Pseudocomplexity of purification for free scalar field theories

I. INTRODUCTION

Quantum entanglement continues to attract attention across multiple disciplines. Recently, a generalization of entanglement entropy called pseudo entropy was proposed in [1,2]. There, instead of starting from a usual density matrix, the authors define a matrix using two pure quantum states jψ 1 i and jψ 2 i,

which is dubbed the transition matrix. The interest in this is twofold. First, in the case of postselection in quantum experiments, the transition matrix becomes important once one specifies the initial state as jψ 1 i and the final state as jψ 2 i, given that they are not orthogonal to each other. In the postselection experiment, the transition matrix then plays the role of the density matrix while computing the weak expectation value hOi ¼ TrðOτ 1j2 Þ of an observable O.

The second motivation comes from the AdS=CFT correspondence [3,4], where pseudo entropy is proposed to be the dual CFT quantity to the area of a minimal hypersurface in Euclidean AdS spacetime. In holography, this is constructed in the path integral technique by dividing a Euclidean time slice into two regions corresponding to the two states. Dividing the entire system into A and its complement A C , the pseudo entropy of A (relative to A C ) is defined as where τ 1j2 A ¼ Tr A C ðτ 1j2 Þ is called the reduced transition matrix, in analogy with existing literature on entanglement entropy derived from the reduced density matrix. When the two states are the same, the transition matrix reduces to the usual density matrix, and pseudo entropy reduces to usual entanglement entropy.

Quantum circuit complexity is another quantum information (QI) theoretic quantity that has been at the center of interest for the last few years. Recently, circuit complexity has been explored in the context of quantum field theory . 1 Complexity is usually defined as the number of elementary structural components needed to construct either an evolved state or operator from a simple initial state or operator. Here, we try to compute the circuit complexity of the reduced transition matrix. This reduced transition matrix is an analogue of a mixed state corresponding to a reduced density matrix, although it can be non-Hermitian, in general. The methods that we use for computing circuit complexity are those for the complexity of purification [28][29][30][31][32][33][34].

First, given a transition matrix, which is written in density matrix form, a state can be associated with this operator by writing the transition matrix in the vectorized form. Then, we compute the circuit complexity for this state using Nielsen's method [35][36][37]. This vectorization means changing the bra part of the transition matrix to a ket part, which amounts to a doubling of the Hilbert space [38][39][40]. Hence, the corresponding transition state looks like

which, as mentioned in [2], for a transition matrix, turns the corresponding state into a tensor product of the two states. Then, the corresponding complexity, following the approach of [5], of the state dual to the transition matrix is

Note that the complexity defined using canonical purification in (1.4) is simply the direct sum of two complexities of two individual states jψ 1;2 i. The direct sum is representative of the fact that the vectorized version is a tensor product. It is quite unlikely that this will have any nontrivial holographic interpretation, as typically holographic duals involve some sort of minimization. Motivated by this, we want to define some notion of pseudocomplexity of purification [Cðτ

1j2

A Þ] in this paper, extending the notion of complexity of purification [29][30][31][32][33][34] for this case. We first purify the reduced transition matrix and compute the minimal complexity among all possible purifications. 2 We do these computations for free scalar field theories and Lifshitz field theories in (1 þ 1) dimensions.

Physically, pseudocomplexity is related to the idea of postselection as suggested in [1,2,41] while defining pseudo entropy. The idea is that if, along some physical process, one starts with a state jψ 1 i and somehow the final state is postselected to be jψ 2 i, which is not necessarily a result of a simple unitary evolution with a simple Hamiltonian, the process can still be approximated as a result of an operator jψ 2 ihψ 1 j operating on the initial state. In general, jψ 2 i might be a result of many unitary and nonunitary steps along the way. 3 Hence, this operator jψ 2 ihψ 1 j might very well be a non-Hermitian operator.

This operator is considered to be a transition matrix, similar to a density matrix in this treatment. The reason is that this operator does not have any nontrivial information about the system other than the outer product of the initial and the postselected state, similar to a density matrix. Then, the reduced transition matrix, defined after a reduction in degrees of freedom, can associate a notion of entropy similar to the entanglement entropy of a reduced density matrix. Similarly, the pseudocomplexity should measure the complexity of this operator, whereas the pseudocomplexity of purification should measure the complexity of forming such a reduced transition matrix once the two states jψ 1;2 i are given. Another way of looking at this problem is to relate it to the averaged number of maximally entangled pairs of qubits to be distilled from the intermediate state once the final state is postselected [1]. While pseudo entropy measures this number, pseudocomplexity of purification measures the amount of work (of course, in terms of resources or gates) needed for this distillation starting from the reduced transition matrix.

