Aberystwyth University

Lindbladian purification Arenz, Christian; Burgarth, Daniel; Giovannetti, Vittorio; Nakazato, Hiromichi; Yuasa, Kazuya

Published in: Quantum Science and Technology DOI: 10.1088/2058-9565/aa6759 Publication date: 2017 Citation for published version (APA): Arenz, C., Burgarth, D., Giovannetti, V., Nakazato, H., & Yuasa, K. (2017). Lindbladian purification. Quantum Science and Technology, 2(2), [024001]. https://doi.org/10.1088/2058-9565/aa6759

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Lindbladian purification

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1 2 Lindbladian purification 3 1, 2 1 3 4 4 4 Christian Arenz, Daniel Burgarth, Vittorio Giovannetti, Hiromichi Nakazato, and Kazuya Yuasa 5 1Institute of Mathematics, Physics, and Computer Science, 6 Aberystwyth University, Aberystwyth SY23 2BZ, UK 2Frick Laboratory, Princeton University, Princeton NJ 08544, US 7 3 8 NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56126 Pisa, Italy 4Department of Physics, Waseda University, Tokyo 169-8555, Japan 9 (Dated: March 17, 2017) 10 11 In a recent work [D. K. Burgarth et al., Nat. Commun. 5, 5173 (2014)] it was shown that a series of 12 frequent measurements can project the dynamics of a quantum system onto a subspace in which the 13 dynamics can be more complex. In this subspace even full controllability can be achieved, although 14 the controllability over the system before the projection is very poor since the control Hamiltonians commute with each other. We can also think of the opposite: any Hamiltonians of a quantum system, 15 which are in general noncommutative with each other, can be made commutative by embedding 16 them in an extended , and thus the dynamics in the extended space becomes trivial 17 and simple. This idea of making noncommutative Hamiltonians commutative is called “Hamiltonian 18 purification.” The original noncommutative Hamiltonians are recovered by projecting the system 19 back onto the original Hilbert space through frequent measurements. Here we generalize this idea 20 to open-system dynamics by presenting a simple construction to make Lindbladians, as well as 21 Hamiltonians, commutative on a larger space with an auxiliary system. We show that the original 22 dynamics can be recovered through frequently measuring the auxiliary system in a non-selective way. Moreover, we provide a universal pair of Lindbladians which describes an “accessible” open quantum 23 system for generic system sizes. This allows us to conclude that through a series of frequent non- 24 selective measurements a nonaccessible generally becomes accessible. This 25 sheds further light on the role of measurement backaction on the control of quantum systems. 26 27 28 I. INTRODUCTION possible unitary transformations on Q, or equivalently, 29 if the dynamical Lie algebra L(H) spans the whole op- 30 erator algebra of Q, this last property of H being also 31 Noncommutativity is one of the key features of quan- referred to as accessibility. 32 tum mechanics. The order in which operations and/or When it comes to open quantum systems, the charac- 33 measurements are performed influences the outcomes of terization of the reachable (realizable) operations, as well 34 an experiment. In particular in the Lie-theoretical ap- as the associated notion of controllability, becomes more 35 proach to quantum control theory [1] the noncommu- complicated since the allowed operations do not possess a 36 tativity plays an important role. The goal of quantum group structure and the notion of dynamical generators 37 control is to steer a quantum system to realize a desired is typically lost [5, 6]. A partial exception is provided 38 transformation on it by shaping classical time-dependent by the subset of Markov processes which are equipped 39 fields [2]. Here the noncommutativity of the generators with a semigroup structure and admit the notion of dy- 40 associated with the control fields influences the complex- namical generators, i.e., the super-operators of Gorini- 41 ity