Remote preparation of four-qubit states via two-qubit maximally entangled states

Abstract

In this paper, an efficient scheme of remotely preparing four-qubit quantum states via two-qubit maximally entangled states is proposed. This scheme can be accomplished by using appropriate unitary transformations and some classical communication. The detailed processes for preparation of four-qubit states are presented in the general case and two special cases, respectively. The method of constructing special measurement basis in general case is provided, and the similar methods for real-parameter state case and equatorial state case are also discussed. Meanwhile, the probabilities of successful preparation under each case are calculated in our schemes. The results show that the successful probability is only 1 / 16 in the general case, and the probability can reach up to 100% when the relative phase factors are zero or the amplitude parameters are 1 / 4.

Appendix

The parameter matrix $M_4$ used for constructing projective measurement basis in preparing arbitrary four-qubit state is shown as below. It can be found that the matrix in three-qubit preparation given by Eq. (12) exactly forms the blue elements in the right upper corner in $M_4$.

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The measurement basis $\{|\tau _i\rangle |i=0,1,\ldots ,15\}$ used in preparing four-qubit real-parameter state mentioned in Sect. 3.1 is shown as below

$$\begin{aligned} \left( \begin{array}{c} |\tau _0\rangle \\ |\tau _1\rangle \\ |\tau _2\rangle \\ |\tau _3\rangle \\ |\tau _4\rangle \\ |\tau _5\rangle \\ |\tau _6\rangle \\ |\tau _7\rangle \\ |\tau _8\rangle \\ |\tau _9\rangle \\ |\tau _{10}\rangle \\ |\tau _{11}\rangle \\ |\tau _{12}\rangle \\ |\tau _{13}\rangle \\ |\tau _{14}\rangle \\ |\tau _{15}\rangle \end{array} \right) = \left( \begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }r@{\quad }r@{\quad }r@{\quad }r@{\quad }r} a_0 &{} a_1 &{} a_2 &{} a_3 &{} a_4 &{} a_5 &{} a_6 &{} a_7 &{} a_8\\ a_1 &{} -a_0 &{} a_3 &{} -a_2 &{} a_5 &{} -a_4 &{} -a_7 &{} a_6 &{} a_9\\ a_2 &{} -a_3 &{} -a_0 &{} a_1 &{} a_6 &{} a_7 &{} -a_4 &{} -a_5 &{} a_{10}\\ a_3 &{} a_2 &{} -a_1 &{} -a_0 &{} a_7 &{} -a_6 &{} a_5 &{} -a_4 &{} a_{11}\\ a_4 &{} -a_5 &{} -a_6 &{} -a_7 &{} -a_0 &{} a_1 &{} a_2 &{} a_3 &{} a_{12}\\ a_5 &{} a_4 &{} -a_7 &{} a_6 &{} -a_1 &{} -a_0 &{} -a_3 &{} a_2 &{} a_{13}\\ a_6 &{} a_7 &{} a_4 &{} -a_5 &{} -a_2 &{} a_3 &{} -a_0 &{} -a_1 &{} a_{14}\\ a_7 &{} -a_6 &{} a_5 &{} a_4 &{} -a_3 &{} -a_2 &{} a_1 &{} -a_0 &{} a_{15}\\ a_8 &{} -a_9 &{} -a_{10} &{} -a_{11} &{} -a_{12} &{} -a_{13} &{} -a_{14} &{} -a_{15} &{} -a_0 \\ a_9 &{} a_8 &{} -a_{11} &{} a_{10} &{} -a_{13} &{} a_{12} &{} a_{15} &{} -a_{14} &{} -a_1 \\ a_{10}&{} a_{11} &{} a_8 &{} -a_9 &{} -a_{14} &{} -a_{15} &{} a_{12} &{} a_{13} &{} -a_2 \\ a_{11}&{}-a_{10} &{} a_9 &{} a_8 &{} a_{15} &{} -a_{14} &{} a_{13} &{} a_{12} &{} -a_3 \\ a_{12}&{} a_{13} &{} a_{14} &{} -a_{15} &{} a_8 &{} a_9 &{} -a_{10} &{} -a_{11} &{} -a_4 \\ a_{13}&{}-a_{12} &{} a_{15} &{} a_{14} &{} -a_9 &{} -a_8 &{} a_{11} &{} a_{10} &{} -a_5 \\ a_{14}&{}-a_{15} &{} -a_{12} &{} -a_{13} &{} a_{10} &{} -a_{11} &{} a_8 &{} a_9 &{} -a_6 \\ a_{15}&{} a_{14} &{} -a_{13} &{} -a_{12} &{} a_{11} &{} -a_{10} &{} -a_9 &{} a_8 &{} -a_7 \\ \end{array}\right. \nonumber \\ \left. \begin{array}{r@{\quad }r@{\quad }r@{\quad }r@{\quad }r@{\quad }r@{\quad }r} a_9 &{} a_{10} &{} a_{11} &{} a_{12} &{} a_{13} &{} a_{14} &{} a_{15}\\ -a_8 &{} -a_{11} &{} a_{10} &{} -a_{13} &{} a_{12} &{} a_{15} &{} -a_{14}\\ a_{11} &{} -a_8 &{} -a_9 &{} -a_{14} &{} -a_{15} &{} a_{12} &{} a_{13}\\ -a_{10} &{} a_9 &{} -a_8 &{} a_{15} &{} -a_{14} &{} a_{13} &{} a_{12}\\ a_{13} &{} a_{14} &{} -a_{15} &{} -a_8 &{} a_9 &{}-a_{10} &{} -a_{11}\\ -a_{12} &{} a_{15} &{} a_{14} &{} -a_9 &{} -a_8 &{} a_{11} &{} a_{10}\\ -a_{15} &{} -a_{12} &{} -a_{13} &{} a_{10} &{} -a_{11} &{}-a_8 &{} a_9\\ a_{14} &{} -a_{13} &{} -a_{12} &{} a_{11} &{} -a_{10} &{}-a_9 &{} -a_8\\ a_1 &{} a_2 &{} a_3 &{} a_4 &{} a_5 &{} a_6 &{} a_7\\ -a_0 &{} -a_3 &{} a_2 &{} -a_5 &{} a_4 &{} a_7 &{} -a_6\\ a_3 &{} -a_0 &{} -a_1 &{} -a_6 &{} -a_7 &{} a_4 &{} a_5\\ -a_2 &{} a_1 &{} -a_0 &{} -a_7 &{} -a_6 &{} a_5 &{} -a_4\\ a_5 &{} a_6 &{} a_7 &{} -a_0 &{} a_1 &{} -a_2 &{} a_3\\ -a_4 &{} a_7 &{} a_6 &{} -a_1 &{} -a_0 &{} -a_3 &{} -a_2\\ -a_7 &{} -a_4 &{} -a_5 &{} a_2 &{} a_3 &{} -a_0 &{} a_1\\ a_6 &{} -a_5 &{} a_4 &{} -a_3 &{} a_2 &{} -a_1 &{} -a_0\\ \end{array} \right) \left( \begin{array}{c} |0000\rangle \\ |0001\rangle \\ |0010\rangle \\ |0011\rangle \\ |0100\rangle \\ |0101\rangle \\ |0110\rangle \\ |0111\rangle \\ |1000\rangle \\ |1001\rangle \\ |1010\rangle \\ |1011\rangle \\ |1100\rangle \\ |1101\rangle \\ |1110\rangle \\ |1111\rangle \\ \end{array} \right) \qquad \end{aligned}$$

