Distance Bounds on Quantum Dynamics

PHYSICAL REVIEW A 78, 012308 ͑2008͒

Distance bounds on quantum dynamics

Daniel A. Lidar Departments of Chemistry, Electrical Engineering, and Physics, Center for Quantum Information Science & Technology, University of Southern California, Los Angeles, California 90089, USA

Paolo Zanardi Department of Physics, Center for Quantum Information Science & Technology, University of Southern California, Los Angeles, California 90089, USA

Kaveh Khodjasteh Department of Physics, Center for Quantum Information Science & Technology, University of Southern California, Los Angeles, California 90089, USA and Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire 03755, USA ͑Received 29 March 2008; published 7 July 2008͒ We derive rigorous upper bounds on the distance between quantum states in an open-system setting in terms of the operator norm between Hamiltonians describing their evolution. We illustrate our results with an example taken from protection against decoherence using dynamical decoupling.

DOI: 10.1103/PhysRevA.78.012308 PACS number͑s͒: 03.67.Lx, 03.67.Pp

I. INTRODUCTION extensively ͑Sec. II͒. We mention the trace norm and opera- When an open quantum system undergoes a dynamical tor norm, and a norm that mixes them via a duality between evolution generated by a Hamiltonian, how far does its state the Schrödinger and Heisenberg pictures. We then briefly evolve from itself compared to the magnitude of the Hamil- review the accepted distance measures between states so as tonian? This is a fundamental question that is central to to quantify the meaning of an expression such as ʈ␳ ͑ ͒ ␳0ʈ͑ ͒͑ ͒ quantum-information science ͓1͔ and quantum control ͓2,3͔. S t − S t Sec. III . Next, we discuss how to compute the In order to make it more precise, suppose that the unitary effective Hamiltonian ⍀͑t͒ using the Magnus expansion or propagator of the evolution, U͑t͒, which is related to the Thompson’s theorem, and introduce a generalized effective ˙ i superoperator generator ͑Sec. IV͒. Since in many applica- Hamiltonian H͑t͒ via the Schrödinger equation U=−ប HU,is written in terms of an effective Hamiltonian ⍀͑t͒ as U͑t͒ tions one is interested not in the distance between states gen- =exp͓−it⍀͑t͒/ប͔. The effective Hamiltonian can in turn by erated by unitary, closed-system evolution, but instead in the calculated from the Hamiltonian H͑t͒ using a Dyson or Mag- distance between states of systems undergoing nonunitary, nus expansion. If the system S is “open,” it is coupled to a open-system dynamics, we prove an upper bound on the dis- bath B and its time-evolved state is given via the partial trace tance between such system-only states in terms of the dis- ␳ ͑ ͒ ͑ ͒␳͑ ͒ ͑ ͒† ͓ ͔ ϩ ͑ ͒ operation by S t =TrB U t 0 U t 4,5 . In the context of tance between the full “system bath” states Sec. V .We quantum information, the question we have stated pertains to then come to our main result: an upper bound on the distance the problem of “quantum memory”: i.e., what is between system states in terms of the distance between ͑ef- ʈ␳ ͑ ͒ ␳ ͑ ͒ʈ ʈ⍀͑ ͒ʈ ʈ ͑ ͒ʈ ͒ ϩ S t − S 0 compared to t or H t ? When the issue fective Hamiltonians describing the system bath dynamics is quantum computation or general quantum control, one is ͑Sec. VI͒. We find the intuitive result that the distance is interested in comparing two time-evolved states: the “ideal” upper bounded by a function depending on the spectral di- ␳0͑ ͒ ameter of the difference between the effective Hamiltonians. state S t , which is error free and is described by an “ideal” effective Hamiltonian ⍀0͑t͒͑e.g., for a quantum algorithm͒, We present a discussion of our result in terms of an example ␳ ͑ ͒ and the “actual” state S t , which undergoes the full noisy borrowed from decoherence control using dynamical decou- dynamics described by the total effective Hamiltonian ⍀͑t͒. pling ͑Sec. VII͒. We conclude in Sec. VIII with some open Then the question becomes the following: what is questions. ʈ␳ ͑ ͒ ␳0͑ ͒ʈ ʈ⍀͑ ͒ ⍀0͑ ͒ʈ ʈ ͑ ͒ 0͑ ͒ʈ S t − S t compared to t − t or H t −H t ? The memory question can, of course, be seen as a special case of the computation question. II. UNITARILY INVARIANT NORMS Here we prove bounds that answer these fundamental questions. Our bounds have immediate applications to prob- Unitarily invariant norms are norms satisfying, for all uni- ͓ ͔ lems in decoherence control ͓6͔ and fault-tolerant quantum tary U,V 8 , computation ͓7͔, as they quantify the sense in which a dis- ͑ ͒ tance between effective Hamiltonians describing the evolu- ʈUAVʈ = ʈAʈ . ͑1͒ tion should be made small, in order to guarantee a small ui ui distance between a desired and actual state. To begin, we first recall the definition and key properties We list some important examples. of so-called unitarily invariant norms, as we use such norms ͑i͒ The trace norm:

͑ ͒ ʈ⌫ ⌳⌫†ʈ ʈ ͑⌳␳͒ † †ʈ ʈAʈ ϵ Tr͉A͉ = ͚ s ͑A͒, ͑2͒ i.e., V is unitary ; then, 1 2 ϱ,1=sup␳ V2V1 V1V2 1 1 i ʈ⌳␳ʈ ʈ⌳ʈ i =sup␳ 1 = ϱ,1. Therefore the OT norm is also submul- tiplicative. However, note that unlike the case of the standard ͉ ͉ϵͱ † ͑ ͒ ͑ where A A A and si A are the singular values eigenval- operator norm, there is no simple expression for ʈ⌳ʈϱ in ͉ ͉͒ ,1 ues of A . terms of the singular values of ⌳. ͑ ͒ V denotes the Hilbert-Schmidt dual ء⌳ ii The operator norm: Let be an inner product space On the other hand, if ʈ ʈϵͱ͚ ͉ ͉2͗ ͘ equipped with the Euclidean norm x i xi ei ,ei , of ⌳, i.e., Vۋ͚ ෈V V ͕ ͖ ⌳ V where x= ixiei and =Sp ei . Let : . The opera- X͔͒ ∀ ␳,X, ͑12͒͑ء⌳tor norm is Tr͓⌳͑␳͒X͔ =Tr͓␳ ʈ⌳xʈ then one can prove the following identity ͓see, e.g., Sec. 2.4 ʈ⌳ʈ ϵ ͑⌳͒ ͑ ͒ ϱ sup = max si . 3 of Ref. ͓10͔ or Eq. ͑9͒ of Ref. ͓11͔͔: x෈V ʈxʈ i 13͒͑ .͒ء⌳͑ X͒ʈϱ = maxs͑ء⌳ϱ: = sup ʈءʈ⌳ʈϱ = ʈ⌳ʈ Therefore ʈ⌳xʈՅʈ⌳ʈϱʈxʈ. In our applications V=L͑H͒ is the ,1 i ʈXʈϱ=1 i space of all linear operators on the Hilbert space H, ⌳ is a superoperator, and x=␳ is a normalized quantum state: For example, for completely positive maps ͓1,4,5͔ ⌳␳ ʈ␳ʈ ␳ ͚ ␳ † ͑ ͕ ͖ ͒ 1 =Tr =1. = iAi Ai where Ai are the Kraus operators , it is easily ⌳ ␳ ͚ †␳ ͑ ͒ ⌳ iAi Ai. The duality 12 between and = ء iii͒ The Frobenius, or Hilbert-Schmidt, norm: verified that͑ is the duality between the Heisenberg and Schrödinger ء⌳ ʈ ʈ ϵ ͱ † ͑ ͒2 ͑ ͒ A 2 Tr A A = ͱ͚ si A . 4 pictures. i All unitarily invariant norms satisfy the important property III. DISTANCE AND FIDELITY MEASURES of submultiplicativity ͓9͔: Various measures are known that quantify the notion of ʈABʈ Յ ʈAʈ ʈBʈ . ͑5͒ ui ui ui distance and fidelity between states. For example, the dis- ␳ ␳ The norms of interest to us are also multiplicative over ten- tance measure between quantum states 1 and 2 is the trace sor products and obey an ordering ͓9͔ distance ʈA Bʈ = ʈAʈ ʈBʈ , i =1,2,ϱ, 1 i i i D͑␳ ,␳ ͒ϵ ʈ␳ − ␳ ʈ . ͑14͒ 1 2 2 1 2 1 ʈAʈϱ Յ ʈAʈ Յ ʈAʈ , 2 1 The trace distance is the maximum probability of distinguishing ␳ from ␳ : namely, D͑␳ ,␳ ͒ ʈ ʈ Յ ʈ ʈ ʈ ʈ ʈ ʈ ʈ ʈ ͑ ͒ 1 2 1 2 AB ui A ϱ B ui, B ϱ A ui. 6 ͑͗ ͘ ͗ ͘ ͒ ͗ ͘ ␳ =max0ϽPϽI P 1 − P 2 , where P i =Tr iP and P is a pro- There is an interesting duality between the trace and operator jector, or more generally an element of a positive operator- norm ͓9͔: valued measure ͑POVM͓͒1͔. The fidelity between quantum states ␳ and ␳ is ʈ ʈ ͕͉ ͑ † ͉͒ ʈ ʈ Յ ͖ ͑ ͒ 1 2 A 1 = max Tr B A : B ϱ 1 , 7 ͑␳ ␳ ͒ϵʈͱ␳ ͱ␳ ʈ ͱͱ␳ ␳ ͱ␳ ͑ ͒ F 1, 2 1 2 1 = 1 2 1, 15 ʈ ʈ ͕͉ ͑ † ͉͒ ʈ ʈ Յ ͖ ͑ ͒ A ϱ = max Tr B A : B 1 1 , 8 ␳ ͉␺͗͘␺͉ ␳ ͉␾͗͘␾͉ which reduces for pure states 1 = and 2 = to F͉͑␺͘,͉␾͒͘=͉͗␺͉␾͉͘. The fidelity and distance bound each ͉ ͑ ͉͒ Յ ʈ ʈ ʈ †ʈ ʈ †ʈ ʈ ʈ ͑ ͒ Tr BA A ϱ B 1, B ϱ A 1. 9 other from above and below ͓12͔: In the last three inequalities A and B can map between spaces 1−D͑␳ ,␳ ͒ Յ F͑␳ ,␳ ͒ Յ ͱ1−D͑␳ ,␳ ͒2, ͑16͒ of different dimensions. If they map between spaces of the 1 2 1 2 1 2 same dimension, then so that knowing one bounds the other. Many other measures ͓ ͔ ʈ ʈ ͉ ͑ † ͉͒ ͑ ͒ exist and are useful in a variety of circumstances 1 . A 1 = max Tr B A . 10 B†B=I

