Stability of Bi-Partite Correlations in Symmetric N-Qubit States Under Deterministic Measurements

Abstract

In this paper, we analyze the distribution of bi-partite correlations in pure symmetric N-qubit states during local deterministic measurements, which ensure the same value of the reduced purities in the outcome states. It is analytically shown that all reduced purities grow in the process of deterministic measurements. This allows us to characterize the stability of bi-partite entanglement during the optimal correlation transfer under single-qubit measurements in the asymptotic limit $N\gg 1$.

1. Introduction

Different facets of quantum correlations [1,2,3] in the symmetric multi-qubit case have been extensively analyzed in the last decade [4,5,6,7,8,9,10]. Several global characteristics, requiring, in the general case, intricate optimization procedures, such as, e.g., geometrical entanglement [11,12,13] and localizable entanglement [14,15,16,17], become more accessible for such states as a consequence of their invariance under particle permutations. Valuable information about the structure of pure symmetric states can be extracted by analyzing the outcomes of local measurements [18,19]. One of the simplest but still important properties of N-qubit systems is the degree of correlation between any two of its parts, which, in the case of a pure state $|\psi \rangle$, can be assessed by the reduced m-th order purities,

$$p_m^{\left(N\right)}=\mathrm{Tr}{\left({\widehat{\rho }}_m^{\left(N\right)}\right)}^2,{\widehat{\rho }}_m^{\left(N\right)}={\mathrm{Tr}}_{N-m}|\psi \rangle \langle \psi |,$$

(1)

where Tr is the trace operator and ${\mathrm{Tr}}_l$ is the partial trace with respect to l qubits.

For symmetric states $p_m^{\left(N\right)}$, $m=1,\dots ,N-1$, exhibit global features of the system due to the invariance of partitions ${\mathcal{H}}_N={\mathcal{H}}_{N-m}\otimes {\mathcal{H}}_m$ concerning the choice of m particles out of N. The purities (1) are constrained as ${(m+1)}^{-1}\le p_m\le 1$. The lower bound corresponds to the complete entanglement between an m-particle cluster and the rest of the system, while the upper bound is reached only for the coherent-like states (the only symmetric factorized states). An appropriately normalized sum of the purities (1) quantifies the total amount of quantum correlations stored in a multipartite state [20,21,22], and also describes the spread of the (discrete) Q-function in the discrete phase-space [23]. In the case of symmetric states, such a measure acquires the form of a weighted sum,

$${\delta }_{\psi }^{\left(N\right)}=2^{-N}\sum _{m=0}^N{C}_m^N{p}_m^{\left(N\right)},$$

(2)

where the Binomial coefficients $C_m^N$ take into account the number of possibilities to choose m qubits out of N. ${\delta }_{\psi }^{\left(N\right)}=1$ for any factorized state, and the lower values of (2) correspond to more correlated states.

Under a local von Neumann measurement involving the detection of a single-particle observable $\mathbf{n}·\widehat{\sigma }$, being $\widehat{\sigma }=({\widehat{\sigma }}_x,{\widehat{\sigma }}_y,{\widehat{\sigma }}_z)$, the Pauli operators, an N-partite symmetric state, is probabilistically mapped into one of two possible $N-1$ partite symmetric states, which is, in principle, distinct from the original state entanglement properties. A particular case of such an operation is a deterministic measurement protocol [24] consisting of a (probabilistic) projection of an initial N-qubit state into two orthogonal states (eigenstates of $\mathbf{n}·\widehat{\sigma }$),

$${|\psi \rangle }_N\Rightarrow \left\{\begin{array}{cc}|{\psi }_{\mathbf{n}}{\rangle }_{N-1}& \\ |{\psi }_{{\mathbf{n}}_{\perp }}{\rangle }_{N-1}\end{array}\right.,N-1{\langle {\psi }_{{\mathbf{n}}_{\perp }}|{\psi }_{\mathbf{n}}\rangle }_{N-1}=0,$$

where the measurement basis $\mathbf{n}$ is chosen in such a way that the resulting pure $N-1$ qubit states have the same and the maximum possible degree of correlation, according to a fixed measure. The deterministic procedure ensures an optimal correlation transfer through local measurements in the sense that a required physical property of a single-qubit measurement is independent of the particular outcome.

A relevant question in this respect is how the m-order purities are altered in the course of the deterministic measurements, i.e., when the purities $p_m^{(N-1)}\left({\mathbf{n}}_d\right)$ of the output states are selected as the hallmark of quantum correlations, ${\mathbf{n}}_d$ being the optimal measurement basis. The robustness of the purities quantifies the structural stability of symmetric states under optimal local measurements, allowing the characterization of the efficiency of conversion of high-order correlations contained in an initial N-qubit state into their lower-order counterparts captured by outcome $\left(N-1\right)$ partite states.

