Wigner separability entropy and complexity of quantum dynamics
We propose the Wigner separability entropy as a measure of complexity of a quantum state. This quantity measures the number of terms that effectively contribute to the Schmidt decomposition of the Wigner function with respect to a chosen phase space decomposition. We prove that the Wigner separability entropy is equal to the operator space entanglement entropy, measuring entanglement in the space of operators, and, for pure states, to twice the entropy of entanglement. The quantum to classical correspondence between the Wigner separability entropy and the separability entropy of the classical phase space Liouville density is illustrated by means of numerical simulations of chaotic maps. In this way, the separability entropy emerges as an extremely broad complexity quantifier in both the classical and quantum realms.
pacs:05.45.Mt, 03.65.Sq, 05.45.Pq
Measuring complexity in a simple and unified way has been a major and long quest in both quantum and classical dynamics. While there exists a direct connection between chaos and algorithmic complexity of trajectories in classical physics ford (), the problem is particularly elusive for quantum mechanics alicki (), where the notion of trajectory is forbidden by the Heisenberg uncertainty principle and complexity can be attributed not only to the lack of integrability but also to the tensor-product structure of the Hilbert space, that is, to entanglement.
The phase space representation of quantum mechanics is a very convenient framework to investigate quantum complexity, in that one can compare classical and quantum dynamical evolutions of distributions in phase space. In this context, the number of Fourier harmonics of the Wigner function has already been used to characterize the complexity of a quantum state, both for single-particle harmonics () and many-body vinitha () quantum dynamics, in particular to detect quantum phase transitions. This complexity measure can be equally applied to classical and quantum mechanics, with the Liouville density used instead of the Wigner function in the classical case. However, this quantity has the disadvantage of being basis-dependent. Moreover, knowledge of the whole Fourier harmonics spectrum of the Wigner function seems in general not necessary to compute expectation values of physically relevant observables. For instance, even though the number of harmonics grows exponentially in time for both integrable and non-integrable quantum chaotic Ising chains vinitha (), the resources required to simulate both local and extensive observables grow exponentially in the chaotic case but only polynomially for the integrable model prosenznidaric ().
Recently, a new complexity indicator has been introduced for classical dynamics, the separability entropy prosen1 (), measuring the logarithm of the effective number of terms in the Schmidt (or singular value) decomposition of the Liouville density, with respect to an arbitrary phase space decomposition. This quantity estimates the minimal amount of computational resources required to simulate the classical Liouvillian evolution and grows linearly in time for dynamics that cannot be efficiently simulated. In this paper, we extend this notion of complexity to the quantum realm, by defining the Wigner separability entropy as the number of terms that effectively contribute to the Schmidt decomposition of the Wigner function. We prove that such quantity is equal to the operator space entanglement entropy prosenOSEE (), constructed from the Schmidt decomposition of the density operator in the space of Hilbert-Schmidt operators and quantifying the complexity of time-dependent density-matrix renormalization group simulations. Furthermore, for pure states the Wigner separability entropy is twice the entanglement entropy. We illustrate the quantum to classical correspondence for the separability entropy and its link to the entanglement entropy by means of numerical simulations of quantum chaotic maps.
The paper is organized as follows. In Sec. II we define the Wigner separability entropy and prove its relation to the operator space entanglement entropy and, for pure states, to the entanglement entropy. The dynamical evolution of is studied for quantum chaotic map models in Sec. III. We finish with concluding remarks in Sec. IV.
Ii Wigner separability entropy
Given a system described in a -dimensional compact phase space by the Wigner function (with the normalization constraint ) and an arbitrary phase space decomposition, , into two set of coordinates, , we can write the Schmidt (singular value) decomposition of the Wigner function:
with , and orthonormal bases for and , respectively, and the Schmidt coefficients (singular values) satisfying . We then define the Wigner separability entropy as
The coefficients satisfy and are the Schmidt coefficients of the singular value decomposition of . That is, is normalized in : . Note that for pure states
The Wigner separability entropy quantifies the logarithm of the number of terms that effectively contribute to decomposition (1) and therefore provides a measure of separability of the Wigner function with respect to the chosen phase space decomposition.
The main advantage of defining the separability entropy in phase space by means of the Wigner function is that such quantity can be directly translated to classical mechanics. The classical analog of the Wigner separability entropy is the s-entropy defined in Ref. prosen1 (), where the classical phase space distribution is used instead of the Wigner function .
