Asymmetry activation and its relation to coherence under permutation operation

Asymmetry is a root of the variety of various physical phenomena. Consider a perfectly symmetric state. Such a state does not reflect any changes so that it can realize no variety. To realize rich variety, the state needs to have sufficient asymmetry. In physics, people consider that symmetry breaking is needed for our world. Therefore, to explain various phenomena, many researchers introduced spontaneous symmetry breaking, which plays an important role in various areas in physics, nuclear star, superfluidity, superconductivity, cold atoms, Higgs boson, Nambu-goldstone boson. When the degree of symmetry breaking is large, it realizes rich phenomena. Since the amount of symmetry breaking can be restated as the amount of asymmetry, it is a central topic to study the amount of asymmetry.

Fortunately, quantum information theory has strong tools to evaluate the amount of asymmetry [1, 2]. The variety generated by group symmetry under a given state can be measured by the number of distinguishable states generated by group action to describe the symmetry. This number is evaluated by the difference between the von Neumann entropy of the averaged state with respect to the group action and the von Neumann entropy of the original state. This fact can be shown by a simple application of quantum channel coding theorem [3, 4] to the family of the states translated by the group action although this idea can be backed to various studies [5, 6, 7]. Therefore, the difference between the von Neumann entropies of these two states can be considered as the amount of the asymmetry of the given state. Now, we focus on the permutation symmetry on qubits similar to the recent paper [8]. When we apply the permutation of the state $\ket{1^{M}\,0^{N}}$, the averaged state $\rho_{mix,N+M,M}$ is permutation-invariant, and has no asymmetry. However, it still works as a resource for asymmetry when it is attached to another qubits state $\ket{1^{l}\,0^{k-l}}$. When we have only the state $\ket{1^{l}\,0^{k-l}}$, the amount of asymmetry is measured by the von Neumann entropy $S(\rho_{mix,k,l})$ of $\rho_{mix,k,l}$. When the state $\rho_{mix,N+M,M}$ is attached, the amount of asymmetry is $S(\rho_{mix,k+N+M,l+M})-S(\rho_{mix,N+M,M})$, which is larger than the asymmetry $S(\rho_{mix,k,l})$ of the added state. In this paper, we focus on the increase of the amount of asymmetry by adding an invariant state, and call this phenomena an asymmetry activation. In fact, an asymmetry activation with various groups often happens behind of symmetry breaking. In this paper, we study the asymmetry activation under quantum information-theoretical objects because no systematic research studied the asymmetry activation in the area of quantum information.

Now, we discuss how quantum theory can improve the asymmetry activation because the above state is classical. To address this problem, we focus on the coherence, which is a key concept in quantum theory [9]. That is, we consider a permutation-invariant state with great coherence. We focus on the Dicke state $\ket{\Xi_{N+M,M}}$ with the same weight as the classical state $\rho_{mix,N+M,M}$. This state has the largest coherence among invariant states with the same weight as the state $\rho_{mix,N+M,M}$ because any invariant state with the same weight has the same diagonal elements as the state $\rho_{mix,N+M,M}$. A Dicke state has been playing an important role in the calculation of entanglement measures [10, 11, 12, 13, 14], quantum communication, and quantum networking [15, 16]. The experiments [15, 16] are motivated by the fact the overlap of symmetric Dicke states with biseparable states is close to 1/2 for large $N$ [17]. Multipartite entanglement has been detected in an ensemble of thousand of atoms realizing a Dicke state [18]. Some typical Dicke states have been realized in trapped atomic ions [19]. Recently, the multi-qubit Dicke state with half-excitations has been employed to implement a scalable quantum search based on Grover’s algorithm by using adiabatic techniques [20]. These studies show the importance of Dicke states

In particular, the Dicke state can be used for quantum metrology, which has been tested experimentally in cold gases of thousands of atoms [21, 22]. The Dicke states are optimal in linear interferometers, in the sense, that they saturate the inequality $F_{Q}[J_{x},\rho]+F_{Q}[J_{y},\rho]+F_{Q}[J_{z},\rho]\leq N(N+2)/4$, which are saturated also by GHZ states [23, 24]. Here, $F_{Q}$ is the quantum Fisher information corresponding to a unitary dynamics. Also the reference [25] studied the estimation of the same direction with Dicke states. These studies consider the information only for the direction of the rotation by the special unitary group. They do not focus on the direction of permutation because the Dicke state is invariant for permutation. It is a completely new idea to use the information of the direction of permutation by modifying the Dicke state. To execute this new idea, using non-negative integers $n,m,k,l$ as $M:=m-l$, $N:=n-m-k+l$, we consider the asymmetry of the following state.

