Measurement-induced resetting in open quantum systems
MMeasurement-induced resetting in open quantum systems
Andreu Riera-Campeny , Jan Ollé , and Axel Masó-Puigdellosas Física Teòrica: Informació i Fenòmens Quàntics. Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra,Spain Institut de Física d’Altes Energies (IFAE). The Barcelona Institute of Science and Technology (BIST). Universitat Autònomade Barcelona, 08193 Bellaterra, Spain Física Teòrica: Grup de Física Estadística. Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra,Spain
We put forward a novel approach tostudy the evolution of an arbitrary openquantum system under a resetting pro-cess. Using the framework of renewalequations, we find a universal behavior forthe mean return time that goes beyondunitary dynamics and Markovian measure-ments. Our results show a non-trivial be-havior of the mean switching times withthe mean measurement time τ , which per-mits tuning τ for minimizing the meantransition time between states. We com-plement our results with a numerical anal-ysis, which we benchmark against the cor-responding analytical study for low dimen-sional systems under unitary and open sys-tem dynamics. A reset is a process that brings a system in anygiven state to a normal condition or reset state.Resets might be beneficial to the user since theyallow for a safe fresh start under controlled con-ditions. This conceptually simple idea appears inmany areas of physics from information manage-ment protocols to the modelling of animal searchprocesses. For example, a computer can use a re-set to avoid crashing and a bird that recurrentlycomes back to its nest enhances the rate of findinghidden food.Classical stochastic processes with resets haveattracted a lot of attention lately. It has beenshown that the inclusion of resetting can be ben-eficial for many practical purposes. For instance,the first passage of a diffusive random walker un-der Markovian resetting has been shown to at-tain an optimum value for a finite reset rate [1]. This optimization capacity has been observed fordifferent types of walkers and resetting mecha-nisms [2–16], even when a time penalty is consid-ered after the resets [17–19], for enzymatic inhibi-tion [20, 21] or the completion of a Bernoulli trialprocess [22]. Also, general analyses of the com-pletion time of stochastic processes under resetshave been done in [23–26]. An in-depth reviewon stochastic resetting can be found in [27].In the quantum world, a Markovian masterequation describing the evolution of a system un-dergoing unitary dynamics with a fixed reset statewas introduced for the first time in [28]. There,the authors studied the prevalence of entangle-ment in open quantum systems. The same reset master equation has been used further as a modelfor open quantum dynamics [29, 30]. Recently,the spectral properties of the generator of the re-set master equation as well as the connection be-tween classical and quantum reset processes havebeen investigated [31]. Moreover, equivalent re-sults can also be found studying a Markovianresetting process added on top of the otherwiseunitary dynamics, without the explicit use of thereset master equation [32].Despite the remarkable progress done in theaforementioned works, it is not clear under whichcircumstances such resetting dynamics wouldarise. Here, we consider an inherent resetting pro-cess built already in the postulates of quantummechanics: the measurement-induced collapse ofthe wave-function. This collapse occurs when anexternal agent performs a measurement of thequantum state inducing its decoherence. Then,the post-measurement (or collapsed) state can beinterpreted as a reset state of the dynamics.It is worth