It is interesting to establish a connection between the Wigner separability entropy and the operator space entanglement entropy prosenOSEE (), constructed from the Schmidt decomposition of the density operator acting on the Hilbert space . Since , the density operator is a Hilbert-Schmidt operator, that is, an operator with finite Hilbert-Schmidt norm . Therefore, can be interpreted as a vector belonging to the Hilbert space of Hilbert-Schmidt operators, with the Hilbert-Schmidt inner product . Therefore, given , the density operator has a Schmidt decomposition,
where and are orthonormal (, ) bases for and , respectively, and the Schmidt coefficients satisfying . The operator space entanglement entropy prosenOSEE () is then given by
In what follows, we prove that , that is, the operator space entanglement entropy equals the Wigner separability entropy. This result follows from the fact that the Weyl correspondence establishes an isomorphism between Hilbert-Schmidt operators and functions on classical phase space pool (); folland (). Since the density operator is the integral kernel of a unique linear one-to-one transformation mapping the operators , , i.e. , and vice versa for the inverse transformation, , and since the Weyl transform of the density operator is the Wigner function, it follows that and have the same Schmidt coefficients, and therefore
In order to explicitly illustrate the above result, we consider a density operator written in terms of a set of bosonic creation-annihilation operators, , , and define the Wigner function as
where and are -dimensional complex variables, the integration runs over the complex -planes for , is the displacement operator
and are the coherent states
with being an eigenstate of the annihilation operator , i.e., , and being the vacuum state . Using the singular value decomposition
Finally, we derive
Eq. (15) is precisely the singular value decomposition of the Wigner function (the orthogonality of the Schmidt vectors (16,17) can be checked straightforwardly) and has the same Schmidt coefficients as the the singular value decomposition (11) of the density operator, so we conclude that (7).
When the density operator describes a pure state, , there exists a simple relation between the Wigner separability entropy and the entanglement content of the state . The Schmidt decomposition of reads nielsen (); qcbook ()
with () orthonormal basis for (), , and . On the other hand, we can also write the Schmidt decomposition of the operator :
where and are the reduced density operators for subsystems 1 and 2 and is the von Neumann entropy. Since for a pure state von Neumann entropy of the reduced density matrix quantifies the entanglement of nielsen (); qcbook (),
we can conclude that the Wigner separability entropy is twice the entanglement entropy :
For pure states , and therefore the Wigner separability entropy is equal to the quantum mutual information
Quantum mutual information measures total correlations, both of classical and quantum nature, between subsystems 1 and 2 and for pure states classical correlations are equal to quantum correlations, measured by vedral ().
Iii Wigner separability entropy for chaotic maps
In order to illustrate the quantum to classical correspondence for the separability entropy, we study the evolution in time of and of its classical counterpart for three quantum maps: (i) the quantum baker’s map, (ii) the perturbed quantum cat map, and (iii) a model consisting of two coupled perturbed cat maps. The first model (i) exhibits an atypical failure of the quantum to classical correspondence due to discontinuity in the mapping producing drastic quantum diffraction effects. The second model (ii) is typical in that the Wigner separability entropy follows the classical separability entropy up to the Ehrenfest time scale , until Wigner function evolution (whose time derivative is given by the Moyal bracket) is well approximated by the evolution of the classical Liouville density (whose time derivative is given by the Poisson bracket, the first order in expansion of the Moyal bracket), i.e.
Note that for chaotic dynamics with Lyapunov exponent , the Ehrenfest time scales as Zaslavsky (). The third model (iii) illustrates for pure states the connection between the separability entropy and the entanglement entropy.
We study the quantum versions of classical chaotic maps on the 2-torus . Quantization imposes the Planck constant to coincide with the inverse of the dimension of the Hilbert space, i.e., . The baker map has been studied in detail in many works (see for example Saraceno1 (); Saraceno2 ()). The classical baker map is defined by
where the discrete time measures the number of map iterations. The quantum analogue of the classical baker map is a unitary operator acting on an -dimensional Hilbert space (assuming to be even). In the position()-representation its matrix reads:
where is the -dimensional antiperiodic Fourier matrix:
The classical perturbed cat map reads Ozorio ()
where and are taken , and where . Throughout this paper we have used
satisfying the symplectic condition , and . The perturbed quantum cat map in the -representation is given by the unitary matrix whose elements read
Finally, we have considered a 2D model (in 4D phase space ) consisting of two coupled perturbed cat maps. In order to do this in a symplectic way we have used a symmetric coupling in the position coordinates Wood (). In this case the classical map is given by (superscripts indicate which ‘cat’ the coordinates refer to)
where and () are again taken , and where we have used the same perturbation as before (for both maps). Here, is the coupling term.
The quantized version of the 2D perturbed cat map is obtained by multiplying a separable (tensor) product of quantized 1D perturbed cat maps with a simple exponentiated coupling matrix (which is diagonal in the -representation). More explicitly, unitary matrix is given by
and . In the following calculations we take .