of the resulting dynamics. For instance, for two com- Kossakowski-Lindblad-Sudarshan (GKLS) form [7, 8] 42 muting Hamiltonians H1 and H2, which can be switched (Lindbladians in the following). Still, also in this case, 43 on and off by external control fields, the resulting uni- determining which dynamics can be activated by control- tary evolution is just equivalent to the one generated by ling a given collection L := , , . . . of Lindbladians 44 {L1 L2 } 45 a linear combination of H1 and H2. On the contrary, by is a difficult unsolved problem. One would be tempted to properly concatenating transformations induced by two tackle it by studying the Lie algebra L(L) generated by L 46 L(L) 47 noncommuuting Hamiltonians one can produce effective and the corresponding Lie group e . However, at vari- evolutions associated with generators which are linearly ance with the closed system scenario, linear combinations 48 independent of the original ones, enabling the system and commutators of elements of L will in general produce 49 to explore more “directions.” For a finite-dimensional super-operators which are no longer allowed dynamical 50 closed quantum system Q, the set of effective evolutions generators (e.g., they cannot be cast in the GKLS form), 51 that can be implemented in this way is formed by the or said it differently, eL(L) will include transformations 52 unitaries of the Lie group eL(H) generated by the dy- which are unphysical. Furthermore, even for the elements 53 namical Lie algebra L(H) associated to the set of con- of eL(L) which are physically allowed, it is in general not 54 trol Hamiltonians H := H1, H2, . . . , i.e., the real vector clear if it would be possible to implement them by simply 55 space spanned by all po{ssible linear}combinations of the playing with the control fields. In view of these facts for 56 elements of H and their iterated commutators [1, 3, 4]. open quantum systems one distinguishes the physical no- 57 Accordingly a close system Q characterized by a control tion of controllability, i.e., the ability of using L and the 58 set H is said to be fully controllable if eL(H) includes all classical fields which activate them to perform all physi- 59 60 Accepted Manuscript AUTHOR SUBMITTED MANUSCRIPT - QST-100121.R1 Page 2 of 7

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1 2 cally allowed quantum transformations, from the weaker commutative on an extended space by means of an auxil- 3 notion of accessibility, which in this case corresponds to iary system, which, through frequent non-selective mea- 4 have L(L) equal to whole Lie algebra generated by arbi- surements, yields the original noncommutative dynam- 5 trary Lindbladians. In the following, we will refer to this ics. Moreover, we present a universal pair of Lindbladi- 6 as the ‘GKLS algebra’ noting however that it contains ans that generate the full GKLS dynamical Lie algebra 7 many elements which are not of GKLS form. Differently for generic system sizes, the analysis providing us with 8 from the closed quantum system case, it is indeed possi- a short and elementary proof of the generic accessibility 9 ble that a control set L is accessible but not controllable. [7]. Applying hence the Lindbladian purification proce- Still studying the accessibility of a collection of Lind- dure to such a universal set, we then show that almost all 10 bladians is a well posed mathematical problem, which open systems become accessible, even though their gen- 11 can also shed light on the controllability issue, with ac- erators are commutative with each other, by performing 12 cessibility being a necessary condition for controllabil- frequent non-selective measurements on a part of the sys- 13 ity. Furthermore accessibility implies that the reachable tem. 14 set has non-zero volume and therefore has physical rele- This article is organized as follows. Along the lines 15 vance: the short time dynamics explores a high dimen- of [12] we begin in Sec. II by reviewing the definition 16 sional space and is therefore of