Parameterizing density operators with arbitrary symmetries to gain advantage in quantum state estimation

Quantum state estimation lies at the heart of many quantum information tasks, variational quantum algorithms and certification of quantum devices. However, the problem of estimating unknown quantum states is generally hard because the number of independent parameters needed for a full description grows exponentially with the number of qubits. Thus, many techniques have been developed for particular classes of states that satisfy certain conditions, which imposes constraints that reduce the number of independent parameters needed for a full specification of the state. An important example of such conditions is the existence of physical symmetries in the generated states. In some cases, symmetries have their origin in the physical properties of the system under consideration, as is the case of the permutational invariance of identical particle systems [1, 2]. In other cases, symmetric states can be generated intentionally to carry out certain specific tasks [3, 4].

Recently, a method for estimating quantum states with arbitrary symmetries using the MaxEnt technique was developed in [5, 6]. One of the main advantages of this method is that one obtains a substantial reduction in the number of observables needed to perform a reasonable estimation of the state. However, it has the drawback that the numerical optimization is carried out over the whole space of density operators. Thus, while the number of measurements is substantially reduced, the fact that symmetric states need less parameters is not used to take advantage during the numerical computation.

Another example of a relevant method for quantum state estimation that relies on symmetries is the permutationally invariant quantum tomography [7]. This method extracts the permutationally invariant part of an unknown quantum state, which has much less independent parameters than the full state. In fact, the number of measurements needed to determine the permutationally invariant part of an unknown state grows polynomially with the number of qubits [8]. The formulation of the problem and the choice of the observables to measure in laboratory implementations depends on the specific properties of the symmetry group [7, 9]. Up to now, there is no similar method for other symmetries.

In conclusion, a reduction in the computational complexity of the problem can be attained expressing a density operator that has the given symmetry as a combination of elements of a certain symmetric subspace of the space of Hermitian matrices. Generalizing the method of permutationally invariant tomography, the aim of this work is to reformulate the task of state estimation with arbitrary symmetries as a non-linear convex optimization problem. For that, we present a method for writing a quantum state with a given symmetry as a linear combination of the elements of a proper subspace of the space of Hermitian matrices. In doing so, we obtain a reduction in the dimensionality of the resulting convex optimization problem. This strategy has the following advantages with regards to previous works:

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  It works for a very general family of symmetries, namely those associated with finite groups and connected Lie groups. Almost all symmetries of interest in physics fall into these two categories. As relevant examples, we can mention: arbitrary rotations, permutationally invariant states (and all its Bosonic and Fermionic subfamilies) and Werner states.

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  Our method only depends on the mathematical peculiarities of the symmetry group via its generators. No further dependence on the particular properties of the group are needed. This means that, in practice, the main problem consists in identifying a set of generators of the Lie algebra (for connected Lie groups) or a set of generators of the group (for finite groups). In most tasks, this is equivalent to the mere specification of the group involved¹¹1For example, when we have a certain rotation, say a rotation in the $\hat{z}$ direction, we express it as $e^{-\frac{i}{\hbar}\sigma_{z}\theta}$, with $\theta\in\mathbb{R}$. Then, the Lie algebra generators are given by the set $\{\sigma_{z}\}$. Werner and permutationally-invariant state generators can be computed easily, as shown in [6].. This feature simplifies the mathematical formulation of the problem considerably in comparison with other methods.

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  The empirically-implemented observables need not possess the symmetry of the state. Only a mild “non-orthogonality” condition is required. Thus, our estimation technique can, in principle, be easily adapted to the particularities of very specific measurement set-ups (a feature that would allow to save a lot of experimental resources in many cases).

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  These features result in a reduction of both the number of independent parameters, that define the dimension of the convex optimization problem, and the number of measured observables.

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  Finally, the above methodology can be formulated as a simple algorithm, so that it can be used in practice without the need of delving deep into the technical details.

This paper is organized as follows. In Section 2 we show how to write a density operator having an arbitrary symmetry as a linear combination of a proper subspace of the space of Hermitian operators. Next, in Section 3, and after analysing the linear inversion method, we show how to get advantage of this decomposition to formulate a convex optimization problem with a reduced number of parameters. We sketch how to define the symmetric part of a state (an approach similar to [9] but considering arbitrary symmetries instead) in Section 4. Then, we test the performance of our method in a more realistic scenario using states with different symmetries in Section 5. Finally, in Section 6, we summarize our results.

