Engineered spin chains for perfect quantum state transfer

For the application to quantum information processing technology, quantum state transfer (QST) between two directly interacting qubits, as well as between two distant ones interacting through the intermediary of different quantum systems, was widely considered. As the intermediary system, it was proposed to use photons in an optical fibrr for the case of the QST between two quantum dots placed inside two microcavities connected through the optical fibre [1–4], or a network of interacting spins in the case of the QST between two spin-qubits interacting with this network [5–17] (for a list of references, see the review of Bose [10]). It is known that the perfect QST from one end of a homogeneous N-qubit chain to another is possible only for N=2 and 3 [7]. In order to achieve the perfect QST, it was proposed to use engineered spin chains with untunable but modulated coupling between adjacent spins, and a special version of engineered spin chains that allowed the perfect QST to be found in [7].

The purpose of this work was to establish a wide class of engineered inhomogeneous mirror-symmetric N-spin chains with modulated untunable couplings that can provide the perfect QST between two ends of each chain. It includes as special cases the spin chains proposed in [7]. In section 2, the conditions sufficient for the engineered mirror-symmetric spin chains allowing the perfect QST are derived. From these conditions, the algebraic equations determining the physical parameters of these spin chains are established. By solving these equations, we can obtain the sets of values of these physical parameters. Examples of the chains with even numbers of spins are presented in section 4, and those with odd numbers of spins are presented in section 5. Section 6 is the conclusion.

Following the idea in [8, 9], we have studied mirror-symmetric engineered inhomogeneous N-spin chains. The Hamiltonian of each chain is invariant under its mirror inversion I through the centre of this chain. In the Hilbert space of state vectors of a chain of N spins, we introduce basis vectors |i₁ i₂ ... i_N 〉, where the index i_n labels the quantum state of nth spin, i_n =0 for its ground state and i_n =1 for the excited state. N basis vectors of the form

are state vectors of the quantum states with the excitation of only one spin—single excitation states. The mirror inversion I has following action on these state vectors

Consider first a chain of an even number of spins, N=2N₀, N₀ being an integer. Instead of the basis vectors |n〉 in the Hilbert subspace V⁽¹⁾of state vectors of single excitation states, it is convenient to use symmetric and antisymmetric normalized combinations,

as the basis vectors of V⁽¹⁾. Each set of unit vectors | φ_n ⁽⁺⁾〉 or | φ _n ⁽⁻⁾〉 spans a N₀-dimensional vector subspace V⁽⁺⁾ or V⁽⁻⁾, respectively, of V⁽¹⁾. Each of them is invariant under the mirror inversion, | φ _n ⁽⁺⁾〉 are the even state vectors while | φ _n ⁽⁻⁾〉 are the odd ones:

V⁽¹⁾ is decomposed into the direct sum of V⁽⁺⁾ and V⁽⁻⁾.

The mirror symmetry of the spin chain means that each vector subspace V⁽⁺⁾ and V⁽⁻⁾ is invariant with respect to the action of the Hamiltonian

In each vector subspace, V⁽⁺⁾ or, V⁽⁻⁾ there are N₀ eigenvectors | ψ_k ⁽⁺⁾〉 or | ψ _k ⁽⁻⁾〉 of H:

Each of them is a linear combination of N₀ unit vectors (2):

Coefficients A_kn ^(±) satisfy the following conditions:

and therefore formula (6) has the inverse one:

In order to study the transfer of the excitation from one spin of the chain to another, we now consider the time evolution of the state vector of a single excitation state |n〉. Using formulae (2), (5), (6) and (9), we obtain

Suppose that eigenvalues E_k ^{( ±)}, k=1,2, ...,N₀ can be expressed in the form

with some value E₀ of energy, some value τ of time and arbitrary integers ν_k and μ_k . From equations (7), (8) and (10) it follows that

This means that, up to a common phase factor ^{e −i E₀τ}, the dynamical evolution of the chain during the time interval τ leads to the mirror inversion (1): the excited state from the nth spin perfectly transfers to the (N+1−n)th spin, n=1, 2,...,N₀. In other words, the dynamical evolution of the chain during the time interval τ generates the perfect QST between two mirror-symmetric spins, in particular between two spins at two ends of the chain. Formulae (11) are the conditions sufficient for the perfect QST in a chain of 2N₀ spins.

