Efficiency of Multi-Qubit W states i n Information Processing

1. Efficiency of Multi-Qubit W states in Information Processing Atul Kumar IPQI-2014 IIT Jodhpur 25.02.2014

2. IPQI-2014 • A pure state of two-qubits is said to be entangled if • Apart from being central to the foundational aspects of quantum physics, entanglement has also been used as an efficient resource in communication protocols to perform tasks such as quantum teleportation, quantum cryptography, quantum secret sharing, quantum secure direct communication etc. • Our focus for this work is quantum teleportation and entanglement swapping

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5. IPQI-2014 Pictorial representation of original scheme 2-animation-final2.exe D. Bouwmeesteret al, Nature 390 (1997), 575

6. IPQI-2014 • For two-qubit systems, the maximally entangled resources are Bell States; • For three-qubit systems, maximally entangled GHZ States and non-maximally entangled W states • For multiqubit systems some of the entangled resources are • Cluster states • Brown States

7. IPQI-2014 • Teleportation of a single qubit using three-qubit W state as a resource • The state to be teleported • Teleportation protocol is probabilistic using standard measurements and unitary transformations • Probability of teleportation depends on the unknown coefficients of state to be teleported • Alternately, one can also realize teleportation with success probability of 2/3 independent of and Shi and Tomita, Phys. Lett. A 296, 161 (2002)

8. IPQI-2014 • In general, quantum teleportation using partially entangled states is always probabilistic • However, Agrawal and Pati proposed a new class of W type states for deterministic quantum teleportation of a single qubit • Projection basis used to realize teleportation protocol for • n=1 P. Agrawal and A. K. Pati, Phys. Rev. A 74,062320 (2006)

9. IPQI-2014 • How is it possible? • We are performing a three qubit projective measurement to achieve the task • What if we perform standard two qubit and single qubit measurements? • Teleportation of single qubit is still probabilistic • Hence, to achieve teleportation of a single qubit using three-qubit W type of states one has to perform multiqubit measurements • Distinguishing these measurements is an issue • But nevertheless one can achieve perfect teleportation! S. Adhikari and S. Gangopadhyay, IJTP 48, 403 (2009)

10. IPQI-2014 • In this work, we address the following questions • Generalization of W type of states for perfect information transfer protocols • Given a four qubit W type state shared between two users, is it possible to let these users share a two qubit entangled state using entanglement swapping? • If so what is the degree of entanglement of the finally shared resource between the two users? • Comparison between the three and four-qubit W states in terms of concurrence of finally shared two-qubit states

11. IPQI-2014 • We generalize the three-qubit W type state for the case of four qubits And then to the case of k qubits

12. IPQI-2014 • In order to teleport an unknown state Alice and Bob must share the four qubit W-like state such that Alice has qubits 1, 2 and 3 and Bob has qubit 4 • Performing above four-qubit measurements on Alice’s qubits, perfect teleportation can be achieved

13. IPQI-2014 • Projection basis required if the shared state is a k qubit shared entangled resource • However, for practical cases we have analyzed the efficiency of W-type states for bi-partite entanglement sharing between the sender and the receiver

14. IPQI-2014 • For this we use the four-qubit state as • such that qubits 1, 2 and 3 are with Alice and qubit 4 is with Bob • We further consider that Alice has a two-qubit entangled state • The idea is to establish an entanglement between Alice’s qubit a and Bob’s qubit 4 Concurrence

15. IPQI-2014 • For this Alice measures her qubits in Bell basis • We consider two cases where Alice projects her qubits onto Bell states and or and • After these measurements, Alice shares one of the following states with Bob or • The concurrence of the shared bi-partite states between Alice and Bob can be given as or

16. IPQI-2014 • For n=1, both set of measurements yield identical results • For or the degree of entanglement • is unity i.e. the shared state is maximally entangled! • Hence, for a given value of users can in fact share maximum entanglement • Above two cases are compared to ascertain the measurements to be performed for a given value of state parameter • We found different ranges of the state parameter for a given n to obtain concurrence of the shared state

17. IPQI-2014 n=10 C alpha

18. IPQI-2014 • Why do we need to perform the two-qubit or single qubit measurements if Alice can share the initially entangled two qubit state with Bob? • Initially entangled pair with Alice • If Alice sends the qubit b to Bob through amplitude damping channel where the channel is represented by Kraus operators • The shared state in this case would be Concurrence

19. IPQI-2014 • For a specific case of Concurrence Case I n=1 Case II Two qubit pure state Two qubit state after ADC

20. IPQI-2014 n=1 n=2 n=10 n=5

21. IPQI-2014 • We have also compared the efficacy of our states with three-qubit W-type states • Hence, one can share a bi-partite maximally entangled state for certain value of state parameters • For certain ranges of state parameter four-qubit W-type states are more efficient resources in comparison to the three-qubit W-type states

22. IPQI-2014 n=10 C 4 qubit state 3 qubit state 4 qubit state 3 qubit state alpha

23. IPQI-2014 Acknowledgement Mr. Parvinder Singh Dr. Satyabrata Adhikari IIT Jodhpur Thank You