high complexity. of Hamiltonian purification and presenting an explicit 17 It turns out that almost all control sets L are acces- construction for purifying an arbitrary number of Lind- 18 bladians and Hamiltonians. In Sec. III we consider the 19 sibile [7]. Analogously to the case of closed systems, the key ingredient of this result can be identified with accessibility of controlled master equations. Concluding 20 remarks are given in Sec. IV, and some details on the 21 the noncommutativity of the elements of L. But what about models where L includes only mutually commutat- derivation of the projected dynamics and the proof of 22 accessibility are provided in the Appendices. 23 ing Lindbladians? Is there a way to expand their algebra 24 L(L) to cover the full GKLS algebra? For close quantum 25 systems it has been observed that one can substantially change the dimension of the dynamical Lie algebra L(H) II. LINDBLADIAN PURIFICATION AND 26 NON-SELECTIVE ZENO MEASUREMENTS through frequently observing a part of the system [9], or 27 by tampering it with a strong dissipative process that ex- 28 hibits decoherence-free subspaces [10] (the gain being ex- To begin with we first review the definition of Hamil- 29 ponential in some cases). As a matter of fact, on the ba- tonian purification [12]. Suppose that we have n control 30 sis of the quantum Zeno effect [11], starting with a set of Hamiltonians, which are switched on and off to steer a d- 31 commuting control Hamiltonians H, noncommutativity dimensional quantum system Q. Let H = H1, . . . , Hn 32 { } can be enforced through frequently projecting out part of be the set of the control Hamiltonians acting on the 33 Hilbert space of Q, and H˜ = H˜ , . . . , H˜ be a cor- the system onto a subspace where accessibility and hence Hd { 1 n} 34 full controllability is achieved. Also it has been observed responding set of Hamiltonians acting on an extended 35 that the projection trick can be reversed: specifically, Hilbert space dE of dimension dE (> d), which includes 36 H ˜ starting from a set of noncommutative Hamiltonians H, d as a proper subspace. We call H a purifying set of H H 37 one can construct a new set H˜ formed by commutative if all the elements of H˜ commute with each other, 38 elements on an extended Hilbert space which under pro- 39 H˜iH˜j = H˜jH˜i, i, j 1, . . . , n , (1) jection reduces to the original one. This mechanism was ∀ ∈ { } 40 studied in great detail in [12], where, borrowing from the and they are related to those from H through 41 notion of purification of mixed quantum states [13], the 42 term Hamiltonian purification was introduced. Hj = P H˜jP, j 1, . . . , n , (2) 43 ∀ ∈ { } One may then ask whether a similar procedures can 44 with P being the projection onto . For a generic set H be applied to the algebra of a set L of Lindbladians, d 45 consisting of n linearly independeHnt Hamiltonians it can namely, if it is possible to enlarge L(L) by means of some 46 be shown [12] that there always exists an H˜ where the projection mechanisms and, on the contrary, if Lind- 47 (min) bladian purification is always achievable. In this arti- minimal dimension dE of the extended Hilbert space is 48 cle we address these issues showing that indeed any set bounded above by d(min) nd. For instance for the case 49 E ≤ L of Lindbladians can be “purified,” i.e., can be made with n = 2 Hamiltonians H1 and H2, Proposition 1 of 50 commutative with each other, by embedding them in Ref. [12] states that a purifying set can be constructed on 51 a larger space (note that the term “pure” was already d = d d with an auxiliary single qubit Hilbert 52 H E H ⊗ H A used in [14] for Markovian generators in a slightly dif- space dA , the purifications and the projector being 53 ferent way). To this end, we need to employ a different H ˜ 54 scheme from those for the Hamiltonian purification in- H1 = H1 112 + H2 σx, 55 ⊗ ⊗ troduced in [12], since the naive application of the lat- H˜2 = H2 112 + H1 σx, (3) 56 ter trivially violates some structural properties of GKLS ⊗ ⊗ 57 generators (more details in the following). Our construc- 112 + σz 58 tion allows one to make Lindbladians and Hamiltonians P = 11d , (4) 59 ⊗ 2 60 Accepted Manuscript Page 3 of 7 AUTHOR SUBMITTED MANUSCRIPT - QST-100121.R1

