Universal Dynamical Decoupling: Two-Qubit States and Beyond

# Universal Dynamical Decoupling: Two-Qubit States and Beyond

Musawwadah Mukhtar Department of Physics, National University of Singapore, 117542, Republic of Singapore    Thuan Beng Saw Department of Physics, National University of Singapore, 117542, Republic of Singapore    Wee Tee Soh Department of Physics, National University of Singapore, 117542, Republic of Singapore    Jiangbin Gong Department of Physics, National University of Singapore, 117542, Republic of Singapore Centre for Computational Science and Engineering,
National University of Singapore, 117542, Republic of Singapore
NUS Graduate School for Integrative Sciences and Engineering, Singapore 117597, Republic of Singapore
July 3, 2019
###### Abstract

Uhrig’s dynamical decoupling pulse sequence has emerged as one universal and highly promising approach to decoherence suppression. So far both the theoretical and experimental studies have examined single-qubit decoherence only. This work extends Uhrig’s universal dynamical decoupling from one-qubit to two-qubit systems and even to general multi-level quantum systems. In particular, we show that by designing appropriate control Hamiltonians for a two-qubit or a multi-level system, Uhrig’s pulse sequence can also preserve a generalized quantum coherence measure to the order of , with only pulses. Our results lead to a very useful scheme for efficiently locking two-qubit entangled states. Future important applications of Uhrig’s pulse sequence in preserving the quantum coherence of multi-level quantum systems can also be anticipated.

###### pacs:
03.67.Pp, 03.65.Yz, 07.05.Dz, 33.25.+k

## I Introduction

Decoherence, i.e., the loss of quantum coherence due to system-environment coupling, is a major obstacle for a variety of fascinating quantum information tasks. Even with the assistance of error corrections, decoherence must be suppressed below an acceptable level to realize a useful quantum operation. Analogous to refocusing techniques in nuclear magnetic resonance (NMR) studies, the dynamical decoupling (DD) approach to decoherence suppression has attracted tremendous interest. The central idea of DD is to use a control pulse sequence to effectively decouple a quantum system from its environment.

During the past years several DD pulse sequences have been proposed. The so-called “bang-bang” control has proved to be very useful BangBang1 (); BangBang2 (); exp2 () with a variety of extensions. However, it is not optimized for a given period of coherence preservation. The Carr-Purcell-Meiboom-Gill (CPMG) sequence from the NMR context can suppress decoherence up to UDDexact (). In an approach called “concatenated dynamical decoupling” CDD1 (); CDD2 (), the decoherence can be suppressed to the order of with pulses. Remarkably, in considering a single qubit subject to decoherence without population relaxation, Uhrig’s (optimal) dynamical decoupling (UDD) pulse sequence proposed in 2007 can suppress decoherence up to with only N pulses UDDexact (); UDDprl (); key-12 (). In a UDD sequence, the th control pulse is applied at the time

 Tj =Tsin2(jπ2N+2), j=1,2⋯,N. (1)

In most cases UDD outperforms all other known DD control sequences, a fact already confirmed in two beautiful experiments UDDvsCPMG (); detailpra (); Du-Liu (). As a dramatic development in theory, Yang and Liu proved that UDD is universal for suppressing single-qubit decoherence univUDD (). That is, for a single qubit coupled with an arbitrary bath, UDD works regardless of how the qubit is coupled to its bath.

Given the universality of UDD for suppression of single-qubit decoherence, it becomes urgent to examine whether UDD is useful for preserving quantum coherence of two-qubit states. This extension is necessary and important because many quantum operations involve at least two qubits. Conceptually there is also a big difference between single-qubit coherence and two-qubit coherence: preserving the latter often means the storage of quantum entanglement. Furthermore, because quantum entanglement is a nonlocal property and cannot be affected by local operations, preserving quantum entanglement between two qubits by a control pulse sequence will require the use of nonlocal control Hamiltonians.