In this paper, we analyze the stability of the symmetric N-qubit purities (1) under deterministic measurements. We prove that all the deterministic purities $p_m^{(N-1)}\left({\mathbf{n}}_d\right)=p_m^{(N-1)}\left({\mathbf{n}}_{d\perp }\right)$, $m=1,\dots ,N-1$, are greater than their counterparts in the original state. The optimal measurement directions, i.e., those that minimize the deterministic purities $p_m^{(N-1)}\left({\mathbf{n}}_d\right)$, depend on the original state and, in general, are different for every order m. However, for a wide class of states, there is a measurement direction that simultaneously optimizes all the deterministic purities in the outcome $N-1$ partite states. The relative variation of the m-order purities, characterizing the robustness of an N-qubit state under the optimal measurement, either turn to zero or tend to a constant value in the macroscopic limit $N\gg 1$, and this indicates the stability/instability of the distrubution of quantum correlations in a given state.

2. Purities of Measured States

An arbitrary pure symmetric state expanded in the Dicke basis,

$$|k;N\rangle ={\left(C_k^N\right)}^{-1/2}\sum _{\mathrm{perm}}\mathit{P}\left({|1\rangle }_1{\dots |1\rangle }_k{|0\rangle }_{k+1}\dots {|0\rangle }_N\right),k=0,\dots ,N,$$

(3)

as

$${|\psi \rangle }_N=\sum _{k=0}^N{a}_{k,N}|k,N\rangle ,\sum _{k=0}^N{\left|a_{k,N}\right|}^2=1,$$

(4)

are characterized by the reduced m-partite density matrices ${\widehat{\rho }}_m^{\left(N\right)}$ (1),

$${\widehat{\rho }}_m^{\left(N\right)}=\sum _{k_1,k_2}{a}_{k_1,N}{a}_{k_2,N}^{*}{\left[C_{k_1}^N{C}_{k_2}^N\right]}^{-1/2}{\widehat{\Omega }}_{k_1,k_2}^{\left(m\right)},$$

(5)

where

$${\widehat{\Omega }}_{k_1,k_2}^{\left(m\right)}=\sum _{n=0}^m{C}_{k_2-n}^{N-m}\sqrt{C_n^m{C}_{k_1-k_2+n}^m}\left|k_1-k_2+n,m\right.〉\left.〈n,m\right|={\widehat{\Omega }}_{k_2,k_1}^{\left(m\right)†},$$

with $\left|n,m\right.〉$ being the Dicke basis, and the corresponding purities $p_m^{\left(N\right)}$ (see Appendix A for explicit expressions), whose weighted sum (2) describes one of the global facets of quantum correlations contained in the state.

In particular, it is immediate to obtain:

(a)

for W state (the entangled quantum state of N qubits) $|k=1;N\rangle$,

$$p_m^{\left(N\right)}={\displaystyle \frac{N^2-2Nm+2m^2}{N^2}},{\delta }_{\mathrm{W}}^{\left(N\right)}={\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2N}};$$

(b)

for the Greenberger–Horne–Zeilinger state (GHZ state) $|\mathrm{GHZ}\rangle \sim |000\dots \rangle +|111\dots \rangle$,

$$p_m^{\left(N\right)}={\displaystyle \frac{1+{\delta }_{m,N}}{2}},{\delta }_{\mathrm{GHZ}}^{\left(N\right)}={\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2^N}},$$

where ${\delta }_{a,b}$ is the Kronecker delta.

A single-qubit measurement implies a projection of ${|\psi \rangle }_N$ into a state,

$$|\mathbf{n}(\theta ,\varphi )\rangle =cos{\displaystyle \frac{\theta }{2}}|0\rangle +e^{i\varphi }sin{\displaystyle \frac{\theta }{2}}|1\rangle ,$$

(6)

yielding a symmetric ($N-1$)-partite pure state,

$$\begin{array}{cc}\hfill |{\psi }_{\mathbf{n}}{\rangle }_{N-1}& { = }_1\langle \mathbf{n}(\theta ,\varphi )|\psi \rangle {\mathcal{P}}_{\mathbf{n}}^{-1/2}=\sum _{k=0}^{N-1}{a}_{kN-1}\left(\mathbf{n}\right)|k,N-1\rangle ,\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill a_{kN-1}\left(\mathbf{n}\right)& =& \left[b_{kN}cos{\displaystyle \frac{\theta }{2}}+c_{k+1N}{e}^{-i\varphi }sin{\displaystyle \frac{\theta }{2}}\right]{\mathcal{P}}_{\mathbf{n}}^{-1/2},\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill b_{kN}& =& a_{kN}\sqrt{{\displaystyle \frac{N-k}{N}}},c_{kN}=a_{kN}\sqrt{{\displaystyle \frac{k}{N}}},\hfill \end{array}$$