We now deﬁne another norm, which we call the “operator- IV. GENERATORS OF THE DYNAMICS trace” norm ͑O-T norm͒: A. Effective superoperator generators ʈ⌳␳ʈ ʈ⌳ʈ ϵ 1 ʈ⌳␳ʈ ͑ ͒ We shall describe the evolution in terms of an effective ϱ,1 sup = sup 1, 11 ␳෈L͑H͒ ʈ␳ʈ ʈ␳ʈ 1 1=1 superoperator generator L͑t͒, such that ␳͑t͒ϵetL͑t͒␳͑0͒. ͑17͒ ۋ V and V is a normed space equipped with the ␳͑0͒ۋwhere ⌳:V trace norm. Note that if ⌳␳ is another normalized quantum ʈ⌳ʈ ʈ⌳␳ʈ Յʈ⌳ʈ The advantage of this general formulation is that it incorpo- state, then ϱ,1=1. Also, by deﬁnition, 1 ϱ,1. Moreover, it follows immediately from the unitary rates nonunitary evolution as well. Nevertheless, here we ␳͑ ͒ invariance of the trace norm that the OT norm is unitarily focus primarily on the case of unitary evolution t † † ˙ i invariant. Indeed, let ⌫·=V·V be a unitary superoperator =U͑t͒␳͑0͒U͑t͒ , with U=−ប HU, for which we have

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i The interaction picture propagator U˜ ͑t,0͒ can be formally L͑t͒ =− ͓⍀͑t͒,·͔. ͑18͒ ប expressed as −iA␳ iA t This follows immediately from the identity e e ˜ ͑ ͒ T͓ ͑ ͵ ˜ ͑ ͒ ͔͒ ͑ ͒ n U t,0 = exp − i H s ds 26 −i͓A,·͔␳ϵ͚ϱ ͑−i͒ ͓ ␳͔ =e n=0 n! nA, , satisﬁed for any Hermitian op- 0 ͓ ␳͔ erator A, where nA, denotes a nested commutator—i.e., ͓ ␳͔ ͓ ͓ ␳͔͔ ͓ ␳͔ϵ␳ nA, = A, n−1A, —with 0A, . ϵexp͓− it⍀˜ ͑t͔͒, ͑27͒