The amount of asymmetry is the von Neumann entropy $S(\operatorname{av}_{\pi}(\ket{\Xi_{n,m|k,l}}))$, where $\operatorname{av}_{\pi}$ expresses the averaged state with respect to the permutation.

However, the calculation of this von Neumann entropy is not simple while $S(\rho_{mix,k,l})$ is calculated to $\log\binom{k}{l}$, which is approximated to $lh(k/l)$, where $h(x):=-x\log x-(1-x)\log(1-x)$ is the binary entropy. This calculation is closely related to Schur-Weyl duality, one of the key structure in the representation theory. When we make a measurement given by the decomposition defined by Schur-Weyl duality for the system whose state is $\ket{\Xi_{n,m|k,l}}$, we obtain the distribution of this measurement outcome. This distribution takes a key role in this calculation, The paper [26] discovered notable formulas for this distribution by using Hahn and Racah polynomials, which are ${}_{3}F_{2}$- and ${}_{4}F_{3}$-hypergeometric orthogonal polynomials, respectively. It is possible to numerically calculate this von Neumann entropy by using these tools.

Even though its numerical calculation is possible, it is quite difficult to understand its behavior. To grasp its trend, we consider two kinds of asymptotics, i.e., Type I and Type II limits. Type I limit assumes that $k$ and $l$ are fixed and $m$ is linear for $n$, i.e., $m=\xi n$ with fixed ratio $\xi$, under $n\to\infty$. Type II limit does that $k,l$ and $m$ are linear for $n$ under $n\to\infty$. Type I limit corresponds to the situation of the law of small numbers, and Type II limit corresponds to the situation of the central limit theorem. For the relation of Type I and Type II limits with the law of small numbers and the central limit theorem, see Appendix A. Based on these formulas, the paper [27] derived the asymptotic approximation of the distribution in the limit $n\to\infty$.

In Type I limit, we show that the von Neumann entropy $S\bigl{(}\operatorname{av}_{\pi}[\Xi_{n,m|k,l}]\bigr{)}$ behaves as

The same scaling $\log n$ appears in the bosonic coherent channel when the total power is fixed and the number $n$ of modes increases [28]. Figure 1 shows an example of this approximation. When we use the decohered state $\rho_{mix,N+M,M}$, the amount of asymmetry is calculated as

The amount of asymmetry in (3) is a constant for $n$ while the case with Dicke state $\ket{\Xi_{N+M,M}}$ has the order $\log n$ as (2). In particular, in this scenario, while the size of the added system is finite, the amount of activated asymmetry goes to infinity. Therefore, the Dicke state $\ket{\Xi_{N+M,M}}$ significantly improves the amount of asymmetry over the decohered state $\rho_{mix,N+M,M}$ under Type I limit.

In Type II limit, i.e., the limit $n\to\infty$ with $k,l$ and $m$ linear for $n$, we need to treat many ratios and parameters, which are summarized in Table 1. While the fixed ratios 1 and 2 are natural, the fixed ratio 3 is useful for our purpose.

Then, we show that

Since the amount of asymmetry of the added system is $\log\binom{k}{l}$ and is approximated to $n(\alpha+\gamma)h(\frac{\alpha}{\alpha+\gamma})+o(n)$, the amount of the asymmetry activation is approximated to $n(h(\mu)-(\alpha+\gamma)h(\frac{\alpha}{\alpha+\gamma}))+o(n)$.

When we use the decohered state $\rho_{mix,N+M,M}$, the amount of asymmetry is calculated as

Hence, the amount of the asymmetry activation is approximated to $n(h(\xi)-(\beta+\delta)h(\frac{\beta}{\beta+\delta}))-(\alpha+\gamma)h(\frac{\alpha}{\alpha+\gamma}))+o(n)$. For this derivation, we employ a kind of central limit theorem obtained in [27]. Since the inequality $h(\mu)\geq h(\xi)-(\beta+\delta)h(\frac{\beta}{\beta+\delta})$ holds as shown later, the Dicke state $\ket{\Xi_{N+M,M}}$ significantly improves the amount of asymmetry over the decohered state $\rho_{mix,N+M,M}$ even under Type II limit. In addition, similar characterization is also possible from the viewpoint of the number of distinguished states.

