Quantum speed limits in open system dynamics
Bounds to the speed of evolution of a quantum system are of fundamental interest in quantum metrology, quantum chemical dynamics and quantum computation. We derive a time-energy uncertainty relation for open quantum systems undergoing a general, completely positive and trace preserving (CPT) evolution which provides a bound to the quantum speed limit. When the evolution is of the Lindblad form, the bound is analogous to the Mandelstam-Tamm relation which applies in the unitary case, with the role of the Hamiltonian being played by the adjoint of the generator of the dynamical semigroup. The utility of the new bound is exemplified in different scenarios, ranging from the estimation of the passage time to the determination of precision limits for quantum metrology in the presence of dephasing noise.
pacs:03.65.-w, 03.65.Yz, 03.67.Lx
How fast can a quantum system evolve? Quantum mechanics acts as a legislative body imposing speed limits to the evolution of quantum systems. While these limits are both ultimate and fundamental, at the same time, their existence is at the center of a surge of activity, as a result of their manifold applications, including the identification of precision bounds in quantum metrology GLM11 (), the formulation of computational limits of physical systems Lloyd00 (), and the development of quantum optimal control algorithms Caneva09 ().
Bounds on the speed of evolution are intimately related to the concept of the passage time , which is the required time for a given pure state to become orthogonal to itself under unitary dynamics Schulman08 (). One of the early answers to this problem was provided by Mandelstam and Tamm (MT), who showed that the passage time can be lower-bounded by the inverse of the variance in the energy of the system so that
where , whenever the dynamics under study is governed by an Hermitian Hamiltonian MT45 (); Fleming73 (); AA90 (); Vaidman92 (); Uhlmann92 (); Uffink93 (); DL11 (); kok (); Zwierz12 (). A simple geometric interpretation of this result was provided by Brody using the Fubini-Study metric in the Hilbert space spanned by the initial state and its orthogonal complement Brody03 (). Indeed, the passage time problem can be posed as a quantum brachistochrone problem. From this perspective, a particularly exciting result was found: whenever the Hamiltonian is non-Hermitian PT-symmetric, the passage time can be made arbitrarily small without violating the time-energy uncertainty principle Bender07 (); Mostafazadeh07 (). A second bound, due to Margolus and Levitin (ML), takes the simpler form where the zero of energy is generally shifted to the ground state energy so that ML98 (). This bound has been applied to ascertain fundamental computational limits in nature Lloyd00 (); LT09 ().
Despite the growing body of literature on the subject, the analysis has almost exclusively been focused on unitary dynamics of isolated quantum systems. An analogous bound for open quantum systems is highly desirable, since ultimately all systems are coupled to an environment breuer (); rivas (). As an example, such a bound on the evolution of an open system would help to address the robustness of quantum simulators and computers against decoherence CZ12 (), as well as the relevance of the specific nature of the noise, and in particular whether or not it is Markovian, in phase estimation problems of interest in metrology and precision spectroscopy Huelga97 (); nmm ().
The MT bound can be derived by considering the time evolution of the overlap between the initial state and the quantum state at time subject to a unitary evolution . It can be shown that the MT-limit (eq. 1) is achievable, as for a suitable Hamiltonian we can satisfy the differential equation which for is easily seen to result in thus matching the MT bound vaidman ().
In the case of open system dynamics we need to consider general non-unitary quantum evolutions and have the freedom to choose a variety of distance measures between quantum states. One natural choice here is the fidelity between two mixed states and , which is given by . The quantum speed limit then provides a lower bound on the time that is required to achieve, for a given initial state and a target fidelity , the condition subject to an open system evolution. Ideally, such bounds should reduce to the MT bound in the case of unitary dynamics on pure states and/or be easy to compute.
Bounds on may be derived by taking inspiration from the variational characterization of the fidelity Jozsa (), where the maximization is over all () on a larger Hilbert space that are purifications of the mixed states () on the smaller system S, that is (). Then for any specific purification the inequality holds. A general time evolution of a subsystem can always be generated by a joint unitary dynamics of the system with an environment such that
where . Such a dynamics will be generated by a suitable Hamiltonian but it should be noted that the choice of , and thus is not unique.