In this work, by exploiting a central result in Yang and Liu’s universality proof univUDD () for UDD in single-qubit systems and by adopting a generalized coherence measure for two-qubit states, we show that UDD pulse sequence does apply to two-qubit systems, at least for preserving one pre-determined type of quantum coherence. The associated control Hamiltonian is also explicitly constructed. This significant extension from single-qubit to two-qubit systems opens up an exciting avenue of dynamical protection of quantum entanglement. Indeed, it is now possible to efficiently lock a two-qubit system on a desired entangled state, without any knowledge of the bath. Encouraged by our results for two-qubit systems, we then show that in general, the coherence of an arbitrary -level quantum system, which is characterized by our generalized coherence measure, can also be preserved by UDD to the order of with only pulses, irrespective of how this system is coupled with its environment. Hence, in principle, an arbitrary (but known) quantum state of an -qubit system with levels can be locked by UDD, provided that the required control Hamiltonian can be implemented experimentally. To establish an interesting connection with a kicked multi-level system recently realized in a cold-atom laboratory Jessen (), we also explicitly construct the UDD control Hamiltonian for decoherence suppression in three-level quantum systems.

This paper is organized as follows. In Sec. II, we first briefly outline an important result proved by Yang and Liu univUDD (); we then present our theory for UDD in two-qubit systems, followed by an extension to multi-level quantum systems. In Sec. III, we present supporting results from some simple numerical experiments. Section IV discusses the implications of our results and then concludes this paper.

## Ii UDD Theory for two-qubit and general multi-level systems

### ii.1 On Yang-Liu’s Universality Proof for Single-Qubit Systems

For our later use we first briefly describe one central result in Yang and Liu’s work univUDD () for proving the universality of the UDD control sequence applied to single-qubit systems. Let and be two time-independent Hermitian operators. Define two unitary operator as follows:

 U(N)±(T) = e−i[C±(−1)NZ](T−TN) (2) × e−i[C±(−1)(N−1)Z](TN−TN−1)⋯ × e−i[C∓Z](T2−T1)e−i[C±Z]T1.

Yang and Liu proved that for satisfying Eq. (1), we must have

 (U(N)−)†U(N)+=1+O(TN+1), (3)

i.e., the product of and differs from unity only by the order of for sufficiently small . In the interaction representation,

 ZI(t) ≡ eiCtZe−iCt (4) = ∞∑p=0(it)pp![C,[C,...[C,Z]]]p folds,

hence the above expression for can be rewritten in the following compact form

 U(N)±(T)=e−iCTJ[e−i∫T0±FN(t)ZI(t)dt], (5)

where is the final time, denotes the time-ordering operator, and

 FN(t)=(−1)j, for  t∈(Tj,Tj+1). (6)

As an important observation, we note that though Ref. univUDD () focused on single-qubit decoherence in a bath, Eq. (3) was proved therein for arbitrary Hermitian operators and . This motivated us to investigate under what conditions the unitary evolution operator of a controlled two-qubit system plus a bath can assume the same form as Eq. (2).

### ii.2 Decoherence Suppression in Two-qubit Systems

Quantum coherence is often characterized by the magnitude of the off-diagonal matrix elements of the system density operator after tracing over the bath. In single-qubit cases, the transverse polarization then measures the coherence and the longitudinal polarization measures the population difference. Such a perspective is often helpful so long as its representation-dependent nature is well understood. In two-qubit systems or general multi-level systems, the concept of quantum coherence becomes more ambiguous because there are many off-diagonal matrix elements of the system density operator. Clearly then, to have a general and convenient coherence measure will be important for extending decoherence suppression studies beyond single-qubit systems.

Here we define a generalized polarization operator to characterize a certain type of coherence. Specifically, associated with an arbitrary pure state of our quantum system, we define the following polarization operator,

 P|Ψ⟩≡2|Ψ⟩⟨Ψ|−I, (7)

where is the identity operator. This polarization operator has the following properties:

 P2|Ψ⟩ = I, P|Ψ⟩|Ψ⟩ = |Ψ⟩, P|Ψ⟩|Ψ⊥⟩ = −|Ψ⊥⟩, (8)

where represents all other possible states of the system that are orthogonal to . Hence, if the expectation value of is unity, then the system must be on the state . In this sense, the expectation value of measures how much coherence of the -type is contained in a given system. For example, in the single-qubit case, measures the longitudinal coherence if is chosen as the spin-up state, but measures the transverse coherence along a certain direction if is chosen as a superposition of spin-up and spin-down states. Most important of all, as seen in the following, the generalized polarization operator can directly give the required control Hamiltonian in order to preserve the quantum coherence thus defined.