(9)

which is detected with the probability

$${\mathcal{P}}_{\mathbf{n}}=\sum _{k=0}^{N-1}{\left|b_{kN}cos{\displaystyle \frac{\theta }{2}}+c_{k+1N}{e}^{-i\varphi }sin{\displaystyle \frac{\theta }{2}}\right|}^2,$$

while the projection into the orthogonal state $|{\mathbf{n}}_{\perp }(\theta ,\varphi )\rangle$, ${\mathbf{n}}_{\perp }(\theta ,\varphi )=\mathbf{n}(\theta +\pi ,\varphi )$ produces the outcome

$$|{\psi }_{{\mathbf{n}}_{\perp }}\rangle =\sum _{k=0}^{N-1}{a}_{kN-1}\left({\mathbf{n}}_{\perp }\right)|k,N-1\rangle ,$$

(10)

with the probability ${\mathcal{P}}_{{\mathbf{n}}_{\perp }}=1-{\mathcal{P}}_{\mathbf{n}}$.

Bearing in mind that the resulting state is independent of the measured qubit, we suppose that the first qubit is measured. It is worth noting that the density matrix of the measured state (7) can be represented in the following compact form,

$${\widehat{\rho }}^{(N-1)}\left(\mathbf{n}\right)=|{\psi }_{\mathbf{n}}{\rangle }_{N-1}\langle {\psi }_{\mathbf{n}}|={\widehat{\rho }}_{N-1}^{\left(N\right)}+{\mathrm{Tr}}_1\left(\widehat{\rho }{\widehat{\Sigma }}_{\mathbf{n}}\right),$$

(11)

where the following operator,

$$\begin{array}{ccc}\hfill {\widehat{\Sigma }}_{\mathbf{n}}& =& {\displaystyle \frac{1}{2{\mathcal{P}}_{\mathbf{n}}}}\left(\mathbf{n}·\widehat{\sigma }+\widehat{I}\right)-\widehat{I},\hfill \end{array}$$

(12)

$$\begin{array}{ccc}\hfill {\widehat{\Sigma }}_{\mathbf{n}}{\mathcal{P}}_{\mathbf{n}}& =& -{\widehat{\Sigma }}_{{\mathbf{n}}_{\perp }}{\mathcal{P}}_{{\mathbf{n}}_{\perp }},\hfill \end{array}$$

(13)

is applied to the first (measured) qubit. Then, the m-partite density matrices (5) of the measured state take the form

$${\widehat{\rho }}_m^{(N-1)}\left(\mathbf{n}\right)={\mathrm{Tr}}_{N-1-m}{\widehat{\rho }}^{(N-1)}\left(\mathbf{n}\right)={\widehat{\rho }}_m^{\left(N\right)}+{\widehat{\zeta }}_m^{(N-1)}\left(\mathbf{n}\right),$$

where in the m-partite measurement the defect operator is

$${\widehat{\zeta }}_m^{(N-1)}\left(\mathbf{n}\right)={\mathrm{Tr}}_{N-m}\left(\widehat{\rho }{\widehat{\Sigma }}_{\mathbf{n}}\right)={\widehat{\zeta }}_m^{(N-1)†}\left(\mathbf{n}\right),$$

(14)

(certainly, the operator ${\widehat{\Sigma }}_{\mathbf{n}}$ can be applied to any $N-m$ traced qubits). Thus, the purities of the original and the measured states are related as follows,

$$p_m^{(N-1)}\left(\mathbf{n}\right)=p_m^{\left(N\right)}+q_m^{(N-1)}\left(\mathbf{n}\right)+2\mathrm{Tr}\left({\widehat{\rho }}_m^{\left(N\right)}{\widehat{\zeta }}_m^{(N-1)}\left(\mathbf{n}\right)\right),$$

(15)

where $q_m^{(N-1)}\left(\mathbf{n}\right)=\mathrm{Tr}{\left({\widehat{\zeta }}_m^{(N-1)}\left(\mathbf{n}\right)\right)}^2$ is the “purity” of the operator (14).