Now we can make use of the fact that
for any choice of purification of and and any choice of unitary dynamics that generates on the subsystem. This implies that any choice of purification and unitary evolution will achieve at an earlier time than , i.e. . As a consequence, for any choice of , and thus we obtain a lower bound on . If is given, then we can compute in the state and immediately provide a lower bound on via the MT and . Needless to say, performing the optimization over all possible purifications and all possible is a challenging task that will be very hard to perform in the general case. Two routes are suggested themselves. Firstly, well chosen , and thus will lead to excellent bounds for reasonably simple cases. Secondly, analytical lower bounds on may also be obtained by studying different distance measures that are easier to handle and thus admit closed formulae for lower bounds.
Here we follow this second approach to find an analytical and easy to compute lower bound on the speed of evolution in open quantum systems. We shall derive a bound analogous to the seminal result by MT where the energy variance of the initial state is replaced by a more general measure taking into account the coupling to the environment. We shall pay particular attention to the dynamics governed by a dynamical semigroup in which case the evolution of the system is ruled by a master equation of the Lindblad form lindblad (). We shall show in the following that Markovian systems are subjected to a MT-type of bound where the adjoint of the generator of the dynamical semigroup plays the role of the system Hamiltonian in the unitary case.
Decay of an open quantum system
Consider a given system described by a state (from now we drop the upper index S for convenience) coupled to an environment in a state , and assume both system and environment are weakly coupled such that the initial global state can be approximated by . Let the global reversible dynamics be governed by a unitary evolution operator . The reduced dynamics of the system is given by a one-parameter family of dynamical maps , parameterized by the time variable . Whenever the typical time scale of the environment is much smaller than that of the system, one can assume a Markovian dynamics. Under Markovian dynamics, such maps form a quantum dynamical semigroup , (we assume that the open system is not subjected to an external time-dependent field so that the generator of the quantum dynamical semigroup is time independent). Any such map can be represented by a Markovian master equation
where the generator of admits the Lindblad form lindblad ()
such that . In such scenario we might pose the following question: Which is the bound to the speed of evolution from an initial state under the action of a quantum dynamical semigroup ? To answer this question we introduce as a figure of merit the so called relative purity LCW98 ()
which is a generalization of the survival probability often used for a pure state subject to a Hamiltonian , and that has proved useful in studying quantum speed limits in the unitary case LT09 ().
Derivation of the bound from the (Lindblad) master equation
Let us now characterize the decay rate of the relative purity. Note that whenever the generator admits a Lindblad form (i.e. for a Markovian quantum master equation),
where the adjoint of the generator of the dynamical map reads
The rate of change of can then be bounded using the Cauchy-Schwarz inequality for operators, . Then
that is, by making reference exclusively to the initial state and the dynamical map. Let us parametrize with . Upon integration between () and a final , the following bound to the required time of evolution is found
Here, provides an upper bound to the speed of evolution. This generalizes the MT uncertainty relation for open quantum systems governed by a Markovian quantum master equation. The generalization to a time-dependent Lindbladian is straightforward and reads
Derivation of the bound using general quantum channels
To remove the Markovian approximation, we note that any kind of time evolution of a quantum state can be written in the form . In particular, is independent of if the dynamical map is induced from an extended system with the initial condition . Then, the dynamical map is said to be universal. Let such map govern the evolution and consider
Parametrizing , a bound can be derived
where is the Hilbert-Schmidt norm of . Details of the derivation are provided in SM ().
The bound to the speed of evolution presented above is the main result of this paper. In the following we shall analyze some particular cases to illustrate its use, see too SM ().
