N-partite Entanglement Measures of GHZ States in a Non-inertial Frame

Abstract

Entanglement measures of GHZ states in a non-inertial frame are analyzed. The form of the GHZ states of n entangled qubits where q of them are non-inertial observers is studied. Some generalities of the entanglement measures of GHZ states are derived. The entanglement measures depend on a parameter $r\in [0,\pi /4]$ associated with the acceleration $a\in [0,\infty ]$ and on the number of non-inertial observers. It was observed that the negativity $N_{1-(n-1)}$ is the same in GHZ states with the same number of non-inertial observers as long as there is at least one inertial qubit in the $(n-1)$ modes. The whole residual entanglement of GHZ states with q non-inertial observers can be increased up to $\cos ^{2q}(r)$ by increasing the number of inertial entangled qubits in it, i.e. $n\rightarrow \infty$. It is observed that $\cos ^{2q}(r)$ is a good approximation of the whole residual entanglement for $q \gg 1$. Using the latter, it is observed that at infinite acceleration any GHZ state with $q \ge 4$ has an entanglement close to $10\%$ of the fully inertial GHZ state, and with $n \ge 7$, this is less than $1\%$, which should be considered in quantum network protocols. Regarding the entropy, it was found that this is a function of a parameter r, and of the number q of accelerated observers, but not of the number of entangled qubits. Finally, a formula to calculate the entropy of GHZ states in a non-inertial frame also was found.

Appendices

Appendix A. Toolbox for Quantum Basic Operations

This code was developed in Mathematica11.3 in order to help in calculating the state vectors, the density matrices, the tracing over inaccessible regions, the transpose and the entropies in this work.

To define a direct product function it was used:

Defining a state vector using the DirectProduct function (tip: also can be used an infix notation with $\otimes$)

A vector obtained in this way can be stored with SparseArray to optimise calculations with long matrices

For calculating a density matrix can be used the KroneckerProduct function

In order to trace out over the (n-1) first qubits, was developed this code:

For tracing out over last qubit it’s necessary to use this code

Function to transpose the n-1 first qubits of a system.

Formula to transpose the last qubit of a system

The negativity can be calculated using this script

where IsNegativeFunction1D is defines as

Finally, to calculate von Neumann entropy it was developed

Appendix B. Form of GHZ States in a Non-inertial Frame

To better understand the results presented in this paper, it is necessary to examine the form of the GHZ states in a non-inertial frame, where a fully inertial GHZ state vector is a column vector of the form

$$\begin{aligned} | \mathrm GHZ \rangle _{1,2,\dots ,n}= \frac{1}{\sqrt{2}}\left( | 0_10_2\dots 0_n \rangle +| 1_11_2\dots 1_n \rangle \right) = \left( \frac{1}{\sqrt{2}},0,\dots ,0,\frac{1}{\sqrt{2}}\right) ^T \end{aligned}$$

(29)

where $| \mathrm GHZ \rangle _{1,2,\dots ,n}$ has $2^{n}$ entries and there are only two nonzero terms. Then, the n-th observer becomes non-inertial, so the column vector becomes

$$\begin{aligned} | \mathrm GHZ \rangle _{1,2,\dots ,n_{\textrm{I}},n_{\textrm{II}}}=\left( \frac{\cos (r)}{\sqrt{2}},0,\frac{\sin (r)}{\sqrt{2}},0,\dots ,0,\frac{1}{\sqrt{2}},0\right) ^T \end{aligned}$$

(30)

where $| \mathrm GHZ \rangle _{1,2,\dots ,n_{\textrm{I}},n_{\textrm{II}}}$ has $2^{n+1}$ entries and three non-zero terms. Then, as more qubits become non-inertial, the length of the column vector increases to $2^{(n+q)}$ where $2^{q}+1$ terms are nonzero. These terms are of the form $\cos ^{\alpha }(r)\sin ^{\beta }(r)$ and there is $2^{q}$ blocks of zeros of different length that separate them. For example, a heptapartite system with four non-inertial qubits is

$$\begin{aligned}&| GHZ \rangle _{1,2,3,4_{\textrm{I}},4_{\textrm{II}},5_{\textrm{I}},5_{\textrm{II}},6_{\textrm{I}},6_{\textrm{II}},7_{\textrm{I}},7_{\textrm{II}}}=\frac{1}{\sqrt{2}}\left( \cos ^4(r),\underbrace{0}_{2},\cos ^3(r)\sin (r),\underbrace{0}_{8},\right. \\&\left. \cos ^3(r)\sin (r),\underbrace{0}_{2},\cos ^2(r)\sin ^2(r),\underbrace{0}_{32},\cos ^3(r)\sin (r),\underbrace{0}_{2},\cos ^2(r)\sin ^2(r),\underbrace{0}_{8},\right. \\&\left. \cos ^2(r)\sin ^2(r),\underbrace{0}_{2},\cos (r)\sin ^3(r),\underbrace{0}_{128},\cos ^3(r)\sin (r),\underbrace{0}_{2},\cos ^2(r)\sin ^2(r),\underbrace{0}_{8},\right. \\&\left. \cos ^2(r)\sin ^2(r),\underbrace{0}_{2},\cos (r)\sin ^3(r),\underbrace{0}_{32},\cos ^2(r)\sin ^2(r),\underbrace{0}_{2},\cos (r)\sin ^3(r),\underbrace{0}_{8},\right. \\&\left. \cos (r)\sin ^3(r),\underbrace{0}_{2},\sin ^4(r),0,0,\dots ,0,0,\frac{1}{\sqrt{2}},\underbrace{0}_{85}\right) ^T \end{aligned}$$