The rest of this article is organized as follows: In Sec. II we first review a path-integral calculation of Gaussian transition matrix following [41] and propose a definition of pseudocomplexity of purification. We then use our new definition and calculate this quantity in a simple setting of coupled harmonic oscillators on a lattice. The results of our numerical analyses are collected in Sec. III. We conclude in Sec. IV with discussions and an outlook.

II. PSEUDOCOMPLEXITY OF PURIFICATION

The concept of purification originates from quantum information theory (considered to be a close cousin of Schmidt decomposition as a process). The process of purification refers to constructing a pure state from a mixed reduced density (transition) matrix such that if one traces out the auxiliary degrees of freedom added while constructing the purified state, one gets back the original reduced density (transition) matrix. While both the actual and purified states generate the same reduced density matrix, the purified state only has important information about the density matrix because the extra parameters added in the process of purification are random. The only constraint on them is that the purified state should follow the standard quantum mechanical postulates and properties of a pure state. As mentioned already, we consider random purification by all possible values of parameters for which the purified state can be consistently formed. However, we choose a specific purification among them by minimizing the complexity functional [30]. This is motivated by the definition of complexity, where minimal resources play an important part. In terms of quantum gates, this means choosing the minimum number of gates so that the resources needed are minimized. Now we describe our main setup.

A. Setup

We consider a system of scalar fields which is governed by the Hamiltonian

This theory exhibits an anisotropic (Lifshitz) scaling symmetry t → λ z t, x → λx in the m → 0 limit; for z ¼ 1, this is an ordinary relativistic scalar field theory. For practical purposes, it is useful to consider the discretized Hamiltonian

ð2:2Þ

with N being the total number of points in the lattice.

We consider a collection of coupled linear harmonic oscillators on a lattice of space points, labeled by capital Latin indices. Any two Gaussian states jψ α i (α ¼ 1; 2) of this system can be generically expressed in their position representation as

where q A denotes displacement of the Ath oscillator, W ðαÞ AB is a positive-definite matrix, and N α is the normalization constant,

We divide the entire system into a part A and its complement A C ; the lattice points within the subsystem Ω are labeled by lowercase Latin letters, while lowercase Greek indices label those outside it. We adopt the notation

This is a Gaussian integral. Finally, the matrix elements of the reduced transition matrix can be expressed in a simple form [2]:

; ð2:6Þ

where

ð2:7Þ

and

þ W ð2Þ Þ: ð2:8Þ

B. Auxiliary parameters and purification

We use Nielsen's method for the computation of circuit complexity [5,[35][36][37] and choose the F 1 norm for the complexity functional [5,42]. For a pure (or purified) state, this is written in terms of the (unentangled) reference state frequency and the frequency of the normalized version of the coupled oscillator system in the lattice pictures. We start from various numbers of initial oscillators and reduce to transition matrices after tracing out all but one or two of the oscillators. Afterwards, we purify this reduced transition matrix τ 1j2 A by adding parameters corresponding to adding one or two more oscillators (similar to the doubling of degrees of freedom in qubit purifications). However, we do not choose canonical purification since it always returns the original transition state as mentioned in (1.3), the complexity of which is trivial, as explained before. 4 For the reduced transition matrix, the parameters that arise while matching the original reduced transition matrix with the reduced density matrix of the purified state take care of all possible purifications. In this optimal picture of purification, the number of unknown arbitrary parameters is much higher than the number of unknowns needed to purify ρ A 1 or ρ A 2 . This is because the information of not one but two pure states goes into the transition matrix and hence the reduced transition matrix.