Passage time. - Under unitary time evolution, the passage time is the minimum time required for a time evolving state to become orthogonal to its initial value . Let us consider a pure state such that and let . It follows from Eq. (9), that
Alternatively, for , , , a factor smaller. A similar reduction of the bound occurs for time-dependent Hamiltonians, in agreement with DL11 (). The usual definition of the passage time , refers to the orthogonalization measured by the fidelity, as stated above MT45 (); Fleming73 (); Vaidman92 (); Schulman08 ()
Non-Hermitian Hamiltonians. - Non-Hermitian Hamiltonians are ubiquitous in quantum physics and enjoy of a wide range of applications from quantum optics PlenioKnight () to reactive scattering Moiseyev (). Their standard derivation is based on Feshbach’s partitioning theory, that allows to describe the effective dynamics of a quantum system governed by a Hamiltonian , when restricted to a given subspace associated with projector (with complement , such that , , ). The effective Hamiltonian governing the dynamics in the restricted subspace, , is generally non-Hermitian. Under the density matrix . Similarly, in open systems under Markovian dynamics it is customary to split the generator of the dynamical map in two contributions and , i.e. . describes the coherent evolution associated with the non-Hermitian Hamiltonian , while the dissipator is associated with spontaneous decay, and it is a jump operator PlenioKnight (). More generally, let , where and are both Hermitian operators, so that . Noting that upon setting , the bound to the speed of evolution under non-Hermitian Hamiltonians still holds, it follows from Eq. (7) that
where the last line applies exclusively to pure states. Using Eq. (11) with , , one finds times the same expression.
From quantum speed limits to metrological bounds
The ultimate bound to parameter estimation is dictated by the ability to efficiently discriminate neighboring quantum states. In a seminal paper BC94 (), Braunstein and Caves (BC) derived a quantum Cramer-Rao bound for the uncertainty in the (local) estimation of a classical parameter of the form:
where denotes the quantum Fisher information and is the total number of repetitions of the experiment where a -dependence is linearly imprinted via a general evolution. When the dynamics is unitary, an initial preparation of a probe state in a cat (GHZ) state of subsystems allows to saturate the lower bound and achieve a Heisenberg-limited resolution where . If the subsystems are used independently, so that the input state is factorizable as product states, only the standard scaling dictated by the central limit theorem is achievable. This implies that the error bars in the actual estimation of a parameter could be reduced by by means of employing an entangled input probe provided that the system evolves unitarily. Whether or not the standard scaling can be surpassed when the system’s dynamics is open is a most relevant issue where only partial results are known. Motivated by experiments on precision spectroscopy, where a phase difference is estimated which is proportional to the detuning between an external oscillator and a selected atomic frequency, we will focus here on phase estimation problems under dephasing noise. Assuming decoherence to be Markovian and affecting each subsystem independently (local noise assumption), it was shown in Huelga97 () that this type of noise renders product and maximally entangled states metrologically equivalent, and argued that Markovian dephasing would restore the standard scaling with an optimal resolution to be achieved by a type of partially entangled states so that . Subsequent work proved this bound to be achievable asymptotically kitagawa () but only very recently it was proved in all generality that the bound is sharp and coincides with the one imposed by the maximization of the quantum Fisher information Davidovich11 (). The metrological equivalence of product and maximally entangled state preparations under Markovian decoherence can be predicted with the new bound eq. (8), which yields the ratio , where and are the optimal interrogation times when using maximally entangled and product state inputs, respectively. This can be easily shown by writing the dephasing master equation in the interaction picture as
and considering a pure state of the form . Then, and
This yields a minimal orthogonalization time . Repeating the same procedure for a maximally entangled input of the (GHZ) form with , we obtain an optimal interrogation time which leads to when the resolution is estimated operationally as , with denoting a projective population measurement, which is known to be optimal for this specific context. Alternatively, we can estimate the Fisher information in the form
where and is a population projective measurement. Note that this measurement procedure is optimal in this context. The resulting expressions for product and cat states are, respectively
The ratio therefore equals 1 when considering the optimal interrogations times as dictated by the bound eq. (8). Moreover, for pure states and the case of Markovian pure dephasing we have that (Note that this may be generalized to the mixed state case and any form of local noise as the locality implies that number of terms in grows linearly in the number of subsystems ). Then with eq. (8) and the fact that the Fisher information obeys smerzi1 (); smerzi2 (), the limit on the speed of evolution imposes the persistence of the standard scaling no matter how weak the dephasing rate. This is a result that is now firmly established Davidovich11 (); rafal () and that comes out in a rather natural fashion within this new framework.