3. Deterministic Purities

Let us establish a deterministic measurement condition in such a way that an m-partite purity in both outcome states (7) and (10) are the same,

$$p_m^{(N-1)}\left({\mathbf{n}}_d\right)=p_m^{(N-1)}\left({\mathbf{n}}_{\mathbf{d}\perp }\right)=p_{m,d}^{(N-1)}\left({\mathbf{n}}_{\mathbf{d}}\right).$$

(16)

In Appendix B, we prove the following

Theorem 1.

An m-partite deterministic purity is never smaller than the corresponding purity in the original state,

$$p_m^{(N-1)}\left({\mathbf{n}}_d\right)\ge p_m^{\left(N\right)}.$$

(17)

In other words, an m-partite cluster in a deterministically measured state is always less or equally correlated with the rest of the qubits compared to the initial state. This is a strong statement since, without imposing the deterministic condition, the measured state in one of the outputs, in general, may have a smaller purity than the original one.

Corollary 1.

(see Appendix C): Fixing the global correlation measure (2) as a deterministic parameter, i.e.,

$$\begin{array}{ccc}\hfill {\delta }_{\psi }^{(N-1)}\left({\mathbf{n}}_d\right)& =& {\delta }_{\psi }^{(N-1)}\left({\mathbf{n}}_{\mathbf{d}\perp }\right),\hfill \\ \hfill {\delta }_{\psi }^{(N-1)}\left(\mathbf{n}\right)& =& 2^{-(N-1)}\sum _{m=0}^{N-1}{C}_m^{N-1}{p}_m^{(N-1)}\left(\mathbf{n}\right),\hfill \end{array}$$

(18)

one has,

$${\delta }_{\psi }^{(N-1)}\left({\mathbf{n}}_d\right)\ge {\delta }_{\psi }^{\left(N\right)},$$

(19)

where the measurement directions ${\mathbf{n}}_d$ here are, in general, different from those fixed by the purity conditions (16).

Thus, the total amount of correlations quantified by the correlation measure (2) cannot increase in the deterministic measurement scheme.

The optimal measurement protocol consists of the deterministic transmission of a maximum possible amount of correlations from N to $N-1$ partite states, requiring a minimization of purities in the measured states, i.e.,

$$p_{m,d}^{(N-1)}=\underset{\mathbf{n}}{min}{p}_{m,d}^{(N-1)}\left({\mathbf{n}}_{\mathbf{d}}\right).$$

(20)

The optimal deterministic measurement directions ${\mathbf{n}}_d$, ensuring the deterministic conditions (16), in general, depend both on the state and on the partition. A relevant property of an N-partite state is the robustness of the m-partite purities, which can be quantified in terms of relative variances,

$$s_m={\displaystyle \frac{p_{m,d}^{(N-1)}-p_m^{\left(N\right)}}{p_m^{\left(N\right)}}},m=0,\dots ,N-1.$$

(21)

This characteristic is especially useful in the case of a large number of qubits separating from stable states, with ${\lim }_{N\to \infty }{s}_m=0$, in which correlation properties are largely maintained under deterministic projections, from unstable states, with non-vanishing values of the variances (21) for a significant number of partitions.

In Figure 1 we plot the relative variances (21) for the state

$$|\psi \rangle =\sqrt{1-x^2}|k=15;N=31\rangle +x|k=0;N=31\rangle ,$$

where the optimum measurement direction depends on the parameter x. It is worth noting that the measurement of ${\widehat{\sigma }}_y$ (instead of minimizing each of the deterministic purities separately) leads to quite a similar (smoothed) form of the optimal $s_m\left(x\right)$. This relation is not casual and is discussed in Section 4.

4. Minimizing the Deterministic Purities

A search for deterministic measurement directions and especially those directions that minimize the deterministic purities (20) for generic symmetric states is an involved optimization problem. However, for real symmetric states, $\mathrm{Im}\left(a_{k,N}\right)=0$ in Equation (4), there exists a particular measurement direction that optimizes the m-purities alltogether. In Appendix D, we prove the following:

Theorem 2.

The measurements of the ${\widehat{\sigma }}_y$ operator satisfy the deterministic conditions (16) simultaneously for all partitions, and the purities $p_{m,d}^{(N-1)}\left({\mathbf{n}}_{\mathbf{d}}\right)$ reach their extreme values (minima or maxima) for real symmetric N-qubit states (rebits).