We now consider a two-qubit system interacting with an arbitrary bath whose self-Hamiltonian is given by . The qubits interact with the environment via the interaction Hamiltonian for , where , , and are the standard Pauli matrices, and are bath operators. We further assume that the qubit-qubit interaction is given by , where the coefficients may also depend on arbitrary bath operators. A general total Hamiltonian describing a two-qubit system in a bath hence becomes

 H = HE+H1E+H2E+H12 (9) = c0+σ1xcx,1+σ1ycy,1+σ1zcz,1+σ2xcx,2 + σ2ycy,2+σ2zcz,2+σ1xσ2xcxx+σ1xσ2ycxy + σ1xσ2zcxz+σ1yσ2xcyx+σ1yσ2ycyy+σ1yσ2zcyz + σ1zσ2xczx+σ1zσ2yczy+σ1zσ2zczz.

For convenience each term in the above total Hamiltonian is assumed to be time independent (this assumption will be lifted in the end).

Focusing on the two-qubit subspace, the above total Hamiltonian is seen to consist of 16 linearly-independent terms that span a natural set of basis operators for all possible Hermitian operators acting on the two-qubit system. This set of basis operators can be summarized as

 {Xi}i=1,2,⋯,16={σk⊗σl}, (10)

where , with the orthogonality condition . But this choice of basis operators is rather arbitrary. We find that this operator basis set should be changed to new ones to facilitate operator manipulations. In the following we examine the suppression of two types of coherence, one is associated with non-entangled states and the other is associated with a Bell state.

#### ii.2.1 Preserving coherence associated with non-entangled states

Let the four basis states of a two-qubit system be , and . The projector associated with each of the four basis states is given by

 |0⟩⟨0| = P0=14(1+σ1z)(1+σ2z), |1⟩⟨1| = P1=14(1+σ1z)(1−σ2z), |2⟩⟨2| = P2=14(1−σ1z)(1+σ2z), |3⟩⟨3| = P3=14(1−σ1z)(1−σ2z). (11)

As a simple example, the quantum coherence to be protected here is assumed to be .

We now switch to the following new set of 16 basis operators,

 Y1 = P|0⟩=2P0−I = 12(−I+σ1z+σ2z+σ1zσ2z), Y2 = P0+P1=12(I+σ1z), Y3 = P0−P1+2P2=12(I−σ1z+2σ2z), Y4 = P0−P1−P2+3P3 = 12(I−σ1z−σ2z+3σ1zσ2z), Y5 = |1⟩⟨3|+|3⟩⟨1|=12(σ1x−σ1xσ2z), Y6 = −i|1⟩⟨3|+i|3⟩⟨1|=12(σ1y−σ1yσ2z), Y7 = |2⟩⟨3|+|3⟩⟨2|=12(σ2x−σ1zσ2x), Y8 = −i|2⟩⟨3|+i|3⟩⟨2|=12(σ2y−σ1zσ2y), Y9 = |1⟩⟨2|+|2⟩⟨1|=12(σ1xσ2x+σ1yσ2y), Y10 = −i|1⟩⟨2|+i|2⟩⟨1|=12(σ1yσ2x−σ1xσ2y), Y11 = |0⟩⟨1|+|1⟩⟨0|=12(σ2x+σ1zσ2x), Y12 = −i|0⟩⟨1|+i|1⟩⟨0|=12(σ2y+σ1zσ2y), Y13 = |0⟩⟨2|+|2⟩⟨0|=12(σ1x+σ1xσ2z), Y14 = −i|0⟩⟨2|+i|2⟩⟨0|=12(σ1y+σ1yσ2z), Y15 = |0⟩⟨3|+|3⟩⟨0|=12(σ1xσ2x−σ1yσ2y), Y16 = −i|0⟩⟨3|+i|3⟩⟨0|=12(σ1xσ2y+σ1yσ2x). (12)