When the Dicke state $\ket{\Xi_{N+M,M}}$ is decohered, the state changes to the classical state $\rho_{mix,N+M,M}$. Since both states are invariant for permutation, their difference is the existence/non-existence of the coherence. Therefore, the difference between (2) and (3) and the difference between (4) and (5) can be considered as the improvement of asymmetry by using the coherence. In addition, the formula (4) has a very simple form and is characterized by the binary entropy of the quantity $\mu$. Since the quantity $\mu$ has not appeared in physics, the formula (4) means that the quantity $\mu$ is a newly discovered quantity that has a physical role related to asymmetry. Therefore, we can expect that the new quantity $\mu$ takes certain roles in other topics.

The remaining part of this paper is organized as follows. As a preparation, Section 2.1 discusses the general theory for the amount of asymmetry, and prepares several formulas for the number of distinguished states. Also, it explains the general theory for asymmetry activation. Section 3 gives our problem setting, and introduces a key distribution to determine the amount of asymmetry. Section 4 discusses the amount of asymmetry for the decohered state. Section 5 reviews the asymptotic behavior of the distribution related to Schur-Weyl duality, which was obtained in the paper [27]. Section 6 derives the asymptotic behavior of the von Neumann entropy of the average state, which expresses the asymptotic behavior. Section 7 considers the asymptotic behavior of the number of distinguishable states. Section 8 gives another example for the asymmetry activation. Section 9 gives the conclusion and discussion.

Appendix B gives the proofs of the statements given in Section 3. Appendix C gives the derivation of the statement given in Section 4. Appendix D gives the derivation of the statements given in Sections 6 and 7. Appendix E shows an asymptotic formula for the von Neumann entropy presented in Section 6.

This paper investigates how an invariant state activates the amount of asymmetry. To address this question, we focus on the $n$-fold tensor product $\mathcal{H}=(\mathbb{C}^{2})^{\otimes n}$ of the qubit system $\mathbb{C}^{2}=\mathbb{C}\ket{0}\oplus\mathbb{C}\ket{1}$, and consider it as the representation space of the permutation group $G=\mathfrak{S}_{n}$ acting by permutation of tensor factors as

Then, we apply the general theory given in Subsection 2.2 when ${\cal H}_{1}=(\mathbb{C}^{2})^{\otimes N+M}$, ${\cal H}_{2}=(\mathbb{C}^{2})^{\otimes k}$, $G_{1}=\mathfrak{S}_{N+M}$, $G_{2}=\mathfrak{S}_{k}$, $G_{3}=\mathfrak{S}_{N+M+k}$ and $\rho_{2}=\ket{1^{l}\,0^{k-l}}\in(\mathbb{C}^{2})^{\otimes k}$ is defined as

while the state $\ket{1^{l}\,0^{k-l}}$ is asymmetric with respect to the permutation.

As one possible choice of $\rho_{1}$, we choose a Dicke state on $(\mathbb{C}^{2})^{\otimes(N+M)}$, which is a typical invariant state because it is given as the permutation invariant state with fixed weights $N$ and $M$ as

Remember that the state $\ket{\Xi_{n,m|k,l}}$ is defined as

In contrast, the decohered state of the Dicke state is given as

To study the asymmetry, we focus on the $\operatorname{SU}(2)$-$\mathfrak{S}_{n}$ Schur-Weyl duality on the tensor product system $\mathcal{H}=(\mathbb{C}^{2})^{\otimes n}$ depicted as

The Schur-Weyl duality claims that we have the following decomposition of $(\mathbb{C}^{2})^{\otimes n}$ into irreducible representations (irreps for short):

Here $\boxtimes$ denotes the tensor product of linear spaces, equipped with $\operatorname{SU}(2)$-action on the left and $\mathfrak{S}_{n}$-action on the right factor, $\mathcal{V}_{(n-x,x)}$ denotes the $\mathfrak{S}_{n}$-irrep corresponding to the partition $(n-x,x)$, and $\mathcal{U}_{(n-x,x)}$ denotes the highest weight $\operatorname{SU}(2)$-irrep of dimension $n-2x+1$. Using the hook formula (see [30, I.5, Example 2; I.7, (7.6)] for example), we can compute the dimension of $\mathcal{V}_{(n-x,x)}$ as

where $\square\in(n-x,x)$ denotes a box in the Young diagram corresponding to the partition $(n-x,x)$. Using this formula, we have the completely mixed state $\rho_{\text{mix},\mathcal{V}_{(n-x,x)}}=\frac{1}{\dim\mathcal{V}_{(n-x,x)}}\operatorname{\mathsf{id}}_{\mathcal{V}_{(n-x,x)}}$. The highest weight $\operatorname{SU}(2)$-irrep of dimension $2j+1$ has the standard basis $\{\ket{j,m}\mid m=-j,-j+1,\dotsc,j\}$, where $j$ and $m$ express the total and z-direction angular momenta, respectively.