mentioning that the type of dynam-ics we propose in this work resemble the arising a r X i v : . [ qu a n t - ph ] J a n n the so-called quantum walks [33–36]. There, aquantum system under unitary evolution is recur-rently measured, often stroboscopically, and theoverall dynamics are analyzed. Particularly, theprobability of reaching a target state [37, 38] andthe time needed to do so [39–41] are the studiedmagnitudes.Nevertheless, there are several open questionsthat require further research. For instance: (i) Isit possible to go beyond the stroboscopic and theMarkovian regime? (ii) Is it possible to go beyondthe unitary dynamics paradigm? (iii) Does reset-ting display any universal behavior? (iv) When isstochastic quantum resetting beneficial? In thiswork, we put forward a formalism based on re-newal equations in order to tackle questions (i)–(iv). We consider an N -dimensional, open quan-tum system at a given initial state ρ (0) = ρ , whose evolution is formally described bya completely-positive and trace-preserving map ρ ( t ) = E ( t )[ ρ ] , which we call for simplicity aquantum evolution E ( t ) . The most general formof a quantum evolution is given by the Kraus de-composition [42] E ( t )[ ρ ] = X k A k ( t ) ρ A k ( t ) † , (1) where A k are the Kraus operators and fulfill thecompleteness relation P k A k ( t ) † A k ( t ) = at alltimes t . The simplest case of quantum evolutioncorresponds to having a single Kraus operatorA ( t ) . In that case, the completeness relation im-plies that the evolution is unitary. From now on,we restrict ourselves to time-homogeneous evo-lutions E ( t + s ) = E ( t ) E ( s ) for the sake of thediscussion.Additionally, we consider that the evolution E ( t ) can be interrupted at stochastic times t = t , t , · · · in which a measurement M isperformed. At this point, the dynamics arerestarted, being the outcome of the measure-ment the new initial state. The stochastic times t = t , t , · · · are such that the differences ∆ t i = t k − t k − are sampled from a given probabilitydensity function ϕ ( t ) , the measurement time dis-tribution. The outputs m i for i = 1 , · · · , N of the measurement M are assumed to be non- ⇢ ( t = )
Acknowledgments
Acknowledgments. – We thank John Calsamiglia,Ramon Muñoz-Tapia, Vicenç Méndez and AlbertSanglas for enlightening discussions. We alsothank Eli Barkai for his insightful comments.ARC acknowledges financial support fromthe Spanish MINECO/ AEI FIS2016-80681-P, PID2019-107609GB-I00, from the CatalanGovernment: projects CIRIT 2017-SGR-1127,AGAUR FI-2018-B01134, and QuantumCAT001-P-001644 (RIS3CAT comunitats), co-financed by the European Regional DevelopmentFund (FEDER). JO acknowledges financialsupport from the grants FPA2017-88915-Pand SEV-2016-0588 from MINECO. IFAE ispartially funded by the CERCA program ofthe Generalitat de Catalunya. AMP acknowl-edges financial support from the Ministerio deEconomia y Competitividad through Grant No.CGL2016-78156-C2-2-R.All authors have contributed equally to thiswork.
A Derivation of Eqs. (7)–(8) in the main text
It is possible to write renewal equations for the survival probabilities starting from the different statesin the measurement basis as: Q ? ( t ) = ¯ ϕ ( t ) + X j = ? Z t dt ϕ ( t ) p ( j, t | ? ) Q j ( t − t ) dt , (21) Q i ( t ) = ¯ ϕ ( t ) + X j = ? Z t dt ϕ ( t ) p ( j, t | i ) Q j ( t − t ) dt , (22) where ¯ ϕ ( t ) = R ∞ t ϕ ( t ) dt is the survival probability of the measurement time distribution. Performingthe Laplace transform at both sides of the Eq. (21) and Eqs. (22) and using the convolution theorem,we get Q ? ( s ) = ˆ¯ ϕ ( s ) + X j = ? L [ ϕ ( t ) p ( j, t | ? )]( s ) Q j ( s ) , (23) Q i ( s ) = ˆ¯ ϕ ( s ) + X j = ? L [ ϕ ( t ) p ( j, t | i )]( s ) Q j ( s ) . (24) We note that Eqs. (24) are self-contained in the subspace orthogonal to | m ? i . Eq. (23) and Eq. (24) can be compactly written by introducing the ( N − -dimensional vector w ?,i ( s ) = L [ ϕ ( t ) p ( i, t | ? )]( s ) and the matrix ( N − × ( N − -dimensional W ij ( s ) = L [ ϕ ( t ) p ( i, t | j )]( s ) . Also, we take the survival robabilities Q i ( s ) as the components of a vector Q ( s ) . In this way, straightforward algebra leads tothe vector equations ˆ Q ? ( s ) = ˆ¯ ϕ ( s ) (cid:18) w T? (cid:16) − W T ( s ) (cid:17) − v (cid:19) , (25)ˆ Q ( s ) = ˆ¯ ϕ ( s )(1 − W T ( s )) − v , (26) where v = (1 , · · · , T .The mean first detection time T i is related to its associated survival probability Q i ( t ) through T i = ˆ Q i ( s = 0) . Therefore, the mean first detection time is found T ? = τ (cid:18) w T? (cid:16) − W T ( s = 0) (cid:17) − v (cid:19) , (27) T = τ (1 − W T ( s = 0)) − v , (28) where we have used τ = ˆ ϕ ( s = 0) . B Derivation of Eq. (9) in the main text
Let M be a non-degenerate quantum measurement defined by the set of projectors {| m i ih m i |} .On the one hand, the transition probabilities p ( i, t | j ) = h m i | E ( t )[ | m j ih m j | ] | m i i obviously fulfillthat P i p ( i, t | j ) = 1 for any evolution E ( t ) . On the other hand, summing over the conditions P j p ( i, t | j ) = h m i | E ( t )[ ] | m i i is in general different from one. However, if the evolution is unitalwe also find P j p ( i, t | j ) = 1 .Again, we consider the matrix W ij ( s ) = L [ ϕ ( t ) p ( i, t | j )]( s ) for i, j = ? and, similarly, the vector w ?,i ( s ) = L [ ϕ ( t ) p ( i, t | ? )]( s ) . For a unital quantum evolution E ( t ) , we find the relation X j W ij ( s ) = X j = ? L [ ϕ ( t )(1 − p ( i, t | ? ))]( s )= ˆ ϕ ( s ) − w ?,i ( s ) . (29) Equivalently, we can write down the matrix form of Eq. (29) as w T? ( s ) = v T (cid:16) ˆ ϕ ( s ) − W T ( s ) (cid:17) , (30) where v = (1 , · · · , is a vector of length N − . C Universal behavior under a quantum evolution E ( t ) with block structure In this appendix, we discuss in more detail the form of the universal behavior T ? = N τ when thequantum evolution E ( t ) is unital and has a block structure. We start considering a general quantumevolution E ( t ) and add an extra label µ = c, d to the measurement states | m i , µ i to denote whetherthey are dynamically connected or disconnected from the target state. More precisely, a disconnectedstate | m i , d i fulfills p ( i, t | ? ) = h m i , d | E ( t )[ | m ? , c ih m ? , c | ] | m i , d i = 0 ∀ t, (31) while the a connected state is a state that does not fulfill the above condition. Obviously, the targetstate is always connected to itself since at time t = 0 the quantum evolution has to be equal to theidentity map E (0) = I . In general, it can happen that for a disconnected state | m i , d i p ( ?, t | i ) = h m ? , c | E ( t )[ | m i , d ih m i , d | ] | m ? , d i 6 = 0 . (32) ven though Eq. (32) may seem unintuitive at first glance, there are well-known examples of it. Forinstance, if we measure a two level atom with spontaneous decay in its energy eigenbasis the groundstate is disconnected from the excited state while the converse is not true.If we now assume E ( t ) to be unital, the possibility stated by Eq. (32) is ruled out. This can be seenas the consequence of two facts:1. Any unital evolution can be written as a linear-affine combination (see [49]): E ( t ) = X l λ l U l ( t ) ρ U l ( t ) † . (33)