Using this new set of basis operators for a two-qubit system, the total Hamiltonian becomes a linear combination of the operators defined above, i.e.,

 H=16∑j=1WjYj, (13)

where are the expansion coefficients that can contain arbitrary bath operators. The above new set of basis operators have the following properties. First, the operator in this set are identical with and hence also satisfies the interesting properties described by Eq. (II.2). Second,

 [Yj,Y1] = 0, for j=1,2,⋯,10; {Yj,Y1}+ = 0, for j=11,12,⋯,16, (14)

where represents the commutator and represents an anti-commutator. Third,

 [10∑i=1AiYi,16∑j=11BjYj] = 16∑j=11CjYj, (10∑i=1AiYi)(10∑j=1BjYj) = 10∑j=1CjYj, (16∑i=11AiYi)(16∑j=11BjYj) = 10∑j=1CjYj. (15)

With these observations, we next split the total uncontrolled Hamiltonian into two terms, i.e., , where

 H0=W1Y1+W2Y2+⋯+W10Y10, (16)

and

 H′=W11Y11+⋯+W16Y16. (17)

Evidently, we have the anti-commuting relation

 {Y1,H′}+=0, (18)

an important fact for our proof below.

Consider now the following control Hamiltonian describing a sequence of extended UDD -pulses

 Hc=N∑j=1πδ(t−Tj)Y12. (19)

After the control pulses, the unitary evolution operator for the whole system of the two qubits plus a bath is given by ( throughout)

 U(T) = e−i[H0+H′](T−TN)(−iY1) (20) × e−i[H0+H′](TN−TN−1)(−iY1) ⋯ × e−i[H0+H′](T3−T2)(−iY1) × e−i[H0+H′](T2−T1)(−iY1) ×  e−i[H0+H′]T1.

We can then take advantage of the anti-commuting relation of Eq. (18) to exchange the order between and the exponentials in the above equation, leading to

 U(T) = (−iY1)Ne−i[H0+(−1)NH′](T−TN) (21) × e−i[H0+(−1)N−1H′](TN−TN−1) ⋯ × e−i[H0+H′](T3−T2) × e−i[H0−H′](T2−T1) × e−i[H0+H′]T1 = (−iY1)Ne−iH0TJ[e−i∫T0FN(t)H′I(t)dt] ≡ (−iY1)NU(N)+(T).

Here is already defined in Eq. (6), the second equality is obtained by using the interaction representation, with , and the last line defines the operator . Clearly, is exactly in the form of defined in Eqs. (2) and (5), with replacing and replacing . This observation motivates us to define

 U(N)−(T)≡e−iH0TJ[e−i∫T0−FN(t)H′I(t)dt], (22)

which is completely in parallel with defined in Eq. (5). As such, Eq. (3) directly leads to

 (U(N)−)†U(N)+=1+O(TN+1). (23)

With Eq. (23) obtained we can now evaluate the coherence measure. In particular, for an arbitrary initial state given by the density operator , the expectation value of at time is given by

 Trace{U(T)ρiU†(T)P|0⟩} (24) = Trace{(−iY1)NU(N)+ρi(U(N)+)†(iY1)NP|0⟩} = Trace{(−iY1)NU(N)+ρiP|0⟩(U(N)−)†(iY1)N} = Trace{(U(N)−)†U(N)+ρiP|0⟩} = Trace{ρiP|0⟩}[1+O(TN+1)],

where we have used , , and the anti-commuting relation between and . Equation (24) clearly demonstrates that, as a result of the UDD sequence of pulses, the expectation value of is preserved to the order of , for an arbitrary initial state. If the initial state is set to be , i.e., , then the expectation value of remains to be at time , indicating that the UDD sequence has locked the system on the state .

