix sector. Clearly, much more equivalent cases could be considered with arbitrary phases instead of only phases, but we will restrict our analysis to these cases.

iii.2 Evolution of Bell states entanglement

In spite of (II.2) and (17), concurrence of states evolved from Bell states could be easily obtained. Short calculations show that concurrence for evolution for Bell state is easily expressed as:

showing that it depends only on one Rabi frequency at time in a very simply way. This expression is consistent with isotropic case reported in delgado1 (); delgado2 (). Note that (III.2) reproduces those results because this expression does not imply that states comeback each period to the original Bell state, only to the same entanglement value. Clearly, Bell states could become separable intermediately if , which is possible if reaches its maximum value when . In isotropic cases, some Bell states are invariant when magnetic field is symmetric or antisymmetric, as was shown in delgado2 (), which in general does not happen here. All this scenario contrasts with evolution of some separable states, by example those of computational basis , whose entanglement expressions depends normally on all or several Rabi frequencies involved, generally resulting in a non periodical behavior. Thus, in some sense, entanglement instead of separability appears as a natural feature of Ising model inclusive with external magnetic fields.

iii.3 Equivalence under rotations

Of course, last evolution operators are related via an homogeneous bipartite rotation in terms of Euler angles sak1 (); morr1 () on Hilbert space and on Fock space :

(28)

where:

expressed in the computational basis. As is expected, different Ising models with magnetic fields in cartesian directions transform between them. Specifically:

(30)

An aspect to remark is that sectors (21) in Ising evolution matrix could reproduce the form of (III.3). Nevertheless that single isolated qubit rotations by magnetic fields are well known, here this is not a trivial aspect because we are using a representation in Bell basis instead of computational basis as in last expression. It means that, under certain conditions, Bell states rotate as a whole. This is not necessarily a physical rotation, but Bell states, by pairs corresponding to rows where that sector is located, are transforming between them under these specific driven Ising interactions as a rotation in a Bloch sphere with main states as those Bell states instead of classical and .

Iv Group structure of evolution operators

Previously has been stated that evolution operators are part of subgroup in defined by their form in (18). But there are a inner structure which can be found in terms of group properties, which are not only related with the form of these operators but instead with their quantum structure related with Ising Hamiltonian (1). In this section, we analyze specific structure and restrictions in (17) to becomes a subgroup, together with traditional operator or matrix product in terms of their physical properties. It means, the physical prescriptions on parameters for each sector with which fulfills a group structure:

where is understood to have the structure in (17). Clearly associativity and existence of identity () are fulfilled because covering structure. Thus, only the product closure and existence of inverse should be analyzed. Because , sector structure is accomplished and analysis of previous conditions can be almost restricted to sectors (note only that exponential factor in each sector is the inverse of its respective factor in other sector, so both should be compatible in addition).

iv.1 Inverse

In spite of sector properties of matrices in , inverse of reduces to obtain inverse of each sector (caring compatibility around of their exponential factors). Because generic sector (21) is unitary, its inverse is:

(31)

thus, conditions for a generic sector (21) mimicking last expression can be obtained by comparing entries with and with in . This comparison shows that there are two possible restrictions to make compatible those four equations:

(33)

with which, only two additional equations remains, by example for entries and :

Equation (IV.1) automatically fulfills the compatibility between sectors because exponential factor becomes real. Combining this condition (IV.1), we get several cases. The most general is obtained noting that in spite of (IV.1), . After, to fulfill (IV.1), it is required that which implies , and . With these conditions, , with form (21), converts into (31). While, a brief analysis of equation (33) into (IV.1), shows that it reduces to a special case of last solution. Thus, the general prescriptions to get the inverse effect of an Ising interaction with a similar interaction but changing physical parameters are:

Note that last prescriptions are compatible with prescriptions of evolution loops in two pulses. Still, there is a pair of particular solutions. The first case, where corresponds to matrices with with which is the form of sectors in diagonal form , whose inverse is simply :

where are independently. With exception of freedom in selection of sign in , these prescriptions agree with (IV.1).

An possible additional case, which appears when equations (IV.1) are solved, is the case with . Nevertheless that this case is completely included in solution (IV.1), it states a specific kind of evolution matrices with of form:

(37)
(38)

Note that when this sector remembers Hadamard-like gates.

Nevertheless that prescriptions to get inverses are well defined, they involves sometimes conditions on which are not possible fulfill in specific experimental designs in terms to find in terms of . Instead, sometimes some of should fulfill strong restrictions (precisely those equivalent to Evolution Loops prescriptions stated before). Last means that inverse evolution for one pulse is not always achievable under general restrictions with other one pulse evolution, despite of inverse is well defined as part of matrices (17).

iv.2 Product closure

To probe the product closure and to obtain the prescriptions on physical parameters, we can restrict our discussion to sectors again, caring the global matching between exponential factors in (21). Thus, it is necessary probe the matching between in:

(39)

with with the form in (21). Demonstration is straight but it requires some transformations which we outline briefly. Again, comparing sectors we note that respective equations are compatible only if:

(40)

with . If this equation fulfills, then only two equations remain, for entries:

(41)