B. Magnus expansion where the second equality serves to define the effective interaction picture Hamiltonian ⍀˜ ͑t͒. In perturbation theory the effective Hamiltonian can be Lemma 1. There exist unitary operators ͕W͑s͖͒ such that evaluated most conveniently by using the Magnus expansion, which provides a unitary perturbation theory, in con- 1 t ⍀˜ ͑t͒ϵ ͵ W͑s͒H˜ ͑s͒W͑s͒†ds. ͑28͒ trast to the Dyson series ͓13͔. The Magnus expansion ex- t ⍀͑ ͒ ⍀͑ ͒ ͚ϱ ⍀ ͑ ͒ 0 presses t as an infinite series: t = n=1 n t , where ⍀ ͑ ͒ 1 ͐t ͑ ͒ This is remarkable since it shows that the time-ordering 1 t = t 0H t1 dt1 and the nth-order term involves an inte- gral over an nth-level nested commutator of H͑t͒ with itself problem can be converted into the problem of finding the at different times. A sufficient ͑but not necessary͒ condition ͑continuously parametrized͒ set of unitaries ͕W͑s͖͒. for absolute convergence of the Magnus series for ⍀͑t͒ in the Proof. The formal solution ͑26͒ can be written as an infi- interval ͓0,t͒ is ͓14͔ nite, ordered product N t t jt ͵ ͑ ͒ Ͻ ␲ ͑ ͒ U˜ ͑t,0͒ = lim ͟ expͫ− i H˜ͩ ͪͬ. ͑29͒ H s ϱds . 19 →ϱ 0 N j=0 N N Thompson’s theorem ͓15,16͔ states that for any pair of operators A and B, there exist unitary operators V and W, such C. Relating the effective Hamiltonian to the “real” A B VAV†+WBW† ͕ ͖N Hamiltonian that e e =e . It follows immediately that if Aj j=0 are Hermitian operators, then it is always possible to find There is also a way to relate the effective Hamiltonian to ͕ ͖N unitary operators Wj such that the real Hamiltonian in a nonperturbative manner. To this j=1 N N end we make use of a recently proven theorem due to Th- ͟ ͓ ͔ ͓ ͚ †͔ ͑ ͒ ompson ͓15,16͔. exp − iAj = exp − i WjAjWj . 30 j=0 j=0 Let us consider a general quantum evolution generated by a time-dependent Hamiltonian H, where V is considered a The proof is nonconstructive, i.e., the unitary operators ͑ ͕ ͖N ͑ ͒ perturbation to H0 in spite of this we will not treat V per- Wj j=1 are not known. Applying this to Eq. 29 yields turbatively͒: N t jt ˜ ͑ ͒ ͫ ͚ ˜ͩ ͪ †ͬ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ U t,0 = lim exp − i WjH W H t = H0 t + V t . 20 j N→ϱ N j=0 N The propagators satisfy t = exp͓− i͵ W͑s͒H˜ ͑s͒W͑s͒†ds͔, ͑31͒ dU͑t,0͒ =−iH͑t͒U͑t,0͒, ͑21͒ 0 dt which is the claimed result with the effective Hamiltonian ⍀˜ ͑ ͒ ͑ ͒ ᭿ dU ͑t,0͒ t defined as in Eq. 28 . 0 =−iH ͑t͒U ͑t,0͒. ͑22͒ An immediate corollary of Lemma 1 is the following: dt 0 0 Corollary 1. The effective Hamiltonian ⍀˜ ͑t͒ defined in We define the interaction picture propagator with respect to Eq. ͑28͒ satisfies, for any unitarily invariant norm, H , as usual, via 0 1 t ʈ⍀˜ ͑t͒ʈ Յ ͵ dsʈV͑s͒ʈ ϵ͗ʈVʈ ͑͘32͒ ˜ ͑ ͒ ͑ ͒† ͑ ͒ ͑ ͒ ui ui ui U t,0 = U0 t,0 U t,0 . 23 t 0

It satisﬁes the Schrödinger equation Յ ʈ ͑ ͒ʈ ͑ ͒ sup V s ui. 33 0ϽsϽt dU˜ ͑t,0͒ ˜ ͑ ͒˜ ͑ ͒ ͑ ͒ =−iH t U t,0 , 24 ͓ ͑ ͔͒ ˜ ͑ ͒ ͑ ͒† ͑ ͒ ͑ ͒ dt Proof. We have Eq. 25 H s =U0 t,0 V t U0 t,0 and ͓Eq. ͑28͔͒ ʈ⍀˜ ͑t͒ʈ =ʈ 1 ͐t W͑s͒H˜ ͑s͒W͑s͒†dsʈ . The result fol- with the interaction picture Hamiltonian ui t 0 ui lows from the triangle inequality. ᭿ H˜ ͑s͒ = U ͑t,0͒†V͑t͒U ͑t,0͒. ͑25͒ We have presented the bound in the interaction picture. 0 0 Clearly, the same argument applies to the Schrödinger pic- See Appendix A for a proof. ture, where instead one obtains

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t ͑ /ʈ ʈ ͒ʈ ʈ ͑ / ͒ 1 =dk, but dB IB ͑k͒ X ͑k͒ = d k k=d, which gives an in- ʈ⍀͑ ͒ʈ Յ ͵ ʈ ͑ ͒ ͑ ͒ʈ ͑ ͒ t ui ds H0 s + V s ui. 34 equality in the wrong direction. t 0