(31)

where the labels under the zeroes indicate the number of actual zeroes within that particular block of zeroes. The blocks of zeros follow a sequence: blocks of 2 zeros alternate every 2 terms, blocks of 8 zeros alternate every 4 terms, blocks of 32 zeros alternate every 8 terms, and so on. Thta is, it is observed that the blocks of zeros are of length $2^1,2^3,...,2^{(2q-1)}$. These alternate every $2^1,2^2,2^3,\dots ,2^{(q-1)}$ terms, respectively. The number of zeros after the $1/\sqrt{2}$ term also presents a regularity. This can be calculated with the formula

$$\begin{aligned} \sum \limits _{i=0}^{q-1} 2^{2 i} \end{aligned}$$

(32)

Now, it is considered the $\cos ^{\alpha }(r)\sin ^{\beta }(r)$ terms. These present a regularity in the $\alpha$ and $\beta$ powers. It is possible to obtain these sequences by noting that for $q=1$, these are $\{1,0\}$, for $q=2$ they are $\{2,1,1,0\}$, for $q=3$ they are

$\{3,2,2,1,2,1,1,0\}$, for $q=4$ it is possible to define the matrix

$$\begin{aligned} M^{(4)}=\begin{pmatrix} 4&{}3&{}3&{}2\\ 3&{}2&{}2&{}1\\ 3&{}2&{}2&{}1\\ 2&{}1&{}1&{}0\\ \end{pmatrix} \end{aligned}$$

(33)

where it is possible to read the correct sequence of powers of cosine. For $q=5$ it can be defined

$$\begin{aligned} M^{(5)}=\begin{pmatrix} 5&{}4&{}4&{}3\\ 4&{}3&{}3&{}2\\ 4&{}3&{}3&{}2\\ 3&{}2&{}2&{}1\\ \end{pmatrix}~~,~~ M^{(4)}=\begin{pmatrix} 4&{}3&{}3&{}2\\ 3&{}2&{}2&{}1\\ 3&{}2&{}2&{}1\\ 2&{}1&{}1&{}0\\ \end{pmatrix} \end{aligned}$$

(34)

for $q=6$ are $\{M^{(6)},M^{(5)},M^{(5)},M^{(4)}\}$, for $q=7$ these are $\{M^{(7)},M^{(6)},M^{(6)},M^{(5)},$$M^{(6)},M^{(5)},M^{(5)},M^{(4)}\}$, and for $q=8$ it is possible to define the matrix

$$\begin{aligned} \begin{pmatrix} M^{(8)}&{}M^{(7)}&{}M^{(7)}&{}M^{(6)}\\ M^{(7)}&{}M^{(6)}&{}M^{(6)}&{}M^{(5)}\\ M^{(7)}&{}M^{(6)}&{}M^{(7)}&{}M^{(5)}\\ M^{(6)}&{}M^{(5)}&{}M^{(5)}&{}M^{(4)}\\ \end{pmatrix} \end{aligned}$$

(35)

where again, it is possible to read the correct sequence of powers of cosine. It is easy to see that this pattern is followed for larger q. After this, the sequence of powers of sines can be obtained by subtracting each term $M_{i,j}^{(q)}$ from 2q.

It is necessary to trace out over the antiparticle states in the region $\textrm{II}$ of the density matrix $\rho _{1,2,\dots ,n_{\textrm{I}},n_{\textrm{II}}}=| \mathrm GHZ \rangle _{1,2,\dots ,n_{\textrm{I}},n_{\textrm{II}}}\langle \mathrm GHZ |$. The partial trace over the inaccessible region $\textrm{II}$ maps all non-zero terms on the diagonal of a $(2^{n-1}$x$2^{n-1})$ matrix ($\rho _{1,2,\dots ,n_{\textrm{I}}}$). This procedure removes all blocks of zeroes and most of the coherences. Therefore, the general form of a traced density matrix is

$$\begin{aligned} \left( \begin{array}{cccccc} \frac{\cos ^{2q}(r)}{2} &{} 0 &{} \dots &{} 0 &{} 0 &{} \frac{\cos ^q (r)}{2} \\ 0 &{} \frac{\cos ^{M^{(q)}_{i,j}}\!(r)\sin ^{2q-M^{(q)}_{i,j}}\!(r)}{2}&{}0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} \ddots &{}0 &{} 0 &{} 0 \\ 0&{} 0 &{} 0 &{} \frac{\sin ^{2q}(r)}{2}&{} 0 &{} \vdots \\ \vdots &{} 0 &{} 0 &{} 0&{} \dots &{} 0 \\ \frac{\cos ^q (r)}{2} &{} 0 &{} \dots &{} 0 &{} 0 &{} \frac{1}{2} \\ \end{array} \right) \end{aligned}$$

(36)

where $M^{(q)}_{i,j}$ refers to the sequence of terms that follow the powers of cosine. It must be noted that the terms $\cos ^q(r)/2$ in the lower left and upper right corners are a property of the density matrices of GHZ states in a non-inertial frame.