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1 ˜ 2 with σx, σz, and 112 the Pauli and the identity opera- identifying the Hamiltonians Hj and the Lindblad op- ˜ { } ˜ ˜ ˜ 3 tors of the auxiliary qubit, respectively. The mapping erators Lj,α of the purifying element j = j + j ˜ { } L K D Hj Hj can finally be realized through the quantum as 4 → 5 Zeno effect [11, 15] by frequently monitoring the extended ˜ 6 system via a von Neumann measurement which projects Hj = nHj j j , the system onto , i.e., ⊗ | 〉〈 | j 1, . . . , n , (10) 7 d � ∈ { } H L˜j,α = √n Lj,α j j , 8 � ⊗ | 〉〈 | iH˜j t/N N iHj t 9 lim (P e− P ) = e− P. (5) N with� j n being an orthonormal basis for . 10 →∞ {| 〉}j=1 Hn 11 The question arises if an analogous construction can be Obviously through such a construction the operators 12 extended to the case of Lindbladians. Specifically con- H˜ and L˜ commute with each other for differ- 13 sider a set L = , . . . , of n GKLS generators op- { j} { j,α} {L1 Ln} ent j, trivially ensuring the requirement (9). Regarding 14 erating on a target system Q, the analog of (5), we focus on non-selective projective 15 measurement [17, 18] operating on the auxiliary system, 16 j = j + j, j 1, . . . , n , (6) L K D ∈ { } i.e., the completely positive and trace preserving (CPTP) 17 mapping of the form 18 with j and j being the Hamiltonian and dissipator contriKbutions,Di.e., the super-operators 19 ( ) = P ( )P , (11) P · · · k · · · k 20 k j( ) = i[Hj, ], (7) � 21 K · · · − · · · 22 given in terms of a complete set of orthonormal projec- j( ) = [2Lj,α( )Lj†,α 23 D · · · · · · tion operators Pk corresponding to measurement out- α { } � Lj†,αLj,α( ) ( )Lj†,αLj,α], (8) comes and satisfying P P = δ P and P = 11. 24 − · · · − · · · k k′ kk′ k k k 25 Notice that if we perform (N + 1) of such non-selective L being the Lindblad operators acting on the Hilbert 26 j,α measurements at regular time intervals t/N�during the space d of Q. We ask whether if it is possible to asso- 27 H ˜ ˜ ˜ evolution driven by a Lindbladian , the system will ciate with L a purifying set L = 1, . . . , n formed by evolve according to the CPTP transfLormation 28 GKLS generators possibly acting{oLn an extLen}ded system, 29 ( ) t/N N which are mutually commuting, i.e., Φ L := ( eL ) 30 t,N P ◦ ◦ P N 31 ˜ ˜ ˜ ˜ 2 i j = j i, i, j 1, . . . , n , (9) = id + ( ) t + O t , (12) 32 L ◦ L L ◦ L ∀ ∈ { } P ◦ L ◦ P N N 2 ◦ P 33 � � �� from which one can recover the original elements via a which in the limit of N converges to 34 projective mapping that should mimics (5) (in the above → ∞ 35 expressions we used the symbol “ ” to indicate the com- ( ) ( ) ( )t Φt,L = lim Φt,LN = e P◦L◦P , (13) 36 ◦ ∞ N ◦ P position of super-operators). →∞ 37 A natural guess for identifying L˜ and the projective 38 where id is the identity map and where we used the idem- mapping would be to simply transporting the purification potent property = = 2 of (11). Equation (13) 39 schemes of Ref. [12] at super-operator level, or equiv- P P ◦ P P 40 can also be derived following a pertubative approach with alently, to represent the js as operators in Liouville 41 L a strong amplitude-damping channel inducing the pro- space [16] and then simply applying to them the Hamilto- jection [19–21]. In our construction Eq. (13) is the 42 nian purication scheme. This simple trick however does formal cPounterpart of the Zeno limit (5): it shows that 43 not work because, for instance, mapping as (3) will take alternating the dynamics induced by a GKLS generator 44 positive operators into non-positive one, hence spoiling with induces on the system an evolution which can 45 one fundamental property of GKLS generators. Another bLe effectPively described in terms of an effective dynami- 46 problem comes from the fact that for Markovian open cal