VI. DISTANCE BOUND V. DISTANCE BEFORE AND AFTER PARTIAL We are now ready to prove our main theorem, which pro- TRACE vides a bound on the distance between states in terms of the distance between effective superoperator generators. As an Since we are interested in the distances between states immediate corollary, we obtain the bound in terms of the undergoing open system dynamics, we now prove the fol- effective Hamiltonians. lowing. How much does the deviation between L͑t͒ and L0͑t͒ im- Lemma 2. H H Let S and B be finite-dimensional Hilbert pact the deviation between ␳͑t͒ and ␳0͑t͒? We shall assume ෈H H spaces of dimensions dS and dB, and let A S B. Then tL0͑t͒ for any unitarily invariant norm that is multiplicative over that the desired evolution is unitary—i.e., e i ⍀0͑ ͒ tensor products the partial trace satisfies the norm inequality =U0͑t͒·U0͑t͒†, where U0͑t͒=e− ប t t . Theorem 1. Consider two evolutions: the “desired” unitary d 0 tL0͑t͒ −͑i/ប͒t͓⍀0͑t͒,·͔ ␳ ͑t͒ϵe ␳͑0͒=e ␳͑0͒ۋʈtr Aʈ Յ B ʈAʈ, ͑35͒ evolution ␳͑0͒ B ʈ ʈ tL͑t͒ ␳͑t͒ϵe ␳͑0͒. LetۋIB and the “actual” evolution ␳͑0͒ ⌬ ͑ ͒ϵ ͑ ͒ 0͑ ͒ where I is the identity operator. L t L t −L t . Then This result was already known for the trace norm as a 1 ʈ⌬ ͑ ͒ʈ special case of the contractivity of trace-preserving quantum D͓␳͑t͒,␳0͑t͔͒ Յ minͫ1, ͑et L t ϱ,1 −1͒ͬ. 2 operations ͓1͔. ʈ⌬ ͑ ͒ʈ Յ Proof. Consider a unitary irreducible representation If in addition t L t ϱ,1 1, then ͕U ͑g͒,g෈G͖ of a compact group G on H . Then it follows B B ͓␳͑ ͒ ␳0͑ ͔͒ Յ ʈ⌬ ͑ ͒ʈ from Schur’s lemma that the partial trace has the following D t , t t L t ϱ,1. ͓ ͔ representation 17 : Note that, using the Heisenberg-Schrödinger duality ͑13͒,we ʈ⌬L͑t͒ʈ ʈ⌬L ͑t͒ʈ ⌬L is the dual ء ϱ, where ء can replace ϱ,1 by 1 ͑ ͒ ͵ ͓ ͑ ͔͒ ͓ ͑ ͒†͔ ␮͑ ͒ ⌬L trB X IB = IA UB g X IA UB g d g , generator to . In the case of Hamiltonian evolution we can dB G make this explicit as follows. ͑36͒ Corollary 2. For Hamiltonian evolution, where i 0 i 0 L͑t͒=−ប ͓⍀͑t͒,·͔ and L ͑t͒=−ប ͓⍀ ͑t͒,·͔, and defining where d␮͑g͒ denotes the left-invariant Haar measure normal- ⌬␭͑ ͒ϵ ͉␭ ͑ ͒ ␭ ͑ ͉͒ the spectral diameter t maxi,j i t − j t ized as ͐ d␮͑g͒=1 and d ϵdim͑H ͒. Then ␭ ͑ ͒ ␭ ͑ ͒ ͕␭ ͑ ͖͒ G B B =maxi i t −maxi i t , where i t are the eigenvalues of ⌬⍀͑ ͒ϵ⍀͑ ͒ ⍀0͑ ͒ 1 t t − t , ʈtr Xʈ = ʈtr ͑X͒ I ʈ B ʈI ʈ B B 1 B D͓␳͑t͒,␳0͑t͔͒ Յ minͫ1, ͑et ⌬␭͑t͒/ប −1͒ͬ ͑41͒ d 2 Յ B ͵ ʈ͓I U ͑g͔͒X͓I U ͑g͒†͔ʈd␮͑g͒ ʈI ʈ A B A B B G Յt ⌬␭͑t͒/ប if t ⌬␭͑t͒/បՅ1. ͑42͒ d d = B ͵ ʈXʈd␮͑g͒ = B ʈXʈ. ͑37͒ Corollary 2 means that the “spectral action” t ⌬␭͑t͒/ប plays ʈ ʈ ʈ ʈ IB G IB a key role in bounding the distance between states, a pleas- ᭿ ingly intuitive result. ʈ ʈ ʈ ʈ ͱ ʈ ʈ Proof of Theorem 1. Let us define Hermitian operators In particular, IB 1 =dB, IB 2 = dB, and IB ϱ =1, and since the trace, Frobenius, and operator norms are all multiplica- H0 ϵ itL0͑t͒, H ϵ itL͑t͒, ͑43͒ tive over tensor products, we have, specifically, and consider ͑superoperator͒ unitaries generated by these op- ʈ ʈ Յ ʈ ʈ ͑ ͒ trB X 1 X 1, 38 erators as a function of a new time parameter s ͑we shall hold t constant͒: ʈ ʈ Յ ͱ ʈ ʈ ͑ ͒ trB X 2 dB X 2, 39 U / H0U U/ HU ͑ ͒ d 0 ds =−i 0, d ds =−i . 44 ʈ ʈ Յ ʈ ʈ ͑ ͒ trB X ϱ dB X ϱ. 40 Then U ͑ ͒ −isH0 U͑ ͒ −isH ͑ ͒ Note that not all unitarily invariant norms are multiplicative 0 s = e , s = e , 45 ʈ ʈ over tensor products. For instance, the Ky-Fan k norm · ͑k͒ is the sum of the k largest singular values and is unitarily and we can define an interaction picture via invariant ͓8͔, but it is not multiplicative in this way. For U͑ ͒ U ͑ ͒S͑ ͒ ͑ ͒ Ն Ն ʈ ʈ ʈ ʈ s = 0 s s , 46 example, when dA =dB =d k 2 we have IA ͑k͒ = IB ͑k͒ ʈ ʈ ʈ ʈ ʈ ʈ = IA IB ͑k͒ =k.SoforX=IA IB we have trBX ͑k͒ =d IA ͑k͒ where the interaction picture “perturbation” is