In the case of a chain of an odd number of spins, N=2N₀+1, as well as mirror-symmetric and mirror-antisymmetric state vectors (2), there is one more basis vector invariant under the mirror inversion I:

Invariant basis vector | φ _N0+1⁽⁺⁾〉 and N₀ mirror-symmetric state vectors | φ _n ⁽⁺⁾〉, n=1, 2,...,N₀, span a (N₀+1)-dimensional vector subspace V⁽⁺⁾ invariant with respect to the action of Hamiltonian H. Each of (N₀+1) eigenstates | ψ _k ⁽⁺⁾〉 of H in V⁽⁺⁾,

is a linear combination of (N₀+1) unit vectors |φ _n ⁽⁺⁾ 〉

Mirror-antisymmetric state vectors |φ _n ⁽⁻⁾ 〉, n=1, 2,...,N₀, are basis vectors of a N₀-dimensional invariant vector subspace V⁽⁻⁾. Each eigenstate of |ψ _k ⁽⁻⁾ 〉 of H in V⁽⁻⁾,

can be expressed in the form

Instead of formulae (7) and (8), now we have

and instead of formula (10), we obtain

for n≤N₀, and

Suppose that (N₀+1) eigenvalues E_k ⁽⁺⁾ and N₀ eigenvalues E_k ⁽⁻⁾ are determined by formulae (11), with (N₀+1) arbitrary integers ν_k and N₀ arbitrary integers μ_k . From equations (18) and (19), it follows that

with n≤N₀, and

This means that, up to a common phase factor ^{e − i} E₀ τ, the dynamical evolution of the chain during the time interval τ leads to the mirror inversion (1) for N₀ pairs of mirror-symmetric spins, and at the time moment τ the (N₀+1)th central spin returns to its initial excited state.

Thus, we can conclude that in mirror-symmetric spin chains the perfect QST can take place if the eigenvalues E_k ^{( ±)} of their Hamiltonians satisfy sufficient conditions (11).

In order to satisfy sufficient conditions (11) for the engineered spin chains to allow the perfect QST, their physical parameters must have definite sets of values. In this section, we derive algebraic equations determining the values of these parameters. For certainty, we study a simple model of spin chains with the symmetric XY Heisenberg exchange interactions between two adjacent spins.

Consider first a chain of an even number N=2N₀ of interacting spins with the Hamiltonian of the form

where N₀ energy values Ω_n and N₀ coupling constants J_n , n=1, 2,... , N₀ are the physical parameters to be determined, σ_α ⁽ⁿ⁾ with α=0 3, ± are the 2×2 unit matrix and the matrices σ₃, σ± acting on the state vector of the nth spin. From the Schrödinger equation (5), it follows the system of algebraic equations for coefficients A_kn ^{( ±)} in expressions (6) of eigenstates |ψ _k ^{( ±)} 〉:

The system of algebraic equations (24)–(28) has a nontrivial solution if and only if energies E^{( ±)} have 2N₀ values E_k ^(±), k=1,2,..., N₀, depending on physical parameters Ω_n and J_n , n=1, 2,..., N₀, of the engineered spin chain. Expressions of E_k ^(±) in terms of parameters Ω_n and J_n can be considered as 2N₀ algebraic equations, which determine Ω_n and J_n as functions of E_k ^{( ±)}. From sufficient conditions (11) for E_k ^(±), we derive expressions of physical parameters Ω_n and J_n in terms of constants E₀, τ, ν_k and μ_k , k=1,2, ...^, N₀.

In the case of a chain consisting of an odd number N=2N₀+1 of interacting spins, we have the Hamiltonian of the form

where (N₀+1) energy values Ω_n , n=1, 2,..., N₀+1, and N₀ coupling constants J_n , n=1, 2,...,N₀, are the physical parameters to be determined. From Schrödinger (14) and (16), it follows the system of algebraic equations for coefficient A_kn ⁽⁺⁾ in equation (15) and coefficients A_kn ⁽⁻⁾ in equation (17):

Equations (30)–(33) have exactly the same form as previous equations (24)–(27) in the case of a chain of even number of spins, while instead of equations (28) in the previous case now we have equations (34) and (35).