generator described by the projected super-operator 47 systems described by Lindbladians, the quantum Zeno . It should be stressed that the latter is not in 48 effect, which as we have seen is responsible for the im- GPK◦LLS◦foPrm, i.e., it is not a Lindbladian. Indeed it acts as 49 plementation of the mapping H˜ H , does not take j → j a proper Lindbladian only within the subspace specified 50 place: a Markovian system can leak from one subspace ( )t by the super-projector , but the map e P◦L◦P itself is 51 specified by the projection operator belonging to a mea- not CPTP (an explicit ePxample of this fact is provided in 52 surement outcome even in the limit of infinitely frequent Appendix A). Still we are going to identify (13) with the 53 projective measurements. In spite of these issues how- mechanism that yields the original Lindbladians j L 54 ever a Lindbladian purification scheme can be obtained expressed in the form (6)–(8) from their purifiedLco∈un- 55 with the following simple construction: terparts ˜ of with (10). For this purpose we assume the 56 Lj projectors Pk in (11) to be of the form 57 A purifying set L˜ can always be constructed by introduc- 58 ing an auxiliary Hilbert space n of dimension n and Pk = 11 φk φk , (14) 59 H ⊗ | 〉〈 | 60 Accepted Manuscript AUTHOR SUBMITTED MANUSCRIPT - QST-100121.R1 Page 4 of 7

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1 where φ dA is an orthonormal basis for the auxiliary L := , , . . . , consisting of a drift (unmodu- 2 {| k〉}k=1 {L0 K1 Km} 3 Hilbert space which is chosen to be mutually unbiased lated) term dA 4 [22] against the orthonormal basis j j=1 used for the {| 〉} = + , (20) 5 purification (10). Then, as shown in Appendix B, one L0 K0 D can verify that under the transformation (13) a generic 6 that includes both the dissipative part and the Hamil- 7 density operator ρQ(0) for the original system, obtained D tonian contribution 0( ) = i[H0, ], and of the set by taking the trace over the auxiliary Hilbert space d , 8 H A of Hamiltonian contKrol g· e· n· erat−ors · · · 9 evolves according to 10 ρ (t) = e j tρ (0), (15) k( ) = i[Hk, ], k 1, . . . , m . (21) 11 Q L Q K · · · − · · · ∈ { } 12 recovering hence the original dynamics generated by the As already mentioned in the introduction, for a closed 13 quantum system, i.e., without the dissipative part , the unpurified Lindbladian j. D 14 As a simple exampleLof Lindbladian purification we algebra L(L) associated with L (i.e., the set of all real lin- 15 consider amplitude damping (AD) and pure dephasing ear combinations and iterated commutators of these ele- 16 (PD) in x direction of a single qubit. Within the Born- ments, drift term included) will fully characterize the set 17 of unitary operations that can be implemented through Markov approximation the corresponding Lindbladians m 18 shaping the control functions uk(t) . For an open AD and PD are typically used to describe the main { }k=1 19 nLoise sourcLes in two level systems [13, 23]. The Lindblad quantum system described by the (18), 20 instead, L(L) only characterizes the accessibility of the operators read LAD = σ, LPD = σ + σ† where σ = 0 1 21 | 〉〈 | system. For a detailed analysis of the general structure is the atomic lowering operator and we note that AD and 22 do not commute. According to (10), a puriLfying set of L(L) and simple examples, we refer to [7]. Here we fo- LPD cus instead on studying how the purification mechanism 23 ˜ , ˜ is obtained through the purified Lindblad AD PD can influence the dimension of L(L). In particular we 24 o{pLeratoLrs } 25 shall see how a set of commutative Lindbladians can be turned into a new set of noncommuutative Lindbladians 26 L˜AD = σ σσ†, (16) 27 ⊗ which grant accessibility to the full GKLS algebra via the ˜ LPD = (σ + σ†) σ†σ, (17) projection through frequent measurements. 