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͑ ͒ 0͑ ͒ ⌬␭͑ ͒ ˜V͑s͒ϵU†͑s͒͑H − H0͒U ͑s͒ = itU†͑s͒⌬L͑t͒U ͑s͒. ͑47͒ H t and H t . Let t denote the spectral diameter of 0 0 0 0 ⍀͑t͒−⍀0͑t͒. Then the trace distance between the actual time- ␳ ͑ ͒ ͑ ͒␳͑ ͒ ͑ ͒† Then, using ͑1͒ unitary invariance and ͑2͒ the deﬁnition of evolved system state S t =trBU t 0 U t and the desired ʈʈ ͓ ͑ ͔͒ ␳0͑ ͒ 0͑ ͒␳͑ ͒ 0͑ ͒† the ϱ,1 norm Eq. 11 , one S t =trBU t 0 U t satisﬁes the bound

1 0 0 1 D͓␳͑t͒,␳0͑t͔͒ = ʈetL ͑t͒͑e−tL ͑t͒etL͑t͒ − I͒␳͑0͒ʈ D͓␳ ͑t͒,␳0͑t͔͒ Յ minͫ1, ͑et ⌬␭͑t͒/ប −1͒ͬ ͑53͒ 2 1 S S 2 ͑1͒ 1 ͑2͒ 1 = ʈ͑S͑1͒ − I͒␳͑0͒ʈ Յ ʈS͑1͒ − Iʈϱ , 1 2 1 2 ,1 Յminͫ1, ͑e͑2/ប͓͒͗ʈ⌬H͑t͒ʈϱ͔͘ −1͒ͬ, ͑54͒ 2 ͑48͒ ͗ʈ ʈ ͘ϵ 1 ͐t ʈ ͑ ͒ʈ ⌬ ͑ ͒ where X ϱ t 0ds X s ϱ and H t is the Hamiltonian which explains why we introduced S. We can compute S −͑i/ប͒t ⌬⍀͑t͒ generating U⌬͑t͒ϵe . using the Dyson series of time-dependent perturbation To obtain the second inequality we used ⌬␭͑t͒ theory: =ʈ͓⌬⍀͑t͒,·͔ʈϱ Յ2ʈ⌬⍀͑t͒ʈϱ and Corollary 1. Theorem 2 shows that to minimize the distance between dS/ds =−iV˜ S, the actual and desired evolution it is sufﬁcient to minimize