In this case,

$$a_{kN-1}\left({\mathbf{n}}_d\right)=b_{kN}-i{c}_{k+1N}=a_{kN-1}^{*}\left({\mathbf{n}}_{\mathbf{d}\perp }\right),$$

and the measurement probabilities are the same in both channels,

$${\mathcal{P}}_{\mathbf{n}}={\mathcal{P}}_{{\mathbf{n}}_{\perp }}=1/2.$$

Certainly, the global correlation measure (2) also takes either (local) minimum or maximum value when measuring a single qubit in the $n_y$-direction.

In addition, the deterministic purities $p_{m,d}^{(N-1)}\left(n_y\right)$ achieve the global minima for a broad class of symmetric N-rebit states (see Appendix D). Among such states are all elements of the Dicke basis (3), which are optimally deterministically projected into their superpositions,

$$|k;N\rangle \to \sqrt{{\displaystyle \frac{N-k}{N}}}\left|k;N-1\right.〉\pm i\sqrt{{\displaystyle \frac{k}{N}}}\left|k-1;N-1\right.〉.$$

In particular, for the W state, one has

$$p_{m,d}^{(N-1)}\left(n_y\right)={\displaystyle \frac{N^2-2Nm+2m^2+2m}{N^2}},{\delta }_{\mathrm{W}}^{(N-1)}\left(n_y\right)\approx {\displaystyle \frac{1}{2}}+{\displaystyle \frac{3}{2N}}.$$

One can observe that the relative variances (21),

$$s_m={\displaystyle \frac{2m}{{(N-m)}^2+m^2}},$$

tend to zero for all partitions, which means high stability of the W states under the optimal measurements in the macroscopic limit $N\gg 1$. The situation is essentially different for the Dicke states with a significant portion of excited qubits. In Figure 2, we plot the variances (21) for the states $|k;N\rangle$. One can observe that, for states $|k\sim \alpha N;N\rangle$, $N^{-1}\lesssim \alpha$, and $1-\alpha \lesssim N^{-1}$, the most of partitions, $s_m$ is a nonvanishing constant tending to the maximum value $\sim 1/2$ when $N\to \infty$ at $\alpha \sim 1/2$, which indicates a drastic change of correlation properties in the measured state with respect to the initial one.

From this perspective, it is worth observing in Figure 1 the translation from stable states, close to the factorized state $|000\dots \rangle$ ($1-\left|x\right|\ll 1$) to unstable states close to $|k=15;N=31\rangle$, which is reflected in an abrupt change from a “plato-form” of the variances $s_m$ to their rapid decreasing.

Another typical state where the detection of the ${\widehat{\sigma }}_y$ operator minimizes all the purities is the family $sin\vartheta |000\dots \rangle +cos\vartheta |111\dots \rangle$ (and in particular, the GHZ state), with $p_{m,d}^{(N-1)}\left(n_y\right)=\left(3+cos4\vartheta \right)/4$, for all $N\ge 3$, which is completely stable under optimal measurements, $s_m=0$.

5. Conclusions

The concept of deterministic measurements allows the introduction of novel characteristics (21) describing the stability of symmetric states in the process of an optimal correlation transfer from N to $N-1$ partite states. Despite the optimal measurement direction, minimizing a particular deterministic purity $p_{m,d}^{(N-1)}\left({\mathbf{n}}_{\mathbf{d}}\right)$, in general, depends on the state; there is a class of rebit states where the detection of the ${\widehat{\sigma }}_y$ observable not only ensures the same values of all m-partite purities in both outcome channels but also provides their minimum possible values, i.e., guarantees the maximization of all bi-partite correlations in the measured state. The stability of this type of states can be analyzed analytically in the limit of a large number of qubits, providing, for instance, an unconventional description of the Dicke basis elements (3) with respect to measurement protocols. It is interesting to note (as a result of numerical calculations) that the stability of bi-partite correlations for a broad class of symmetric states can be approximately estimated by measuring the ${\widehat{\sigma }}_y$ operator (see, e.g., Figure 1), which significantly simplifies their analysis.

The robustness of the distribution of quantum correlations undergoing a measurement-induced correlation transfer procedure unveils an aspect of multipartite states alternative, e.g., to the stability of states under decoherence processes. For instance, the GHZ state is measurement-stable but considerably unstable under losses. The parameter (21) becomes especially relevant in the macroscopic limit, indicating the degree of preservation of the global structure of a symmetric state after a sequence of (optimal) local measurements. The robust states are characterized by relative variances vanishing in the limit $N\gg 1$, and thus, admit further measurements without significantly modifying the correlations between its bi-partitions. In contrast, even a single projection of unstable N-partite states into the maximally correlated $N-1$ partite subspaces drastically changes the structure of output states.