28 ⊗ 29 To show this we start by showing that it is possible and a frequent non-selective measurement (11) with pro- 30 to identify a set L formed by just a pair of Lindbladians jectors P = 11 where = 1/√2( 0 1 ) whose algebra L(L) spans the full GKLS algebra. We 31 ± ⊗ |±〉〈±| |±〉 | 〉 ± | 〉 32 recovers the unpurified dynamics. We note here that therefore first prove that the pair Lindblad operators similar to the purified versions (16) 33 and (17) were introduced in [24] for bosonic systems. 0 = i adH0 + 1 2 , (22) 34 L − D| 〉〈 | 35 = i ad 1 1 (23) K − | 〉〈 | 36 III. ACCESSIBILITY 37 with 38 d 1 We now turn our attention to the question on how fre- − 39 H = j j + 1 + h.c., (24) quent non-selective measurements can enrich the algebra 0 | 〉〈 | 40 j=1 41 L(L) of a Markovian open quantum system described by � 42 a collection L of controlled generators. Specifically we where 43 shall focus on systems driven by master equations of the 44 form i ad ( ) = i[H, ], (25) − H · · · − · · · 45 ∂ ρ(t) = (t)ρ(t), (18) L( ) = 2L( )L† L†L( ) ( )L†L, (26) 46 ∂t L D · · · · · · − · · · − · · · 47 does the job, namely, every possible Lindbladian can be 48 where the super-operator (t) = (t) + is provided by L K D generated by linear combinations and iterated commu- 49 a constant dissipative part represented by Lindblad op- tators of (22) and (23). We only sketch the main steps 50 erators Lα, and by a time-dependent Hamiltonian term here, whereas the details can be found in Appendix C. 51 (t)( ) = i[H(t), ] with K · · · − · · · In the following we also use the notations 52 m 53 ( ) = 2B( )A A B( ) ( )A B, (27) H(t) = H + u (t)H , (19) A,B † † † 54 0 k k D · · · · · · − · · · − · · · AdU ( ) = U( )U †. (28) 55 k�=1 · · · · · · 56 m uk(t) k=1 being classical control fields that can be op- We first note that terms of the form i ad j j com- 57 { } − | 〉〈 | erated to switch on and off m control Hamiltonians mute with the dissipative part 1 2 and according to 58 m D| 〉〈 | Hk k=1. This corresponds to having a control set [25] we can generate every element in i adu(d) with 59 { } − 60 Accepted Manuscript Page 5 of 7 AUTHOR SUBMITTED MANUSCRIPT - QST-100121.R1

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1 2 u(d) the Lie algebra of d d hermitian matrices. Us- space. Through the projection by Zeno measurements ing Ad = exp( i ad ×) with U(d) being the unitary for semigroup dynamics the original possibly noncom- 3 U(d) − u(d) 4 group, we can get mutative dynamics can be recovered by frequently mea- 5 suring the auxiliary system in a non-selective way. The AdU 1 2 AdU = U 1 2 U (29) 6 ◦ D| 〉〈 | ◦ † D | 〉〈 | † purification of more general dynamical maps with non- 7 for any U U(d). Now we consider unitaries U that act Markovian dynamics is left for future studies. ∈ (j) Moreover, we have proven that the pair of Lindbladians 8 as U j = k ck k for j = 1, 2 and = 1, 2, 3, 4 . | 〉 ∈I | 〉 I { } (22) and (23) describes an accessible open quantum sys- 9 We numerically verified that, from U 1 2 U , thus cre- � D | 〉〈 | † tem for generic system sizes, which tells us that generally 10 ated together with i adH for all Hamiltonians H = − a nonaccessible open quantum system is turned into an 11 i,j hij i j having support on , all the operators 12 of th∈eIform| 〉〈 | I accessible one by frequent non-selective measurements. 13 � The model has also potential applications in simulating i j , k l + k l , i j , an arbitrary Markovian open system dynamics [26, 27] 14 D| 〉〈 | | 〉〈 | D| 〉〈 | | 〉〈 | i, j, k, l (30) 15 � ∈ I by steering it through control fields. i( i j , k l k l , i j ), 16 � D| 〉〈 | | 〉〈 | − D| 〉〈 | | 〉〈 | Clearly, the presented purification scheme also works for observables and density operators, although, except 17 can�be generated. Doing the same for different quartets 18 = i, j, k, l , we are able to provide linearly indepen- for the partial trace, an operational way that allows us to 19 dIent {operator}s (30) for all i, j, k, l 1, . . . , d . Since any recover the original observables and states is not known 20 Lindbladian can be written in the∈K{ossakows}ki form as a to us. Since the noncommutativity is a unique feature of 21 linear combination of those operators, it means that ev- , and in fact it was argued