In our analysis, the projector $\mathsf{P}_{(n-x,x)}$ onto the isotypical component in the decomposition (28) plays a key role, and is defined as

As the set of the projectors $\{\mathsf{P}_{(n-x,x)}\}_{x}$ forms a projective measurement, we define the probability $p(x\,|\,n,m,k,l)$ as

and denote the corresponding probability distribution by $P_{n,m,k,l}$. Since we can switch $\ket{0}\leftrightarrow\ket{1}$ in the system, the probability mass function (pmf) enjoys the system symmetry

Since the distribution $P_{n,m,k,l}$ describes the averaged state $\operatorname{av}_{\pi}[\Xi_{n,m|k,l}]$ as

our analysis is concentrated in the distribution $P_{n,m,k,l}$. We also denote by $X$ a discrete random variable distributed by the probability mass function (pmf for short) $p(x\,|\,n,m,k,l)$. We choose integers $n,m,k$ and $l$ from the following set;

Then, the von Neumann entropy is given by

which expresses the amount of asymmetry of $\ket{\Xi_{n,m|k,l}}$. Furthermore, we can also consider the number of approximately orthogonal elements among the states $\{\pi(g)\ket{\Xi_{n,m|k,l}}\}_{g\in\mathfrak{S}_{n}}$, which is characterized by a slightly different quantity. As explained in Section 7, the number of distinguishable elements with error probability $\varepsilon>0$ is approximated by the following value:

First, we discuss the case with the decohered state $\rho_{mix,N+M,M}$. Then, we have

To consider the relation with $S\bigl{(}\operatorname{av}_{\pi}[\Xi_{n,m|k,l}]\bigr{)}$, we consider the pinching map ${\cal E}$ defined as

which expresses the decoherence process. Since the pinching map ${\cal E}$ satisfies the covariance condition (10) with respect to permutation, the relation (11) yields the inequality

This inequality also follows from the inequality (21) for the amount of asymmetry activation. Therefore, the Dicke state realizes a larger amount of asymmetry than the decohered state. However, this inequality does not explain how much the Dicke state improves it over the decohered state. To answer this question, we need to consider the asymptotic behavior.

Here, we consider the number $M(\left|1^{l}\,0^{k-l}\right>\!\left<1^{l}\,0^{k-l}\right|\otimes\rho_{mix,N+M,M},\epsilon)$ as well. Counting the possible combinatorics, we find that the number of discriminated states is calculated as $M(\left|1^{l}\,0^{k-l}\right>\!\left<1^{l}\,0^{k-l}\right|\otimes\rho_{mix,N+M,M},0)=\binom{l+M}{k+N+M}/\binom{M}{N+M}$. Since the set $\{\pi(g)\left|1^{l}\,0^{k-l}\right>\!\left<1^{l}\,0^{k-l}\right|\otimes\rho_{mix,N+M,M}\pi(g)^{\dagger}\}_{g\in\mathfrak{S}_{n}}$ is composed of orthogonal states, due to (12), the difference between $M(\left|1^{l}\,0^{k-l}\right>\!\left<1^{l}\,0^{k-l}\right|\otimes\rho_{mix,N+M,M},0)$ and $M(\left|1^{l}\,0^{k-l}\right>\!\left<1^{l}\,0^{k-l}\right|\otimes\rho_{mix,N+M,M},\epsilon)$ with finite error probability $\epsilon$ is a constant. Hence, we discuss the asymptotics of the value given in (37).

In Type I limit, we have

That is, the amount of asymmetry is a constant for $n$ in this case. In Type II limit, we have

The relations (40) and (41) are shown in Appendix C

To derive the asymptotic behaviors of $S\bigl{(}\operatorname{av}_{\pi}[\Xi_{n,m|k,l}]\bigr{)}$ and $\log M({\Xi_{n,m|k,l}},\epsilon)$, we need the asymptotic analysis of our distribution $P_{n,m,k,l}$ under the limit $n\to\infty$. The paper [27] studied the following two cases.

1. 1.

   The limit $n\to\infty$ with $k,l$ and $m/n$ fixed.

2. 2.

   The limit $n\to\infty$ with $m/n$, $k/n$ and $l/n$ fixed.

We call them Type I and II limits, respectively. The details are given in Subsection 5.1 and Subsections 5.2. Hereafter we denote by $\operatorname{Pr}_{n}$ the probability for the distribution $P_{n,n\xi,n\kappa,n\alpha}$, and by $X$ a random variable distributed subject to the distribution. The contents of this section were already shown in the paper [27].