Multistage entanglement swapping using superconducting qubits in the absence and presence of dissipative environment without Bell state measurement

Abstract In recent decades the entangled state generation is of great importance in the quantum information processing and technologies. In this paper, producing the distributed entangled state of superconducting (SC) qubits is considered using an entanglement swapping protocol in three successive stages. The SC qubit pairs $$(i,\,i+1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>i</mml:mi> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mi>i</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> with $$i=1,\,3,\,5,\,7)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mn>5</mml:mn> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mn>7</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , where each pair of the qubits has been placed on a separate chip, are initially prepared in maximally entangled states. The external magnetic fields on capacitively coupled pairs $$(2,\,3)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mn>3</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and $$(6,\,7)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>6</mml:mn> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mn>7</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> are implemented for modulating the frequency of qubits. Then, the SC qubits $$(1,\,4)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mn>4</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and $$(5,\,8)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>5</mml:mn> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mn>8</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> are converted into entangled states via operating proper measurements instead of Bell state measurement (which is generally a hard task). Finally, the distributed entangled state of target SC qubits $$(1,\,8)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mn>8</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> can be obtained by applying external magnetic fields on qubits $$(4,\,5)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mn>5</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and via operating suitable measurements. This process is studied in the absence and presence of thermal decoherence effects. The concurrence, as a measure of entanglement between two target qubits, success probability of the distributed entangled states and the corresponding fidelities are evaluated, by which we find that the state of target SC qubits $$(1,\,8)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mn>8</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is converted to Bell state with maximum entanglement at some moments of time. Under appropriate conditions the maximum of success probability of the obtained states in each stage approaches 1. However, the maxima of concurrence and success probability gradually decrease due to the thermal noise as time goes on. Moreover, compelling amounts of fidelity, success probability and entanglement can be obtained for the achieved entangled states.