2. Any unitary U l ( t ) can be written as U l ( t ) = exp( − i A l ( t )) where A l ( t ) is Hermitian. Hence, | m i , d i is disconnected from | m ? , c i only if for all times t and positive integers n h m i , d | A nl ( t ) | m ? , c i = h m ? , c | A nl ( t ) | m i , d i = 0 , (34) which implies that A l ( t ) (and consequently U l ( t ) ) has a block structure.Then, denoting Π c and Π d the projectors Π µ = P i | m i , µ ih m i , µ | , the unital quantum evolution E ( t ) has a block structure Π ν E ( t )[Π µ ] ∝ δ µν . (35) Together with the unital property, Eq. (35) implies that the quantum evolution is unital block-wise,i.e. E ( t )[Π µ ] = Π µ . Therefore, one could repeat the same derivation of the main text, where only thesubset of connected states are considered. Then, one would arrive to T ? = N c τ. (36) where N c = tr [Π c ] is the number of connected states. Two final remarks are in order: (i) if thedisconnected space is zero-dimensional N c = N and we recover T ? = N τ , and (ii) the block structureof E ( t ) makes the derivation of Eq. (10) not valid since the matrix − W T (0) is not invertible in thatscenario. D The qubit ( N = 2 ): decay dynamics For a qubit under decay dynamics (i.e., under the Lindbladian in Eq. (18) of the main text), thecumbersome expressions for the mean switching time T − and the mean first return time T + for ϕ ( t ) = ϕ exp ( t ) yield T − = τ ( κτ + 1) csc (cid:16) θ (cid:17) (cid:0) ( κτ + 2) + (2 ωτ ) (cid:1) ( κτ + 1) ( κτ ( κτ + 2) + (2 ωτ ) ) − cos( θ ) ( κ ( κτ + 2) − (2 ωτ ) ) , (37) T + = τ (cid:16) θ (cid:17) (cid:0) cos( θ ) (cid:0) κτ ( κτ + 2) − (2 ωτ ) (cid:1) + ( κτ + 1) (cid:0) κτ ( κτ + 2) + (2 ωτ ) (cid:1)(cid:1) ( κτ + 1) ( κτ ( κτ + 2) + (2 ωτ ) ) − cos( θ ) ( κτ ( κτ + 2) − (2 ωτ ) ) , (38) which reduce to the expressions of the unitary case as κ/ω → . E The qutrit ( N = 3 ): unitary dynamics We consider a N = 3 system evolving under the unitary dynamics U ( t ) = e − iωtS x , where S x is arepresentation of the x component of the spin-1 operator. The measurement basis is chosen to be theeigenbasis of the S z operator, which we label by | + 1 i , | i , | − i according to their eigenvalues. There re nine mean times, which we will label as T in out . Whenever | in i = | out i we will have a mean firstreturn time, T ? , and in the other cases we will have mean switching times.Mean switching times are computed using Eq.(8) in the main text, while the remaining ones aresimply τ . We choose ϕ ( t ) = ϕ exp ( t ) to perform the calculation. We show one particular example indetail and then state the final results.Taking | m ? i = | + 1 i , the matrix W is W = L [ ϕ ( t ) p (0 , t | s ) L [ ϕ ( t ) p (0 , t |− )] ( s ) L [ ϕ ( t ) p ( − , t | s ) L [ ϕ ( t ) p ( − , t |− )] ( s ) ! , (39) and so the mean switching times are T T − + ! = τ " − L [ ϕ ( t ) p (0 , t | s = 0) L [ ϕ ( t ) p ( − , t | s = 0) L [ ϕ ( t ) p (0 , t |− )] ( s = 0) L [ ϕ ( t ) p ( − , t |− )] ( s = 0) ! − ! . (40) The transition probabilities are computed using p ( i, t | j ) = |h i | e − iωtS x | j i| , and so the elements insidethe matrix W ( s = 0) explicitly become W ij ( s = 0) = R ∞ dt exp( − t/τ ) |h i | e − iωtS x | j i| /τ .Carrying out the calculation in full yields the final result for | m ? i = | + i , which we find to be T = τ (cid:18)
72 + 2( ωτ ) (cid:19) (41) T − + = 3 τ (cid:18) ωτ ) (cid:19) . (42) In an analogous way, the results for | m ? i = | i are T +0 = τ (cid:18) ωτ ) (cid:19) (43) T − = T +0 . (44) Finally, for | m ? i = | − i we find T + − = T − + (45) T − = T . (46) The analytic three mean return times and the six mean switching times are compared with the resultsof the numerical simulation in Table 1. T in out T ++ T T − + T +0 T T − T + − T − T −− Theory 3 5.5 6 5 3 5 6 5.5 3Simulation 3.06 5.56 5.97 5.02 3.02 5.02 5.94 5.58 3.02
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