ϱ the spectral diameter of the actual and desired effective Hamiltonians, or the time average of the operator norm of S͑ ͒ I S ͑ ͒ s = + ͚ m s , the difference Hamiltonian ⌬H. Techniques for doing so m=1 which explicitly use the Magnus expansion or operator norms include dynamical decoupling ͓18,19͔ and quantum s s1 m−1 S ͑ ͒ ͵ ͵ ͵ ˜ ͑ ͒˜ ͑ ͒ ˜ ͑ ͒ error correction for non-Markovian noise ͓20–24͔. m s = ds1 ds2 dsmV s1 V s2 ¯ V sm . 0 0 0 B. Illustration using concatenated dynamical decoupling ͑49͒ As an illustration, consider the scenario of a single qubit Using submultiplicativity and the triangle inequality, we can coupled to a bath via a general system-bath Hamiltonian then show that ͑see Appendix B for the details͒ ͚ ␴ ͑ HSB=HS IB + ␣=x,y,z ␣ B␣ HS is the system-only Hamil- ʈ ⌬ ͑ ͒ ʈ tonian, ␴␣ are the Pauli matrices, and B␣ I are bath opera- ʈS͑ ͒ Iʈ Յ t L t s ϱ,1 ͑ ͒ B s − ϱ,1 e −1. 50 ͒ tors , with a bath Hamiltonian HB and controlled via a sys- ͑ ͒ Thus, finally, we have from Eq. ͑48͒ tem Hamiltonian HC t , so that the total Hamiltonian is ͑ ͒ ͑ ͒ H t =HC t IB +HSB+IS HB. Suppose that one wishes to 1 ʈt ⌬L͑t͒ʈ preserve the state of the qubit—i.e., we are interested in D͓␳͑t͒,␳0͑t͔͒ Յ ͑e ϱ,1 −1͒. ͑51͒ ⍀0͑ ͒ ͑ ͒ 2 “quantum memory”—so that t =HC t IB +IS HB.It ͓ ͔ ͑ ͒ ʈ⌬ ʈ Յ x was shown in Ref. 19 , Eq. 51 , that by using concatenated If additionally t L ϱ,1 1, then the inequality e −1 ͑ Յ͑ ͒ ͓␳͑ ͒ ␳0͑ ͔͒Յ ʈ⌬ ͑ ͒ʈ dynamical decoupling a recursively defined pulse sequence e−1 x yields D t , t t L t ϱ,1. On the other ͓25͔͒ and assuming zero-width pulses, the Magnus expansion ͓␳͑ ͒ ␳0͑ ͔͒ 1 ʈ͓S͑ ͒ I͔␳͑ ͒ʈ hand, note that D t , t = 2 1 − 0 1 yields the following upper bound: Յ 1 ͓ʈS͑1͒␳͑0͒ʈ +ʈ␳͑0͒ʈ ͔=1. ᭿ 2 1 1 / ʈ⍀͑ ͒ ⍀0͑ ͒ʈ /បՅ ͑␤ / 1 2͒log4 N ͑ ͒ Proof of Corollary 2. For Hamiltonian evolution we have T T − T ui JT T N . 55 ͑1͒ ͑2͒ Here ⍀͑t͒ is the effective Hamiltonian corresponding to the បʈ⌬ ͑ ͒ʈ ʈ͓⌬⍀͑ ͒ ͔ʈ ʈ͓⌬⍀͑ ͒ ͔ ʈ ʈ͓⌬⍀͑ ͒ ͔ʈ -ϱ = t ,· ϱ evolution generated by H͑t͒, T=N␶ ͑the duration of a con ء ·, L t ϱ,1 = t ,· ϱ,1 = t ͉␭ ͑ ͒ ␭ ͑ ͉͒ ϵ ⌬␭͑ ͒ ͑ ͒ catenated pulse sequence with pulse interval ␶͒, and = max i t − j t t , 52 i,j ␤ ϵʈ ʈ ϵ ʈ ʈ ͑ ͒ HB ui, J max B␣ ui 56 ͕␭ ͖ ⌬⍀ ␣ where i are the eigenvalues of . To obtain this result ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ we used a Eq. 13 and b the duality relation 12 , which are measures of the bath and system-bath coupling strength, ͓͑⌬⍀ ␳͔ ͒ ͑␳͓⌬⍀ ͔͒ ͓͑ ⌬⍀ ␳͔ ͒ ∀␳ X =Tr ,X =Tr − , X ,X, respectively. Here we shall take the norm in these last twoء , yields Tr ͓⌬⍀ ͔ ͓ ⌬⍀ ͔ ᭿ definitions as the operator norm. The bound ͑55͒ is valid as . ·, − = ء ·, and hence long as ␤TӶ1 ͓19͔. When this is the case, the right-hand VII. DISCUSSION side of Eq. ͑55͒ decays to zero superpolynomially in the ͑ ͒ A. General bound number of pulses, N. The bound 53 then yields, for suffi- ciently large N, By putting together Eq. ͑35͒ and Corollary 2, we can an- / swer the question we posed in the Introduction. 1 ͑␤ / 1 2͒log4 N D͓␳ ͑T͒,␳0͑T͔͒ Յ ͑e2JT T N −1͒ Theorem 2. Consider a quantum system S coupled to a S S 2 bath B, undergoing evolution under either the “actual” joint / Յ ͑␤ / 1 2͒log4 N ͑ ͒ unitary propagator U͑t͒=e−i/បt⍀͑t͒ or the “desired” joint uni- 2JT T N , 57 0 tary propagator U0͑t͒=e−i/បt⍀ ͑t͒, generated, respectively, by which shows that the distance between the actual and desired