In this section we assume that a given density operator $\rho$ possesses a certain symmetry. This symmetry can be described by a connected Lie group, as is the case of continuous rotations and Werner states [10], or by a finite group, such as discrete rotations with finite elements and permutations. Groups will be denoted by $G$ and their Lie algebras by $\mathfrak{G}$. The set of states that have a certain symmetry form a proper convex subset²²2Except when the symmetry group is the trivial one, i.e., $G=\{I\}$ of the convex set of quantum states [11]. In general, the dimension of this proper convex subset will be much lower than that of the original space, as we will see in Section 5. In order to use this reduction in the dimensionality, we need to rewrite the original state as a linear combination in a proper subspace of the space of Hermitian operators.

Let us start first showing how to do this for connected Lie groups. We will reproduce some of the results presented in [5] to make the presentation more self-contained. In what follows, we consider the scalar product $\langle A;B\rangle=\mbox{tr}(AB^{\dagger})$ in the vector space of complex matrices $\mathbb{C}^{n\times n}$, which is also a Hilbert space. Notice that, if $A$ and $B$ are Hermitian, then $\mbox{tr}(AB^{\dagger})=\mbox{tr}(AB)$. It also defines a real scalar product in the real-Hilbert space of Hermitian matrices.

Given a connected Lie group $G$ with Lie algebra $\mathfrak{G}$, assume that its generators are given by $\{Q_{k}\}_{k=1,\ldots,m}$ (i.e., $\mathfrak{G}$ is the linear span of $\{Q_{k}\}_{k=1,\ldots,m}$). Next, consider a basis for the space of Hermitian matrices: $\{O_{j}\}_{j=1,\ldots,n\times n}$. Then, one can prove that a state $\rho$ that possesses the symmetry ($U\rho U^{\dagger}$ for all $U\in G$) satisfies [5]:

The reason is the following. Assume that a density operator $\rho$ is invariant under the action of $G$: $U\rho U^{\dagger}=\rho$, for all $U\in G$. Now, given that $U$ is unitary and $G$ is connected, then each $U\in G$ can be written as $U=e^{iMt}$, where $M\in\mathfrak{G}$ ($A$ is Hermitian and belongs to the Lie algebra of $G$) and $t\in\mathbb{R}$ (notice that $t=0$ corresponds to $U=Id$, the unit element of the group). Thus,

Equations (2) can be seen as the kernel of a linear functional $\mbox{Tr}\left(i[Q_{k},O_{j}]\ldots\right)$, which is a linear subspace and is thus convex. The density operators that satisfy these equations are just contained in the intersection of all the kernels with the convex set of quantum states. Since the intersection of convex sets is a convex set, Eqs. (2) define a convex set $\mathcal{C}_{G}$ of density matrices. This set is also compact [11]. Notice that, under these conditions, any continuous strictly convex (concave) function defined over $\mathcal{C}_{G}$ will attain a unique maximum (minimum) element. If we add an arbitrary number of mean value equations of the form $\mbox{tr}(\rho O)=o$ (being $O$ an Hermitian matrix and $o\in\mathbb{R}$), we will also obtain a convex set (with the sole condition that the equations be compatible).

Applying the scalar product, an equation of the form

Equation (8) must be interpreted as follows: all states $\rho$ which are invariant under the action of the group $G$ can be written in that way for some unknown parameters $\alpha_{i}$. Equation (8) can be used as the starting point of a convex optimization problem: find the optimal values of the $\alpha_{i}$ in such a way that $\rho$ satisfies some constrains (typically, positivity, unit trace, and a set of mean values experimentally obtained) and maximizes (or minimizes) a convex (or concave) function, such as entropy. Notice that the manifold defined by Eq. (8) is convex, which gives a non-linear convex optimization problem.

The advantage of using such a parameterization is that the set of parameters, for many symmetries, is much smaller than the dimension $d$ of the original space. For example, a permutationally symmetric state $\rho$ is defined by ${d\times d+N-1}\choose{N}$ $-1\sim N^{d^{2}-1}$ independent real parameters, which exhibit a cubic growth with the number $N$ of qubits. In Section 5 we give concrete examples of this reduction in the number of parameters for different symmetries.