So far we have exploited specifically the fact that the system’s dynamics is ruled by a Lindblad master equation. However, our general derivation considers a (linear) dynamical map that is trace preserving and completely positive (CPT) but not necessarily divisible markov (). As a result, the bound could be valid for non Markovian dynamics as long as they admit a representation in terms of a CP map cpt (). We have evaluated the prediction for the optimal interrogation times of product and cat states for a model of non-Markovian dephasing of this type, as proposed in sonia (), and obtained the ratio , just as in the Markov case. This seems to be in contradiction with recent results for models of non Markovian dephasing, which predict a ratio nmm () and raises an interesting conjecture with which we finish this section. There could exist forms of coloured noise for which the metrological equivalence between cats-products input probes still holds. This inequivalence in the achievable resolution of a phase estimation could then be exploited to quantitatively quantify non-Markovianity.
A bound to the speed of evolution under an open-system dynamics has been provided, generalising the classic result by Mandelstam and Tamm known for the unitary case. In the Markovian limit, we have shown that the adjoint of the generator of the dynamical semigroup plays the role of the commutator with the Hamiltonian in the MT bound. Despite the fact that the bound is not tight, in the sense of non coinciding with the unitary solution for closed systems, it allows to naturally predict the inaccessibility of the Heisenberg limit under Markovian noise. Moreover, when using the general form of the bound for universal channels, the new limit on the speed of evolution suggest the inequivalence of different forms of coloured noise for precision spectroscopy. Our results are applicable to a wide variety of scenarios including bounding decoherence rates Zurek91 (), and quantum speed limits in dissipative state preparation PlenioHuelga (), quantum computation and simulation assisted by dissipation frank ().
After the completion of this work, we learned about reference Taddei12 () devoted to quantum speed limits to the global unitary dynamics of a system emmbedded in an environment.
It is a pleasure to thank D. Alonso, D. J. Brody, B. Damski, M. Meister, A. Rivas, and A. Ruschhaupt for fruitful discussions and comments on the manuscript. This work was supported by the U.S. Department of Energy through the LANL/LDRD Program, the Basque Government (IT-559-10), the UPV/EHU UFI 11/55, the STREP PICC, the Alexander von Humboldt Foundation, the Integrated project QEssence and a LANL J. Robert Oppenheimer fellowship.
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Appendix A Derivation of the bound using quantum channels
Any kind of time evolution of a quantum state can be written in the form . In particular, is independent of if the dynamical map is induced from an extended system with the initial contidion . Then, the dynamical map is said to be universal. Let such map govern the evolution and consider
For compactness, let us denote . It follows that
Taking the absolute value at both sides,
Using the Cauchy-Schwarz inequality ,
We next use the fact that . Parametrizing , a bound can be derived
where and is the Hilbert-Schmidt norm of . In the second line, we have used .
Appendix B Decoherence in an isotropic environment
As an illustrative example, let us consider an exactly solvable model of open quantum dynamics. The simplest model of decoherence in an isotropic environment is given
which is governed by a unital, relaxing dynamical map. Consider the initial mixed state
with () and purity . Its time evolution reads
where , and asymptotically tends to
The relative purity evolves according to
The bound, written in the form of Eq. (7), reduces to
this is, which holds given that . Bound Eq. (8) in the text reads
while the exact passage time is given by
We note that so that for an initial pure state the maximum value of .
Appendix C Dynamics in the presence of gain and loss
Brody and Graefe have recently discussed the dynamics of a quantum system in the presence of gains and losses of energy or amplitude BG12 (). In particular, they considered the master equation of the form
In the case of a unitary dynamics for which , the speed of evolution, defined by , has been shown in Brody11 () to be given by the skew-information measure of Wigner and Yanase WY63 (). For they found that the speed of evolution, when is pure, to be given by the expression BG12 ()
where all expectation values are taken with respect to the time-dependent solution of (33). Since (33) is nonlinear in , the bounds derived in this paper is not necessarily applicable. However, it is interesting to note that for an initially pure state, Eq. (33) is tantamount to a dynamical equation . This form allows us to establish a direct connection with the bound Eq. (8) in the text, by noting that the velocity at is precisely given by
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