in [28, 29] 22 ery Lindbladian can be generated through iterated com- that the noncommutativity distinguishes between quan- 23 mutators and linear combinations of the pair of gener- tum and classical mechanics, it is tempting to say that every quantum system can be made classical by purifying 24 ators 0, in (22) and (23). Given that this specific {L K} 25 pair of Lindbladians is accessible, it then follows from the it to a larger space. However, we remark here that this is only the case in a dynamical sense, i.e. the dynamics 26 standard argument (see, e.g., [9]) that almost all pairs can be made commutative but the observable algebra as- 27 are. This was shown previously in a more abstract way sociated with the system still remains noncommutative. 28 by Kurniawan [7]. Now that we have found a pair L = , that 29 {L0 K} 30 describes an accessible quantum system in arbitrary di- 31 mensions, we can make them commutative using a two- ACKNOWLEDGMENTS 32 dimensional (dA = 2) auxiliary Hilbert space, i.e., we can 33 purify them to We like to thank John Gough for fruitful discussions. 34 ˜ 0 = 2i adH 2 2 + √2 1 2 2 2 , (31) This work was supported by the Top Global Univer- 35 L − 0⊗| 〉〈 | D| 〉〈 |⊗| 〉〈 | ˜ sity Project from the Ministry of Education, Culture, 36 = 2i ad 1 1 1 1 . (32) K − | 〉〈 |⊗| 〉〈 | Sports, Science and Technology (MEXT), Japan. DB ac- 37 Obviously on the extended Hilbert space the Lie alge- knowledges support from EPSRC grant EP/M01634X/1. 38 bra associated with the set L˜ = ˜ , ˜ is just two- KY was supported by the Grant-in-Aid for Scientific Re- 39 {L0 K} dimensional, dim L(L˜) = 2, and the system is not ac- search (C) (No. 26400406) from the Japan Society for the 40 cessible. If we perform frequent non-selective projective Promotion of Science (JSPS) and by the Waseda Uni- 41 measurements on the auxiliary system described by the versity Grant for Special Research Projects (No. 2016K- 42 superprojector (11) with P = 11 , where are 215). HN was supported by the Waseda University Grant 43 defined at the end of Sec. I±I, the ⊗or|i±gi〉n〈a±l|dynamic|s±i〉s re- for Special Research Project (No. 2016B-173). 44 covered as (15) and the system becomes accessible. The 45 existence of such a specific setup allows us to conclude [9] 46 that almost all open quantum systems become accessible Appendix A: Projected Lindbladians 47 by Zeno measurements. 48 49 As an example of the fact that the projected counter- 50 IV. CONCLUSIONS part of a Lindbladian does not generate proper 51 quanPtu◦mL◦dPynamics, consider fLor instance the case where 52 We have generalized the work [12] on Hamiltonian describes a qubit amplitude damping with fixed point L 53 purification by establishing a new and simple purifica- 0 0 (this is characterized by a null Hamiltonian term | 〉〈 | 54 tion scheme for Lindbladians, which is also applicable to H = 0 and a unique Lindblad operator Lα = 0 1 ) and | 〉〈 | 55 Hamiltonians. Given n Lindbladians, they can be made where the transformation is the dephasing map [13] P 56 commutative by adding an n-dimensional auxiliary sys- associated with the canonical qubit base, i.e., 57 tem to extend the Lindblad operators with hermitian pro- 58 jectors that form an orthonormal basis for the auxiliary ( ) = 0 0 ( ) 0 0 + 1 1 ( ) 1 1 . (A1) 59 P · · · | 〉〈 | · · · | 〉〈 | | 〉〈 | · · · | 〉〈 | 60 Accepted Manuscript AUTHOR SUBMITTED MANUSCRIPT - QST-100121.R1 Page 6 of 7

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1 Accordingly for an arbitrary ρ we have where is the super-operator 2 T 3 ( )t lim (e P◦L◦P )ρ = 0 0 , (A2) 1 t ( ) = 11 Tr [( )] ( ), (B10) 4 →∞ ◦ P | 〉〈 | A A T · · · dA · · · − P · · · 5 while on the contrary 6 ( )t with TrA[( )] indicating the partial trace over the aux- 7 lim e P◦L◦P ρ = 0 0 + (id )ρ · · · t | 〉〈 | − P iliary system, A and 11A being the identity operator on 8 →∞ = 0 0 + 0 0 ρ 1 1 + 1 1 ρ 0 0 , the associated Hilbert space. 