We have studied the classical and quantum separability entropies. In all cases we have used the initial states given in terms of a Gaussian phase space distribution with dispersion equal to , and its quantum analogue, a coherent state on the torus. Both distributions are centered in the middle of the phase space, i.e., at . We have evolved 10 time steps (iterations) for all maps.
In Fig. 1 we show the separability entropy prosen1 () as a function of time (in units of map steps) for the baker map (25,26). We have used the same discretization number for both, the classical and the quantum simulations and removed the effects of the torus periodicity on the Wigner distributions dittrich (). We have decomposed the 2-torus in coordinates and . While the classical separability entropy saturates to a value of order bit, after an initial short transient growth, due to the exact solvability of its Liouville evolution gaspard (), and drops back towards zero for due to coarse-graining of classical Liouvillean evolution, the Wigner separability entropy grows with time due to quantum interference and diffraction patterns produced by discontinuity of the map. The baker’s map is therefore non generic, in that it exhibits very different classical and quantum results even within the Ehrenfest time range due to diffraction effects.
In order to investigate a case with generic behavior we have taken the smooth (continuous) perturbed cat map (29,30). In Fig. 2 we show the same quantities as in Fig. 1. Now there is quantum to classical correspondence, up to the Ehrenfest time . Note that, similarly to the case of baker map, the classical result also exhibits discretization time scale after which the classical distribution experiences a complexity reduction which is a numerical artifact due to finiteness of . Indeed, due to coarse graining the Liouville distribution becomes homogeneous in phase space, and this implies asymptotically vanishing separability entropy. Finally, and for comparison purposes only, we show the quantum results without removing the effects of the ghost images that appear on account of the torus periodicity (this is shown just for the lower value of ). Wigner separability entropy without ghost images removed exhibits some deviations for shorter times (then, the fact that we have ghost peaks in Wigner distribution is more important than at later times), the removal of ghost images is therefore needed in order to obtain quantum to classical correspondence.
Finally, we have tested the Wigner separability entropy and its relation to pure-state entanglement for two coupled perturbed cat maps (31,32) (in this case we have evaluated the Wigner distributions according to Ref. Konio ()). In practice we have decomposed the four-dimensional phase space in the coordinates and corresponding to each of the two ‘cats’, respectively. We have also compared the results with the classical separability entropy prosen1 (). It is worth mentioning that in order to obtain the classical complexity measure we have used the same phase space decomposition for the four dimensional Liouville distribution that we have previously applied to the four-dimensional Wigner function. All these results are shown in Fig. 3. The agreement between the quantum measures is accurate within the machine precision, i.e. (in the figure we have rescaled von Neumann entropy by a factor of 2), reflecting the match between singular value decompositions for the density matrix and its Wigner distribution. In this case we did not remove ghost images in the Wigner distributions since all the coherences measured by the separability entropy in these coordinate pairs are relevant to measure entanglement. Regarding the classical separability entropy we find a reasonable agreement up to time where due to coarse graining starts to drop. It is important to underline that the classical saturation is entirely due to the phase space coarse graining chosen for the numerical simulations, while the quantum one is an unavoidable phenomenon fixed by the finite size of . Note that, as recently pointed out casati (), the initial growth of entanglement can be reproduced in the semiclassical regime by purely classical computations.
In this paper we have proposed a new measure of complexity of quantum states, the Wigner separability entropy. This quantity turns out to be equal to the operator space entanglement entropy, it quantifies the minimal amount of computational resources required to simulate the quantum dynamical evolution of a system by means of time-dependent density-matrix renormalization group prosenznidaric (). Moreover, due to its relation with the entropy of entanglement for pure states and to the existence of the analogous s-entropy of Liouville densities in classical dynamics, the separability entropy emerges as an extremely broad complexity quantifier in both the classical and quantum realms.
With regard to a previously proposed phase-space quantum complexity indicator, that is, the number of Fourier harmonics of the Wigner function harmonics (); vinitha (), the Wigner separability entropy has the advantage of being basis-independent. More importantly, numerical indications prosenznidaric () suggest that the Wigner separability entropy should be able to distinguish between quantum chaotic and quantum regular motion also for many-body systems. Finally, we point out that the Wigner separability entropy is well defined also for mixed states and could therefore be used to quantify the complexity of decoherent quantum dynamics.
We thank Italo Guarneri for useful discussions. Partial financial support from Conicet, Argentina (GC), and the grants P1-0044 and J1-2208 of Slovenian Research Agency (TP) is greatfully acknowledged.
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