012308-5 LIDAR, ZANARDI, AND KHODJASTEH PHYSICAL REVIEW A 78, 012308 ͑2008͒ state is maintained arbitrarily well. Distance bounds of this dU͑t,0͒ d͓U ͑t,0͒U˜ ͑t,0͔͒ type have also been used in dynamical decoupling applica- = 0 tions involving multiple qubits ͓26,27͔. dt dt dU ͑t,0͒ dU˜ ͑t,0͒ = 0 U˜ ͑t,0͒ + U ͑t,0͒ dt 0 dt VIII. CONCLUSIONS ͑ ͒ ͑ ͒˜ ͑ ͒ ͑ ͒˜ ͑ ͒˜ ͑ ͒ We have presented various bounds on the distance be- =−iH0 t U0 t,0 U t,0 − iU0 t,0 H t U t,0 tween states evolving quantum mechanically, either as closed =−iH ͑t͒U ͑t,0͒U˜ ͑t,0͒ or as open systems. These bounds are summarized in Theo- 0 0 ͑ ͒ ͑ ͒† ͑ ͒ ͑ ͒˜ ͑ ͒ rem 2. We expect our bounds to be useful in a variety of − iU0 t,0 U0 t,0 V t U0 t,0 U t,0 quantum computing or control applications. An undesirable ͓ ͑ ͒ ͑ ͔͒ ͑ ͒˜ ͑ ͒ aspect of Eqs. ͑53͒ is that the operator norm can diverge if =−i H0 t + V t U0 t,0 U t,0 the bath spectrum is unbounded, as is the case, e.g., for an =−iH͑t͒U͑t,0͒, oscillator bath. A brute force solution is the introduction of a high-energy cutoff. However, a more elegant solution is to which is the same differential equation as Eq. ͑21͒. The ini- note ͓28͔͑Lemma 8͒ that every system with energy con- tial conditions of the equations are also the same ͓ ͑ ͒ ͔ ͑ ͒ ͑ ͒ straints ͑such as a bound on the average energy͒ is essentially U 0,0 =I ; thus, Eqs. 23 – 25 describe the propagator ͑ ͒ supported on a ﬁnite-dimensional Hilbert space. An even generated by H t . more satisfactory solution in the unbounded spectrum case is APPENDIX B: DYSON EXPANSION BOUND to ﬁnd a distance bound involving correlation functions. This ͑ ͒ ͑ ͒ can be accomplished by performing a perturbative treatment We prove Eq. 50 . The proof makes use of 1 the tri- ͑ ͒ ͑ ͒ in the system-bath coupling parameter, as is done in the stan- angle inequality, 2 submultiplicativity, and 3 unitary in- dard derivation of quantum master equations ͓4,5͔. variance: ϱ ϱ ͑1͒ ʈS͑ ͒ Iʈ ʈ S ͑ ͒ʈ Յ ʈS ͑ ͒ʈ s − ϱ,1 = ͚ m s ϱ,1 ͚ m s ϱ,1 ACKNOWLEDGMENTS m=1 m=1 ϱ m s sm−1 We thank John Watrous for helpful remarks about uni- ʈ͵ ͵ ˜ ͑ ͒ʈ = ͚ ds1¯ dsm͟ V si ϱ,1 tarily invariant norms and Seth Lloyd for pointing out Ref. m=1 0 0 i=1 ͓28͔. The work of D.A.L. is supported under NSF Grants No. ͑ ͒ ϱ m CCF-0523675 and No. CCF-0726439. 1 s sm−1 Յ ͵ ͵ ʈ ˜ ͑ ͒ʈ ͚ ds1¯ dsm ͟ V si ϱ,1 m=1 0 0 i=1

͑ ͒ ϱ m APPENDIX A: INTERACTION PICTURE 2 s sm−1 Յ ͵ ͵ ʈ˜ ͑ ͒ʈ ͚ ds1¯ dsm͟ V si ϱ,1 We prove that U˜ ͑t,0͒ defined in Eq. ͑23͒ satisfies the m=1 0 0 i=1 ϱ Schrödinger equation ͑24͒ with the interaction picture ͑3͒ s sm−1 ͑ ͒ ͵ ͵ ʈ ⌬ ͑ ͒ʈm Hamiltonian defined in Eq. 25 . This requires proof since all = ͚ ds1¯ dsm t L t ϱ,1 the Hamiltonians considered are time dependent, whereas m=1 0 0 ϱ usually one only considers the perturbation to be time depen- m s ʈ ⌬ ͑ ͒ ʈ dent. To see this we differentiate both sides of Eq. ͑23͒, ʈ ⌬ ͑ ͒ʈm t L t s ϱ,1 = ͚ t L t ϱ,1 = e −1. while making use of Eqs. ͑24͒ and ͑25͒: m=1 m!

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