The construction for finite groups is completely analogous to the one exposed above. Let $G$ be a finite group generated by a set $G_{0}=\{g_{1},g_{1},\ldots,g_{n}\}$ (for finite groups, the existence of this set follows trivially from the definition of generators; see [12], page 9). This means that any element $g\in G$ can be written as a (finite) product of elements of $G_{0}$ (and their inverses). In the Hilbert space, the elements of $G_{0}$ are represented by a set of unitary operators $RG_{0}=\{U_{1},U_{2},\ldots,U_{n}\}$, that, again, generate a representation $RG$ of $G$ in a similar way because any representation establishes a group homeomorphism. Now, the invariance condition of $\rho$ can be stated as

In this section we discuss how to use the decomposition in Eq. (8) to determine a symmetric – but otherwise unknown – quantum state. Even though our results will mainly focus on the convex optimization application, we first show how to such decomposition for the case of linear inversion.

From the above considerations, it follows that, given a group $G$, we can split the vector space of $n\times n$ complex matrices in two orthogonal subspaces, namely, $\mathbb{S}_{G}$ and $\mathbb{S}_{G}^{\bot}$. Thus, it should be obvious that an arbitrary density operator $\rho$ can be, in principle, written as:

Under some circumstances, finding the symmetric part of an unknown density operator can be useful for quantum information theory (see for example the arguments presented in [7] and [9] for the case of permutationally invariant states). Here, we have shown how to generalize the symmetric part of a density operator to a much larger family of symmetries. If we now chose a collection of linearly independent observables $\{A_{i}\}_{i=1,\>\ldots,\>\mbox{dim}(\mathbb{S})}$ that can be written as linear combinations of elements of $\mathbb{S}$, we can perform a tomographic scheme that estimates the symmetric part of an unknown density operator for an arbitrary symmetry defined by $G$.

In this section, we apply the ideas exposed above to estimate multi-qubit quantum states with specific symmetries. To this end, we combine the parameterization from Section 3 with the variational quantum tomography (VQT) technique described in Refs. [15, 16]. We refer to this hybrid approach as group-invariant tomography (GIT). We will study scenarios with a number of measurements equal to the number of parameters of the symmetric state involved. Nevertheless, as our algorithm is based on VQT, it can be used even when the number of parameters is less than quorum [15], as in the case of MaxEnt estimation. Our analysis is focused on permutationally invariant states, Werner states, and states which are invariant under global rotations or under individual qubit rotations.

In Fig. 1, we show the number of independent parameters in the symmetric part of a density operator with the given symmetry as a function of the number of qubits. For comparison, we also include the number of parameters required to estimate a multi-qubit state via standard quantum tomography (SQT).

For an approach to the performance of the GIT in a real experimental scenario, in the following sections we use such a scheme to reconstruct quantum states with specific symmetries by considering simulated measurement results that could be obtained in hypothetical experiments. The simulated lab frequencies were generated modifying the theoretically predicted probabilities with a binomial or Gaussian probability distribution. As figure of merit, we compute the fidelity of reconstruction between the resulting estimated state and the ideal target state.

The GIT scheme was implemented algorithmically in Python following the ideas from Section 2. First, it computes the commutators $[Q_{i},O_{j}]$ for the given symmetry, where $Q_{i}$ is a Lie Generator and $O_{j}$ is a basis element of the space of hermitian $2^{N}\times 2^{N}$ matrices. Then, it uses Gram-Schmidt orthogonalization (GSO) to determine a subset of $c$ linearly independent commutators. This set of $c$ elements is completed with $\{O_{j}\}_{j=1,\dots,2^{2N}}$ and, from this new set of $c+2^{2N}$ ordered elements, a subset of $2^{2N}$ orthogonal elements is obtained applying again GSO. The first $c$ elements generate $\mathbb{S}_{G}^{\bot}$, implying that all remaining elements form an orthogonal basis of $\mathbb{S}_{G}$. Thus, these $r=2^{2N}-c$ matrices are the $S_{i}$’s from Eq. (8). Finally, this expression of $\rho$ is integrated into the VQT formulation [15, 16], that we reproduce here:

In this work, we have shown how to use a priori knowledge of the symmetry of an otherwise unknown state to find a convenient parameterization for quantum state estimation purposes. This idea allows to reduce the number of independent parameters in a convex optimization formulation of the estimation problem, and leads to a concomitant reduction in the number of independent measurements as well.

Unlike previous methods, this approach works for arbitrary symmetries. This protocol can be potentially implemented on experimental scenarios and thus lower the difficulty level in practical implementations. Moreover, symmetries are taken into account in a relatively simple algorithmic way that does not depend on complex mathematical properties of the groups involved. We think that the results presented in this work are a step forward in the development of state estimation protocols, which can be used to reduce both the computational and experimental complexity of the existing methods.