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The measurement basis $\{|\tau _i\rangle |i=0,1,\ldots ,15\}$ used in preparing four-qubit equatorial state mentioned in Sect. 3.2 is shown as below

$$\begin{aligned} |\tau _0\rangle= & {} 1{/}4(|0000\rangle +e^{i\phi _1}|0001\rangle +e^{i\phi _2}|0010\rangle +e^{i\phi _3}|0011\rangle \nonumber \\&+\,e^{i\phi _4}|0100\rangle +e^{i\phi _5}|0101\rangle +e^{i\phi _6}|0110\rangle +e^{i\phi _7}|0111\rangle \nonumber \\&+\,e^{i\phi _8}|1000\rangle +e^{i\phi _9}|1001\rangle +e^{i\phi _{10}}|1010\rangle +e^{i\phi _{11}}|1011\rangle \nonumber \\&+\,e^{i\phi _{12}}|1100\rangle +e^{i\phi _{13}}|1101\rangle +e^{i\phi _{14}}|1110\rangle +e^{i\phi _{15}}|1111\rangle )\nonumber \\ |\tau _1\rangle= & {} 1{/}4(|0000\rangle -e^{i\phi _1}|0001\rangle +e^{i\phi _2}|0010\rangle -e^{i\phi _3}|0011\rangle \nonumber \\&+\,e^{i\phi _4}|0100\rangle -e^{i\phi _5}|0101\rangle +e^{i\phi _6}|0110\rangle -e^{i\phi _7}|0111\rangle \nonumber \\&+\,e^{i\phi _8}|1000\rangle -e^{i\phi _9}|1001\rangle +e^{i\phi _{10}}|1010\rangle -e^{i\phi _{11}}|1011\rangle \nonumber \\&+\,e^{i\phi _{12}}|1100\rangle -e^{i\phi _{13}}|1101\rangle +e^{i\phi _{14}}|1110\rangle -e^{i\phi _{15}}|1111\rangle )\nonumber \\ |\tau _2\rangle= & {} 1{/}4(|0000\rangle -e^{i\phi _1}|0001\rangle -e^{i\phi _2}|0010\rangle +e^{i\phi _3}|0011\rangle \nonumber \\&+\,e^{i\phi _4}|0100\rangle +e^{i\phi _5}|0101\rangle -e^{i\phi _6}|0110\rangle -e^{i\phi _7}|0111\rangle \nonumber \\&+\,e^{i\phi _8}|1000\rangle +e^{i\phi _9}|1001\rangle -e^{i\phi _{10}}|1010\rangle -e^{i\phi _{11}}|1011\rangle \nonumber \\&-\,e^{i\phi _{12}}|1100\rangle -e^{i\phi _{13}}|1101\rangle +e^{i\phi _{14}}|1110\rangle +e^{i\phi _{15}}|1111\rangle )\nonumber \\ |\tau _3\rangle= & {} 1{/}4(|0000\rangle +e^{i\phi _1}|0001\rangle -e^{i\phi _2}|0010\rangle -e^{i\phi _3}|0011\rangle \nonumber \\&+\,e^{i\phi _4}|0100\rangle -e^{i\phi _5}|0101\rangle +e^{i\phi _6}|0110\rangle -e^{i\phi _7}|0111\rangle \nonumber \\&+\,e^{i\phi _8}|1000\rangle -e^{i\phi _9}|1001\rangle +e^{i\phi _{10}}|1010\rangle -e^{i\phi _{11}}|1011\rangle \nonumber \\&+\,e^{i\phi _{12}}|1100\rangle -e^{i\phi _{13}}|1101\rangle +e^{i\phi _{14}}|1110\rangle +e^{i\phi _{15}}|1111\rangle )\nonumber \\ |\tau _4\rangle= & {} 1{/}4(|0000\rangle -e^{i\phi _1}|0001\rangle -e^{i\phi _2}|0010\rangle -e^{i\phi _3}|0011\rangle \nonumber \\&-\,e^{i\phi _4}|0100\rangle +e^{i\phi _5}|0101\rangle +e^{i\phi _6}|0110\rangle +e^{i\phi _7}|0111\rangle \nonumber \\&+\,e^{i\phi _8}|1000\rangle +e^{i\phi _9}|1001\rangle +e^{i\phi _{10}}|1010\rangle -e^{i\phi _{11}}|1011\rangle \nonumber \\&-\,e^{i\phi _{12}}|1100\rangle +e^{i\phi _{13}}|1101\rangle -e^{i\phi _{14}}|1110\rangle -e^{i\phi _{15}}|1111\rangle )\nonumber \\ |\tau _5\rangle= & {} 1{/}4(|0000\rangle +e^{i\phi _1}|0001\rangle -e^{i\phi _2}|0010\rangle +e^{i\phi _3}|0011\rangle \nonumber \\&-\,e^{i\phi _4}|0100\rangle -e^{i\phi _5}|0101\rangle -e^{i\phi _6}|0110\rangle +e^{i\phi _7}|0111\rangle \nonumber \\&+\,e^{i\phi _8}|1000\rangle -e^{i\phi _9}|1001\rangle +e^{i\phi _{10}}|1010\rangle +e^{i\phi _{11}}|1011\rangle \nonumber \\&-\,e^{i\phi _{12}}|1100\rangle -e^{i\phi _{13}}|1101\rangle +e^{i\phi _{14}}|1110\rangle +e^{i\phi _{15}}|1111\rangle )\nonumber \\ |\tau _6\rangle= & {} 1{/}4(|0000\rangle +e^{i\phi _1}|0001\rangle +e^{i\phi _2}|0010\rangle -e^{i\phi _3}|0011\rangle \nonumber \\&-\,e^{i\phi _4}|0100\rangle +e^{i\phi _5}|0101\rangle -e^{i\phi _6}|0110\rangle -e^{i\phi _7}|0111\rangle \nonumber \\&+\,e^{i\phi _8}|1000\rangle -e^{i\phi _9}|1001\rangle -e^{i\phi _{10}}|1010\rangle -e^{i\phi _{11}}|1011\rangle \nonumber \\&+\,e^{i\phi _{12}}|1100\rangle -e^{i\phi _{13}}|1101\rangle -e^{i\phi _{14}}|1110\rangle +e^{i\phi _{15}}|1111\rangle )\nonumber \\ |\tau _7\rangle= & {} 1{/}4(|0000\rangle -e^{i\phi _1}|0001\rangle +e^{i\phi _2}|0010\rangle +e^{i\phi _3}|0011\rangle \nonumber \\&-\,e^{i\phi _4}|0100\rangle -e^{i\phi _5}|0101\rangle +e^{i\phi _6}|0110\rangle -e^{i\phi _7}|0111\rangle \nonumber \\&+\,e^{i\phi _8}|1000\rangle +e^{i\phi _9}|1001\rangle -e^{i\phi _{10}}|1010\rangle -e^{i\phi _{11}}|1011\rangle \nonumber \\&+\,e^{i\phi _{12}}|1100\rangle -e^{i\phi _{13}}|1101\rangle -e^{i\phi _{14}}|1110\rangle -e^{i\phi _{15}}|1111\rangle )\nonumber \\ |\tau _8\rangle= & {} 1{/}4(|0000\rangle -e^{i\phi _1}|0001\rangle -e^{i\phi _2}|0010\rangle -e^{i\phi _3}|0011\rangle \nonumber \\&-\,e^{i\phi _4}|0100\rangle -e^{i\phi _5}|0101\rangle -e^{i\phi _6}|0110\rangle -e^{i\phi _7}|0111\rangle \nonumber \\&-\,e^{i\phi _8}|1000\rangle +e^{i\phi _9}|1001\rangle +e^{i\phi _{10}}|1010\rangle +e^{i\phi _{11}}|1011\rangle \nonumber \\&+\,e^{i\phi _{12}}|1100\rangle +e^{i\phi _{13}}|1101\rangle +e^{i\phi _{14}}|1110\rangle +e^{i\phi _{15}}|1111\rangle )\nonumber \\ |\tau _9\rangle= & {} 1{/}4(|0000\rangle +e^{i\phi _1}|0001\rangle -e^{i\phi _{2}}|0010\rangle +e^{i\phi _{3}}|0011\rangle \nonumber \\&-\,e^{i\phi _{4}}|0100\rangle +e^{i\phi _{5}}|0101\rangle +e^{i\phi _{6}}|0110\rangle -e^{i\phi _{7}}|0111\rangle \nonumber \\&-\,e^{i\phi _{8}}|1000\rangle -e^{i\phi _{9}}|1001\rangle -e^{i\phi _{10}}|1010\rangle +e^{i\phi _{11}}|1011\rangle \nonumber \\&-\,e^{i\phi _{12}}|1100\rangle +e^{i\phi _{13}}|1101\rangle +e^{i\phi _{14}}|1110\rangle -e^{i\phi _{15}}|1111\rangle )\nonumber \\ |\tau _{10}\rangle= & {} 1{/}4(|0000\rangle +e^{i\phi _1}|0001\rangle +e^{i\phi _{2}}|0010\rangle -e^{i\phi _{3}}|0011\rangle \nonumber \\&-\,e^{i\phi _{4}}|0100\rangle -e^{i\phi _{5}}|0101\rangle +e^{i\phi _6}|0110\rangle +e^{i\phi _7}|0111\rangle \nonumber \\&-\,e^{i\phi _8}|1000\rangle -e^{i\phi _9}|1001\rangle +e^{i\phi _{10}}|1010\rangle -e^{i\phi _{11}}|1011\rangle \nonumber \\&-\,e^{i\phi _{12}}|1100\rangle -e^{i\phi _{13}}|1101\rangle +e^{i\phi _{14}}|1110\rangle -e^{i\phi _{15}}|1111\rangle )\nonumber \\ |\tau _{11}\rangle= & {} 1{/}4(|0000\rangle -e^{i\phi _1}|0001\rangle +e^{i\phi _{2}}|0010\rangle +e^{i\phi _{3}}|0011\rangle \nonumber \\&+\,e^{i\phi _{4}}|0100\rangle -e^{i\phi _{5}}|0101\rangle +e^{i\phi _6}|0110\rangle +e^{i\phi _7}|0111\rangle \nonumber \\&-\,e^{i\phi _8}|1000\rangle -e^{i\phi _9}|1001\rangle +e^{i\phi _{10}}|1010\rangle -e^{i\phi _{11}}|1011\rangle \nonumber \\&-\,e^{i\phi _{12}}|1100\rangle -e^{i\phi _{13}}|1101\rangle +e^{i\phi _{14}}|1110\rangle -e^{i\phi _{15}}|1111\rangle )\nonumber \\ |\tau _{12}\rangle= & {} 1{/}4(|0000\rangle +e^{i\phi _1}|0001\rangle +e^{i\phi _{2}}|0010\rangle -e^{i\phi _{3}}|0011\rangle \nonumber \\&+\,e^{i\phi _{4}}|0100\rangle +e^{i\phi _{5}}|0101\rangle -e^{i\phi _6}|0110\rangle -e^{i\phi _7}|0111\rangle \nonumber \\&-\,e^{i\phi _8}|1000\rangle +e^{i\phi _9}|1001\rangle +e^{i\phi _{10}}|1010\rangle +e^{i\phi _{11}}|1011\rangle \nonumber \\&-\,e^{i\phi _{12}}|1100\rangle +e^{i\phi _{13}}|1101\rangle -e^{i\phi _{14}}|1110\rangle +e^{i\phi _{15}}|1111\rangle )\nonumber \\ |\tau _{13}\rangle= & {} 1{/}4(|0000\rangle -e^{i\phi _1}|0001\rangle +e^{i\phi _{2}}|0010\rangle +e^{i\phi _{3}}|0011\rangle \nonumber \\&-\,e^{i\phi _{4}}|0100\rangle -e^{i\phi _{5}}|0101\rangle +e^{i\phi _6}|0110\rangle +e^{i\phi _7}|0111\rangle \nonumber \\&-\,e^{i\phi _8}|1000\rangle -e^{i\phi _9}|1001\rangle +e^{i\phi _{10}}|1010\rangle +e^{i\phi _{11}}|1011\rangle \nonumber \\&-\,e^{i\phi _{12}}|1100\rangle -e^{i\phi _{13}}|1101\rangle -e^{i\phi _{14}}|1110\rangle -e^{i\phi _{15}}|1111\rangle )\nonumber \\ |\tau _{14}\rangle= & {} 1{/}4(|0000\rangle -e^{i\phi _1}|0001\rangle -e^{i\phi _{2}}|0010\rangle -e^{i\phi _{3}}|0011\rangle \nonumber \\&+\,e^{i\phi _{4}}|0100\rangle -e^{i\phi _{5}}|0101\rangle +e^{i\phi _6}|0110\rangle +e^{i\phi _7}|0111\rangle \nonumber \\&-\,e^{i\phi _8}|1000\rangle -e^{i\phi _9}|1001\rangle -e^{i\phi _{10}}|1010\rangle -e^{i\phi _{11}}|1011\rangle \nonumber \\&+\,e^{i\phi _{12}}|1100\rangle +e^{i\phi _{13}}|1101\rangle -e^{i\phi _{14}}|1110\rangle +e^{i\phi _{15}}|1111\rangle )\nonumber \\ |\tau _{15}\rangle= & {} 1{/}4(|0000\rangle +e^{i\phi _1}|0001\rangle -e^{i\phi _{2}}|0010\rangle -e^{i\phi _{3}}|0011\rangle \nonumber \\&+\,e^{i\phi _{4}}|0100\rangle -e^{i\phi _{5}}|0101\rangle -e^{i\phi _6}|0110\rangle +e^{i\phi _7}|0111\rangle \nonumber \\&-\,e^{i\phi _8}|1000\rangle +e^{i\phi _9}|1001\rangle -e^{i\phi _{10}}|1010\rangle +e^{i\phi _{11}}|1011\rangle \nonumber \\&-\,e^{i\phi _{12}}|1100\rangle +e^{i\phi _{13}}|1101\rangle -e^{i\phi _{14}}|1110\rangle -e^{i\phi _{15}}|1111\rangle ) \end{aligned}$$

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