In order to define pseudocomplexity, we begin by introducing purification of the transition matrix τ 1j2 A . We consider a (fictional) auxiliary systemà whose Hilbert space is of the same dimension as our original subsystem A, and we consider two Gaussian states in the enlarged Hilbert space H A ⊗ HÃ,

Then, proceeding exactly as above, we calculate a transition matrix in the enlarged Hilbert space. This particular choice of Gaussian purification is primarily motivated by the study done in [43] in the context of entanglement of purification. This simplifies the numerical analysis as well as the computation of circuit complexity. We consider that this is a purification of our original reduced transition matrix (2.6) for the subsystem A; in this case, the following constraints must be obeyed:

Þ. In our numerical analysis, we adopt the following conditions to satisfy these constraints:

ð2:12aÞ

This particular choice leaves only K ð1Þ and L ð1Þ undetermined.

For the particular model of our choice, the matrices W ðαÞ AB are given by

where N is the total number of lattice points. Here, m α and z α are the mass and dynamical exponent of the respective theory. The matching of J; K; L ðαÞ matrices with their corresponding counterparts introduces a few arbitrary parameters in the purified state. These parameters denote the infinitely many possible Gaussian purifications of the reduced transition matrix. Afterwards, we define the complexity of purification as the complexity of the specific purification for which the complexity functional is minimized in terms of all the parameters of the purified version of the reduced transition matrix. This involves purification parameters for both jψ 1;2 i states, constituting the transition matrix. Finally, we compare the complexity of purification of the reduced transition matrix Cðτ

A Þ with the complexity of purifications for the individual reduced density matrices (ρ 1 A and ρ 2 A ) derived from the initial and final states jψ 1 i and jψ 2 i, respectively.

We propose that the pseudocomplexity of purification is given by

where Ω ðαÞ j denotes the jth eigenvalue of the matrix

L ðαÞ Þ, and the functional is minimized over the parameters associated with the auxiliary systemÃ. This means choosing a particular purification associated with minimizing parameter values.

Let us point out a few attributes of our method of computation. Most of our numerical analyses are performed for the simplest case of a reduced system made of only one oscillator, where the above functional is to be minimized over two unknown parameters. In general, when the reduced system consists of n number of linear harmonic oscillators, the J, K, and L matrices are of dimension n × n. The choice II B always leaves two of them undetermined, and therefore we end up with a total of 2n 2 unknown parameters. On the other hand, to calculate the more wellknown complexity of purification from a reduced density matrix, the number of unknown parameter values over which one is required to perform a minimization scales as n 2 with the reduced system size. It is easy to conclude that the calculation of the new functional (2.14) is at least twice as hard since the transition matrices have information of both the initial and the final state.

However, there exists a way to purify the system mode by mode [30]. Thus, it might be possible to reduce the number of unknown parameters to 2n. Even then, the problem is twice as hard as computing complexity of purification from density matrices, where the associated number scales as n.

It may also be of relevance to note that any purification we consider takes into account only Gaussian states, and the auxiliary system always has the same dimension as the original one, motivated by the usual doubling of Hilbert space typically done in purification using Schmidt decomposition. These are restrictions we invoke in order to keep the numerical job simple. There may exist ways to relax the conditions.

III. RESULTS AND DISCUSSION

A. Pseudocomplexity of purification

In this subsection, we report our numerical results after the minimization with different variables of the theory. In the primary Hamiltonian (2.2), we choose a general version which applies to both the ordinary free scalar QFTs ðz 1 ¼ z 2 ¼ 1Þ and Lifshitz scalar field theories ðz 1 ≠ 1 ≠ z 2 Þ. The variables considered are the dynamical exponents z 1;2 , the masses m 1;2 , the number of oscillators before tracing out, the number of oscillators of the reduced system, and the frequency of the unentangled reference state.

(1) Varying z 2 and m 2 : From the plots of pseudocomplexity of purification against reference frequency in Figs. 1(a) and 1(b), we notice that as the reference frequency is increased, the pseudocomplexity of purification primarily decreases and then shows a polynomial growth and saturation. When the scaling factor z 2 is increased in the same plots, the pseudocomplexity of purification for higher values of reference frequency decreases for higher z 2 . However, the behavior is opposite for small reference frequency values. Hence, there is a crossover in complexity after which the small to large behavior changes. A similar change is observed from Fig. 1(b), where the mass parameter m 2 is varied. We find that at large frequencies, the pseudocomplexity of purification decreases as the mass is increased, whereas, at very small frequencies, the behavior is again opposite. (2) Pseudocomplexity of purification vs z 2 plots: From Fig. 2(a) we observe that as z 2 is increased, the pseudocomplexity of purification primarily decreases and then grows linearly. By looking at multiple plots for various values of z 1 , we find that with increasing z 1 as z 2 is varied, the pseudocomplexity of purification decreases. In this case, there is no change of behavior or crossover between different plots, which we saw earlier. (3) Pseudocomplexity of purification vs m 2 plots: As shown in Fig. 2(b), we find that one of the mass parameters is varied from zero to one, and the pseudocomplexity of purification does not change much. Although the plots are not completely parallel to the x axis, their variance is relatively small compared to what we have observed for other parameters. On the other hand, if we vary one of the scaling parameters z 2 , we find that with increasing values of z 2 , pseudocomplexity of purification increases. (4) Varying number of oscillators before tracing out:

All the above plots are given for the case where the total number of oscillators associated with the states ψ 1;2 before tracing out degrees of freedom is 2. So we trace out one of the two oscillators in each of the states. This begs the question of how the results might possibly change if the number of oscillators is increased. This is more relevant when thinking about quantum field theories, which can be written as the lattice of infinitely many coupled harmonic oscillators. However, we find that the pseudocomplexity of purification does not vary much at all when the number of oscillators of the actual system is varied. This can be seen from Fig. 3. 5Varying number of oscillators in the reduced system: All of our results so far are for the case when there is one oscillator in the reduced transition matrix. The reason is twofold: First and foremost, as we have mentioned before, the number of unknown parameters for pseudocomplexity of purification in terms of which the complexity functional is minimized is much more than the case of usual complexity of purification. This makes the problem much more challenging numerically as the number of parameters grows very quickly as one increases the number of oscillators in the reduced system. Second, from a physical point of view, with the motivation of relating the entanglement entropy to some thermal notion of entropy, one typically chooses the number of oscillators in the reduced system to be small. Although there is no clear motivation for complexity, we stick to this particular choice. The qualitative behavior of the pseudocomplexity of purification with the varying reference frequency remains similar when the subsystem size is increased to include two oscillators as shown in Fig. 4. Hence, we expect all of our results to be universal and hold for larger subsystem sizes.

B. Difference ΔC: Mutual pseudocomplexity of purification

The reduced transition matrix carries complicated mixed information of not one but two pure states ψ 1 and ψ 2 . Hence, while studying the complexity of purification for the reduced transition matrix, it is natural to ask how different it is from the complexities of individual reduced density matrices of those two states. Inspired by the definition of mutual complexity ð△Cðρ AB ÞÞ [30,44,45], which is defined as the difference of complexity between the full state ρ AB and the sum of complexities of the two reduced density matrices ρ A and ρ B , we define the difference as the mutual pseudocomplexity of purification. In the following, we report some properties of the mutual pseudocomplexity of purification based on our numerical analysis.

(1) Subadditivity: From Fig. 5, it is easy to observe that

Therefore, we find that the pseudocomplexity of purification is always subadditive to the sum of the individual complexities of purification of the two relevant states. Our numerical results suggest that the term △Cðτ equality is expected for the case when the two states become the same and hence ρ 1 A ¼ ρ 2 A . (2) Near saturation behavior: It is observed that both the complexity of purification and the pseudocomplexity of purification tend to saturate at a very large reference frequency. By looking at the saturation values of the corresponding plots from Fig. 5, we make the following comment on the saturation of pseudocomplexity of purification in terms of complexity of purification of the individual reduced density matrices,

This means that the quantity ΔCðτ 1j2 A Þ approaches zero at large reference frequencies. This indicates that the dependence of Cðτ 1j2 A Þ on the reference frequency ω is such that for large enough values of ω, the complexity does not distinguish between a transition matrix and the density matrix.

(3) Varying masses: We also study the behavior of

A Þ) with a change in one of the masses m 2 corresponding to the state ψ 2 while keeping the other parameters (m 1 , z 1 , z 2 ) fixed. We look at the behavior in both a small mass difference and a large mass difference regime. In both cases, we find that the difference decreases as one increases the difference between mass parameters; see Fig. 6. In the small mass difference regime, the slope of the plot is linear. This means j△Cðτ The above expression is written in this way because our treatment is symmetric with respect to z 1;2 . In the same plot, we also notice that once z 1 is chosen to be 2 instead of 1, j△Cj increases, but the linear behavior remains unchanged.