In our proof of the UDD applicability in preserving the coherence associated with a non-entangled state, the first important step is to construct the control operator and then the control Hamiltonian . As is clear from Eq. (II.2), each application of the control operator leaves the state intact but induces a negative sign for all other two-qubit states. It is interesting to compare the control operator with what can be intuitively expected from early single-qubit UDD results. Suppose that the two qubits are unrelated at all, then in order to suppress the spin flipping of the first qubit (second qubit), we need a control operator (). As such, an intuitive single-qubit-based control Hamiltonian would be

 Hc,single=π2N∑j=1δ(t−Tj)(σ1z+σ2z). (25)

This intuitive control Hamiltonian differs from Eq. (19), hinting an importance difference between two-qubit and single-qubit cases. Indeed, here the qubit-qubit interaction or the system-environment coupling may directly cause a double-flipping error , which cannot be suppressed by . The second key step is to split the Hamiltonian into two parts and , with the former commuting with and the latter anti-commuting with . Once these two steps are achieved, the remaining part of our proof becomes straightforward by exploiting Eq. (23). These understandings suggest that it should be equally possible to preserve the coherence associated with entangled two-qubit states.

#### ii.2.2 Preserving coherence associated with entangled states

Consider a different coherence property as defined by our generalized polarization operator , with taken as a Bell state

 |~0⟩=1√2[|↑↓⟩+|↓↑⟩]. (26)

The other three orthogonal basis states for the two-qubit system are now denoted as , , . For example, they can be assumed to be , , and . To preserve such a new type of coherence, we follow our early procedure to first construct a control operator and then a new set of basis operators. In particular, we require

 ~Y1 = P|~0⟩=2|~0⟩⟨~0|−I (27) = 12(−I+σ1xσ2x+σ1yσ2y−σ1zσ2z).

We then construct other 9 basis operators that all commute with , e.g.,

 ~Y2 = 12(I+σ1xσ2x), ~Y3 = 12(I−σ1xσ2x+2σ1yσ2y), ~Y4 = 12(I−σ1xσ2x−σ1yσ2y−3σ1zσ2z), ~Y5 = 12(σ1zσ2x−σ1xσ2z), ~Y6 = 12(σ2y−σ1y), ~Y7 = 12(σ2x−σ1x), ~Y8 = −12(σ1yσ2z−σ1zσ2y), ~Y9 = 12(σ1z+σ2z), ~Y10 = −12(σ1xσ2y+σ1yσ2x). (28)

The remaining 6 linearly independent basis operators are found to be anti-commuting with . They can be written as

 ~Y11 = 12(σ1x+σ2x), ~Y12 = −12(σ1yσ2z+σ1zσ2y), ~Y13 = 12(σ1xσ2z+σ1zσ2x), ~Y14 = −12(σ1y+σ2y), ~Y15 = 12(σ1z−σ2z), ~Y16 = 12(σ1xσ2y−σ1yσ2x). (29)

The total Hamiltonian can now be rewritten as , in which

 ~H0=~W1~Y1+~W2~Y2+⋯+~W10~Y10 (30)

and

 ~H′=~W11~Y11+⋯+~W16~Y16. (31)

It is then evident that if we apply the following control Hamiltonian, i.e.,

 ~Hc = N∑j=1πδ(t−Tj)~Y12 (32) = N∑j=1π4δ(t−Tj)(−I+σ1xσ2x+σ1yσ2y−σ1zσ2z),

the time evolution operator of the controlled total system becomes entirely parallel to Eqs. (20) and (21) (with an arbitrary operator replaced by ). Hence, using the control pulse described by Eq. (32), the quantum coherence defined by the expectation value of can be preserved up to , for an arbitrary initial state. If the initial state is already the Bell state (i.e., coincides with the that defines our coherence measure ), then our UDD control sequence locks the system on this Bell state with a fidelity , no matter how the system is coupled to its environment.

The constant term in the control Hamiltonian can be dropped because it only induces an overall phase of the evolving state. All other terms in represent two-body and hence nonlocal control. This confirms our initial expectation that suppressing the decoherence of entangled two-qubit states is more involving than in single-qubit cases.