As in the previous case, the system of algebraic equations (30)–(35) has a nontrivial solution if and only if, (N₀+1) energies E_k ⁽⁺⁾, k=1,2, ... , N₀+1, and N₀ energies E_k ⁽⁻⁾, k=1,2, ... , N₀, have (2N₀+1) values depending on (2N₀+1) physical parameters Ω_n , n=1,2, ... , N₀+1, and J_n , n=1,2,... , N₀, of the engineered spin chain. Expressions of E_k ⁽⁺⁾ in terms of parameters Ω_n , n=1,2,... , N₀+1, and J_n , n=1,2,... , N₀ can be considered as (2N₀+1) algebraic equations, which determine Ω_n and J_n as functions of E_k ^{( ±)}. From sufficient conditions (11) with (N₀+1) arbitrary integers ν_k and N₀, arbitrary integers μ_k we derive expressions of (2N₀+1) physical parameters Ω_n and J_n in terms of constants E₀, τ, ν_k , k=1,2,... , N₀+1. and μ_k , k=1,2,... , N₀.

4.1.  N₀=1^: two-spin-chain

It is known that a system of two identical interacting spins allows the perfect QST (see, for example [7, 18]). The Hamiltonian of this system contains two physical parameters Ω, J and has two eigenstates,

with the corresponding eigenvalues of H:

In this case, formulae (11) become

with integers ν and μ. From equations (37) and (38) follows the relation

with some integers λ.

The dynamical time evolution of the excited states |1〉 and |2〉 of two spins is described by the following expressions:

They show that the perfect QST takes place at the time moments τ determined by equation (39).

4.2.  N₀=2^: four-spin-chain

In a mirror-symmetric four-spin-chain, there are four physical parameters Ω₁, Ω₂ and J₁, J₂. The Hamiltonian has four eigenstates with four eigenvalues:

Considering formulae (41) as algebraic equations for determining parameters Ω₁, Ω₂, J₁, J₂ and solving these equations, we obtain

Formulae (42)–(45) show that coupling constants J₁ and J₂ are invariant with respect to the simultaneous shift of all energy levels by one and the same quantity

while Ω₁ and Ω₂ are shifted by the same amount:

All mirror-symmetric four-spin-chains with physical parameters determined by equations (42)–(45) allow the exact QST if the eigenvalues E₁ ^{( ±)} and E₂ ^{( ±)} satisfy conditions (11). From these conditions it follows that E₁ ⁽⁺⁾−E₂ ⁽⁺⁾, E₁ ⁽⁻⁾−E₂ ⁽⁻⁾ and E₁ ⁽⁺⁾−E₁ ⁽⁻⁾can be represented in the form

with some integers ν, μ and λ, and we have

In the special case ν=μ=2λ+1 from equations (47)–(49), we obtain

in agreement with the result of Christandl et al [7] for a four-spin-chain.

4.3.  N₀=3^: six-spin-chain

In a mirror-symmetric six-spin-chain, there are six physical parameters Ω₁, Ω₂, Ω₃ and J₁, J₂, J₃. The Hamiltonian has six eigenstates with six eigenvalues E_k ^{( ±)}, k=1, 2, 3. They are related to six physical parameters through the six following equations:

By solving this system of six algebraic equations, we derive the expressions of physical parameters Ω_n and J_n , n=1, 2, 3, in terms of six eigenvalues E_k ^{( ±)}, k=1, 2, 3:

The perfect QST is ensured if the eigenvalues E_k ^{( ±)}, k=1, 2, 3, satisfy conditions (11). By means of lengthy algebraic calculations, it can be shown that the expressions of coupling constants J₁, J₂, J₃ are invariant with respect to the shift in all energy levels E_k ^{( ±)} by one and the same quantity

k=1, 2, 3, while all parameters Ω₁, Ω₂, Ω₃ are shifted by the same amount

In [7], it was shown that the perfect QST is possible through a six-spin-chain with the following physical parameters:

with some constant J. It is easy to check that, for this six-spin-chain, six eigenvalues E_k ^{( ±)}, k=1, 2, 3, have the expressions of the form (11) with following constants:

This means that the six-spin-chain found in [7] is a special case of a wide class of six-spin-chains satisfying sufficient conditions (11) for the existence of the perfect QST.

The reasoning presented above can be applied to other chains with even numbers of spins.

5.1.  N₀=1^: three-spin-chain

It is known that a system of three identical interacting spins allows the perfect QST (see [7]). Here, we consider a mirror-symmetric three-spin-chain with two energy parameters Ω₁, Ω₂ and one coupling constant J. It has three eigenstates | ψ _k ⁽⁺⁾〉, k=1, 2 and | ψ _k ⁽⁻⁾〉 with eigenvalues

and

Conditions (11) for the perfect QST at the time moment τ have the form

with integers ν ₁, ν ₂ and μ.