9 | 〉〈 | | 〉〈 | | 〉〈 | | 〉〈 | | 〉〈 | In order to prove (15) let us now focus on the evolu- 10 (A3) ( j ) tion induced by CPTP map Φt,L in (13) associated with 11 which in general is not a valid state. the jth element of L on a gene∞ric density matrix ρ(0) of 12 ( j ) 13 the joint system Q + A, i.e., ρ(t) = Φt,L ρ(0). We are interested in the dynamics of the reduced∞density matrix 14 Appendix B: Derivation of the Projected Dynamics 15 of Q, i.e., 16 We start by noticing that given a generic non-selective ( j ) ρQ(t) = TrA[ρ(t)] = TrA[Φt,L ρ(0)]. (B11) 17 transformation as in (11) and the unitary generator ∞ 18 P K with Hamiltonian H, the following identity holds By taking the first derivative with respect to t and using 19 (B9) we obtain 20 ( )( ) = i [H(k), P ( )P ], (B1) P ◦ K ◦ P · · · − k · · · k 21 k ∂ � ρQ(t) = TrA[( ˜j )ρ(t)] 22 ∂t (k) P ◦ L ◦ P 23 where H = PkHPk. Similarly, given a dissipator D = j(TrA[ ρ(t)]) + 2 Lj,α TrA[ ρ(t)]Lj†,α, 24 characterized by Lindblad operators Lα we have L P T α 25 � ( )( ) (B12) 26 P ◦ D ◦ P · · · (kk′) (kk′) (kk′) (kk′) 27 = 2L ( )L † L †L ( )P α · · · α − α α · · · k′ which finally yields the thesis 28 α,k,k′ � � (kk′) (kk′) Pk ( )L †L , 29 ′ α α ∂ j t − · · · ρQ(t) = jρQ(t) = ρQ(t) = eL ρQ(0), (B13) 30 (B�2) ∂t L ⇒ 31 32 (kk′) by noticing that with Lα = PkLαPk′ . Assume next H and Lα as those 33 associated with the Lindbladian ˜j with (10), and Pk as L TrA[ ρ(t)] = ρQ(t), TrA[ ρ(t)] = 0. (B14) 34 in (14). Since φk is mutually unbiased with respect P T 35 to j the foll{o|win〉}g identity holds, {| 〉} 36 Appendix C: An Accessible Pair of Lindbladians iϕjk 37 j φk = e− / dA, (B3) 38 〈 | 〉 with ϕ generic phases, and henc�e Here we show that the pair of Lindbladians 0, 39 jk {L K} 40 in (22) and (23) generates an accessible system. We use H˜ (k) = P H˜ P = H φ φ = H P , (B4) the notations (25)–(28). First of all, we show that = 41 j k j k j k k j k ⊗ | 〉〈 | i ad commutes with the dissipative part K of (kk′) i(ϕjk ϕ ) j j 1 2 42 L˜ = P L˜ P = e − jk′ L φ φ / d , − | 〉〈 | D| 〉〈 | j,α k j,α k′ j,α k k′ A in (22). Using an identity [30] 43 ⊗ | 〉〈 | L0 �(B5) 44 1 1 (kk′) (kk′) 45 L˜ †L˜ = L† L φ φ = L† L P /d . [ i adH , A] = A i[H,A] A+i[H,A], (C1) j,α j,α j,α j,α ⊗ | k′ 〉〈 k′ | j,α j,α k′ A − D 2D − − 2D 46 (B6) 47 we have Inserting these into (B1) and (B2) we then obtain 48 1 49 [ i ad j j , 1 2 ] = 1 2 i 1 2 δ +i 1 2 δ ( ˜ )( ) = ( )( ), (B7) − | 〉〈 | D| 〉〈 | 2D| 〉〈 |− | 〉〈 | j1 | 〉〈 | j2 50 P ◦ Kj ◦ P · · · Kj ◦ P · · · ˜ 1 51 ( j )( ) = ( j )( ) 1 2 +i 1 2 δ i 1 2 δ . P ◦ D ◦ P · · · D ◦ P · · · − 2D| 〉〈 | | 〉〈 | j1− | 〉〈 | j2 52 + 2 Lj,α ( )L† , (B8) (C2) 53 T · · · j,α α 54 � For j = 1, 2 it trivially vanishes, while for j = 1, 2 we 55 that is 1 ∕ 2 2 get 2 ( 1 i 1 i ) 1 2 = 0, where we have used 56 | ∓ 2| − | ± | D| 〉〈 | ( ˜ )( ) = ( )( ) + 2 L ( )L† , αA = α A. This commutativity implies that we 57 P ◦ Lj ◦ P · · · Lj ◦ P · · · j,αT · · · j,α D | | D α can generate every i adu(d) (see [25]) and thus every 58 � − (B9) AdU 1 2 AdU = U 1 2 U for any U U(d). 59 ◦ D| 〉〈 | ◦ † D | 〉〈 | † ∈ 60 Accepted Manuscript Page 7 of 7 AUTHOR SUBMITTED MANUSCRIPT - QST-100121.R1

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1 (j) 2 Taking unitaries U j = k ck k for j = 1, 2 and H = i,j hij i j on = 1, 2, 3, 4 . The same argu- | 〉 | 〉 ∈I | 〉〈 | I { } = 1, 2, 3, 4 , we have ∈I ment applies to any quartets = i, j, k, l , and all the 3 I { } � opera�tors of the form (30) forIall {are ava}ilable. Then, 4 I 5 every Lindbladian can be given as a linear combination 6 (1) (2) (1) (2) of those operators, i.e., U 1 2 U † = (ci )∗cj ck (cl )∗ i j , k l . 7 D | 〉〈 | D| 〉〈 | | 〉〈 | i,j,k,l + �∈I = [cijkl( i j , k l + k l , i j ) 8 (C3) L D| 〉〈 | | 〉〈 | D| 〉〈 | | 〉〈 | 9 i,�j,k,l We numerically verified that all the operators of the form + ci−jkli( i j , k l k l , i j )] (C4) 10 (30) for = 1, 2, 3, 4 can be obtained by linear com- D| 〉〈 | | 〉〈 | − D| 〉〈 | | 〉〈 | 11 I { } binations of U 1 2 U † and i adH with different U and with some coefficients ci±jkl. 12 D | 〉〈 | − 13 14 15 16 [1] G. Dirr and U. Helmke, Lie Theory for Quantum Control, [14] G. 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