We have also considered the preservation of another Bell state . Following the same procedure outlined above, one finds that the required UDD control Hamiltonian should be given by

 ~Hc = −N∑j=1π4δ(t−Tj)(I+σ1xσ2x+σ1yσ2y+σ1zσ2z), (33)

which is a pulsed Heisenberg interaction Hamiltonian. Such an isotropic control Hamiltonian is consistent with the fact the singlet Bell state defining our quantum coherence measure is also isotropic.

### ii.3 UDD in M-level systems

Our early consideration for two-qubit systems suggests a general strategy for establishing UDD in an arbitrary -level system. Let be the orthogonal basis states for an -level system. Their associated projectors are defined as , with . Without loss of generality we consider the quantum coherence to be preserved is of the -type, as characterized by . As learned from Sec. II-B, the important control operator is then

 V1 = P|0⟩=2P0−I, (34)

with . A UDD sequence of this control operator can be achieved by the following control Hamiltonian

 ~Hc = N∑j=1πδ(t−Tj)V12. (35)

In the -dimensional Hilbert space, there are totally linearly independent Hermitian operators. We now divide the operators into two groups, one commutes with and the other anti-commutes with . Specifically, the following operators

 V2 = P0+P1, V3 = P0−P1+2P2, ⋯ VM = P0−P1−...−PM−2+(M−1)PM−1 (36)

evidently commutes with . In addition, other basis operators, denoted , also commute with . This is the case because we can construct the following basis operators

 |k⟩⟨l|+|l⟩⟨k| (37)

with and . The other basis operators that commute with are constructed as

 −i|k⟩⟨l|+i|l⟩⟨k|, (38)

also with and . All the remaining basis operators are found to anti-commute with . Specifically, they can be written as

 VM+(M−1)(M−2)+2l−1 = |0⟩⟨l|+|l⟩⟨0|; VM+(M−1)(M−2)+2l = −i|0⟩⟨l|+i|l⟩⟨0|, (39)

where .

The total Hamiltonian for an uncontrolled -level system interacting with a bath can now be written as

 HM = H0+H′, H0 = M2−2M+2∑j=1WjVj, H′ = M2∑j=M2−2M+3WjVj, (40)

where are the expansion coefficients that may contain arbitrary bath operators.

With the UDD control sequence described in Eq. (35) tuned on, the unitary evolution operator can be easily investigated using and . Indeed, it takes exactly the same form (with ) as in Eq. (21). We can then conclude that, the quantum coherence property associated with an arbitrarily pre-selected state in an -level system can be preserved with a fidelity , with only pulses. For an -qubit system, . In such a multi-qubit case, our result here indicates the following: if the initial state of an -qubit system is known, then by (i) setting the same as this initial state, and then (ii) setting as the control operator, the known initial state will be efficiently locked by UDD. Certainly, realizing the required control Hamiltonian for a multi-qubit system may be experimentally challenging.

Recently, a multi-level system subject to pulsed external fields is experimentally realized in a cold-atom laboratory Jessen (). To motivate possible experiments of UDD using an analogous setup, in the following we consider the case of in detail. To gain more insights into the control operator , here we use angular momentum operators in the subspace to express all the nine basis operators. Specifically, using the eigenstates of the operator as our representation, we have

 jx = 1√2⎛⎜⎝1111⎞⎟⎠, jy = 1√2⎛⎜⎝−ii−ii⎞⎟⎠, jz = ⎛⎜⎝10−1⎞⎟⎠. (41)

As an example, we use the state to define our coherence measure. The associated control operator is then found to be

 V1=jz+j2z−I. (42)

Interestingly, this control operator involves a nonlinear function of the angular momentum operator . This requirement can be experimentally fulfilled, because realizing such kind of operators in a pulsed fashion is one main achievement of Ref. Jessen (), where a “kicked-top” system is realized for the first time. The two different contexts, i.e., UDD by instantaneous pulses and the delta-kicked top model for understanding quantum-classical correspondence and quantum chaos Jessen (); haake (); Gongprl (), can thus be connected to each other.

For the sake of completeness, we also present below those operators that commute with , namely,

 V2 = I+12jz−12j2z, V3 = −I−12jz+52j2z, V4 = −1√2