The dynamical time evolution of the excited states |1〉 and |3〉 of two spins at two ends is described by two similar formulae:

while that of the excited state |2〉 is described by the following expression:

In the special case Ω ₁=Ω ₂, formulae (62)–(64) become

Formulae (62)–(64) and (65)–(67) clearly show that, due to the dynamical evolution of the spin-chain, the excitation of one spin is gradually transferred to all others. At the time moment τ there takes place the perfect QST between two spins |1〉 and |3〉at two ends of the chain, and the excitation of the central spin |2〉 perfectly returns to this spin.

5.2.  N₀=2^: five-spin-chain

The Hamiltonian of a mirror-symmetric five-spin-chain contains five physical parameters: three energy values Ω₁, Ω₂, Ω₃ and two coupling constants J₁, J₂, and has five eigenstates with five eigenvalues: E_k ⁽⁺⁾, k=1, 2, 3 and E_l ⁽⁻⁾, l=1, 2. From the Schrödinger equations for the eigenstates, it follows a system of five equations determining five eigenvalues in terms of five physical parameters and vice versa:

By solving the systems of equations (68)–(72), we derive expressions of five physical parameters in terms of five eigenvalues:

By means of lengthy algebraic calculations, it can be shown that the expressions of coupling constants J₁, J₂ are invariant with respect to the shift of all energy levels E_k ⁽⁺⁾, k=1, 2, 3 and E_l ⁽⁻⁾, l=1, 2 by one and the same quantity

while all energy values Ω₁, Ω₂, Ω₃ are shifted by the same amount:

Formulae (73)–(77) with eigenvalues E_k ⁽⁺⁾, k=1, 2, 3 and E_l ⁽⁻⁾, l=1, 2 of the form

determine the sets of values of physical parameters of mirror-symmetric five-spin-chains allowing the perfect QST. A special case of this class of five-spin-chains was found in [7]:

5.3. N₀ =2^: seven-spin-chain

The Hamiltonian of a mirror-symmetric seven-spin-chain contains seven physical parameters: four energy values Ω₁, Ω₂, Ω₃, Ω₄ and three coupling constants J₁, J₂, J₃, and has seven eigenstates with seven eigenvalues: E_k ⁽⁺⁾, k=1, 2, 3, 4 and E_l ⁽⁻⁾, l=1, 2, 3. From the Schrödinger equations for the eigenstates, it follows a system of seven equations determining seven eigenvalues in terms of seven physical parameters and vice versa:

By solving the systems of equations (79)–(85), we derive expressions of seven physical parameters in terms of seven eigenvalues:

It can be shown that the expressions of coupling constants J₁, J₂, J₃ are invariant with respect to the simultaneous shift of all energy levels E_k ⁽⁺⁾, k=1, 2, 3, 4 and E_l ⁽⁻⁾, l=1, 2, 3 by one and the same quantity,

while all energy values Ω₁, Ω₂, Ω₃, Ω₄ are simultaneously shifted by the same amount:

Formulae (86)–(92) with eigenvalues E_k ⁽⁺⁾, k=1, 2, 3, 4 and E_l ⁽⁻⁾, l=1, 2, 3 of the form (78) determine the sets of values of physical parameters of mirror-symmetric seven-spin-chains allowing the perfect QST. A special case of this class of seven-spin-chains with , v₁=1, v₂=0, v₃=−1, v₄=−2, μ₁=0, μ ₂=−1, μ ₃=−2 was found in [7].

The reasoning presented above can be applied to other chains with odd numbers of spins.

The transfer of the excited state along the chains of N spins with the XY Heisenberg exchange interaction between nearest-neighbour spins and without decoherence was studied. It was shown that there exists a wide class of mirror-symmetric engineered N-spin-chains allowing the perfect QST. The Hamiltonian of each chain of N spin of this class depends on N physical parameters and has N eigenvalues. The physical parameters are expressed in terms of the eigenvalues. If the eigenvalues satisfy sufficient conditions of the form (11), then the perfect QST is possible. From the conditions (11), one can establish the sets of values of physical parameters of the chains allowing the perfect QST. Some examples were investigated in detail. The presented reasoning can be extended for the application to the study of all mirror-symmetric spin chains as well as to all two- and three-dimensional mirror-symmetric spin networks.

This work was completed in the Max-Planck Institute for the Physics of Complex Systems MPI-PKS, Dresden, Germany. The authors would like to express their gratitude to MPI-PKS for their hospitality and to Professor Peter Fulde for encouragement. We thank the Vietnam Academy of Science and Technology for their support.