Quantum Information Processing and Diagrams of States September 17th 2009, AFSecurity

Sara Felloni

[email protected] / [email protected]

Quantum Hacking Group: http://www.iet.ntnu.no/groups/optics/qcr/

UNIK – University Graduate Center

NTNU, Department of Electronics and Telecommunications

Sara Felloni, September 17th 2009 Quantum Information Processing and Diagrams of States - p. 1/31 Quantum Information Processing ➲ Quantum Computing ➲ Quantum Information Processing ➲ Quantum Algorithms ➲ Quantum Information Quantum Information Processing Protocols ➲ Possibilities of Quantum Computing ➲ Quantum Information Representations ➲ Single qubit states ➲ Generation of single-qubit states

Diagrams of States: Practical Applications

Entanglement Purification

Conclusions and Bibliography

Sara Felloni, September 17th 2009 Quantum Information Processing and Diagrams of States - p. 2/31 Quantum Computing

...is the study of the information processing tasks performed by Quantum Information Processing means of quantum-mechanical systems: ➲ Quantum Computing ➲ Quantum Information ▲ Processing The elementary physical carrier of information is a qubit – ➲ Quantum Algorithms ➲ Quantum Information described by a 2-dimensional Hilbert space Protocols ➲ Possibilities of Quantum ▲ n Computing The state of a n-qubit register lives in a 2 -dimensional ➲ Quantum Information Representations Hilbert space – tensor product of n 2-dimensional spaces ➲ Single qubit states ➲ Generation of single-qubit ▲ states By imitating classical computations, quantum

Diagrams of States: Practical computations comprise three steps in sequence: Applications ● Preparation of the initial state of the register Entanglement Purification ● Conclusions and Bibliography Computation by quantum gate arrays – unitary transformations of the register state ● Output of the final result – by probabilistic measurement of all or part of the register

Sara Felloni, September 17th 2009 Quantum Information Processing and Diagrams of States - p. 3/31 Quantum Information Processing

...exploits the peculiar properties of quantum systems: Quantum Information Processing ▲ ➲ Quantum Computing The superposition principle – superpositions of different ➲ Quantum Information Processing input states are processed simultaneously ➲ Quantum Algorithms ➲ Quantum Information ▲ Protocols Quantum interference – information behaves and ➲ Possibilities of Quantum Computing propagates as interactions of wave patterns ➲ Quantum Information Representations ▲ ➲ Single qubit states Quantum entanglement – non-classical correlations exist ➲ Generation of single-qubit states between observable physical properties of spatially

Diagrams of States: Practical separated systems Applications

Entanglement Purification ...in order to: Conclusions and Bibliography • solve several computational problems more efficiently than by classical computers • open up new possibilities for communication and cryptography • efficiently simulate physical systems

Sara Felloni, September 17th 2009 Quantum Information Processing and Diagrams of States - p. 4/31 Quantum Algorithms

...can currently be enumerated under few classes: Quantum Information Processing ➲ Quantum Computing ▲ ➲ Quantum Information Basic/early quantum algorithms – explorations of the Processing ➲ Quantum Algorithms features later exploited by more powerful algorithms, ➲ Quantum Information Protocols Deutsch’s algorithm for global properties of functions sets ➲ Possibilities of Quantum Computing ▲ ➲ Quantum Information Algorithms based on amplitude amplification – Grover’s Representations ➲ Single qubit states search in unstructured databases, quantum amplitude ➲ Generation of single-qubit states estimation and quantum counting Diagrams of States: Practical ▲ Applications Algorithms with a super-polynomial speed-up – the

Entanglement Purification quantum Fourier transform, Shor’s factoring and period

Conclusions and Bibliography finding algorithm ▲ Algorithms for the simulation of dynamical systems – natural exploitation of the inherent quantum-mechanical behavior

Sara Felloni, September 17th 2009 Quantum Information Processing and Diagrams of States - p. 5/31 Quantum Information Protocols

...include several practical applications in communication and Quantum Information Processing information theory: ➲ Quantum Computing ➲ Quantum Information Processing ▲ ➲ Quantum Algorithms Quantum cryptography – no-cloning allows eavesdropping ➲ Quantum Information Protocols detection in ideal key generation/distribution protocols ➲ Possibilities of Quantum Computing ▲ ➲ Quantum Information Quantum cloning – imperfect cloning of information allows Representations ➲ Single qubit states state estimation and (partial) eavesdropping ➲ Generation of single-qubit ▲ states Dense coding – entanglement enhances communication of Diagrams of States: Practical classical information Applications

Entanglement Purification ▲ Quantum teleportation – entanglement and classical Conclusions and Bibliography communication allow reconstruction of quantum information ▲ Entanglement purification – distillation of nearly perfect (from many imperfect) entangled states ▲ Optimal compression of quantum information, reliable transmissions in noisy channels, quantum error correction...

Sara Felloni, September 17th 2009 Quantum Information Processing and Diagrams of States - p. 6/31 Possibilities of Quantum Computing

... have not yet been completely determined. Quantum Information Processing ➲ Quantum Computing ➲ Quantum Information Known Not yet known Processing ➲ Quantum Algorithms ▲ Appropriately devised ▲ The ultimate efficiency of ➲ Quantum Information Protocols quantum computations can quantum computers ➲ Possibilities of Quantum Computing be more efficient than ▲ ➲ Quantum Information A general approach to Representations their classical counterparts ➲ Single qubit states devise quantum algorithms ➲ Generation of single-qubit states and protocols

Diagrams of States: Practical Applications Analyzing in depth the functioning of known quantum Entanglement Purification computations helps understand how to successfully exploit

Conclusions and Bibliography ▲ the computational resources offered by quantum-mechanical systems ▲ the hidden structure of the considered problems

Sara Felloni, September 17th 2009 Quantum Information Processing and Diagrams of States - p. 7/31 Quantum Information Representations

...have a key role in helping understand the functioning of Quantum Information Processing quantum algorithms and protocols: ➲ Quantum Computing ➲ Quantum Information Processing Quantum Circuits Diagrams of States ➲ Quantum Algorithms ➲ Quantum Information ▲ Horizontal lines represent ▲ Horizontal lines represent Protocols ➲ Possibilities of Quantum single qubits quantum states (of the Computing ➲ Quantum Information computational basis) Representations ▲ Quantum gates are ➲ Single qubit states ➲ Generation of single-qubit applied from left (input) to ▲ State transformations states

Diagrams of States: Practical right (output) corresponding to quantum Applications gates are illustrated from ▲ They naturally provide Entanglement Purification left (input) to right (output) physical feasibility of Conclusions and Bibliography computations and a ▲ They are to be used in straightforward link to addition to mathematical physical implementation formalism and quantum circuits – too synthetic to visualize the information processing

Sara Felloni, September 17th 2009 Quantum Information Processing and Diagrams of States - p. 8/31 Single qubit states

...can be represented by the Bloch sphere representation: Quantum Information Processing ➲ Quantum Computing α ➲ Quantum Information |Ψi = α|0i + β|1i = Processing β ➲ Quantum Algorithms " # ➲ Quantum Information Protocols Spherical coordinates: ➲ Possibilities of Quantum Computing θ ➲ Quantum Information θ iΦ θ cos 2 Representations |Ψi = cos 2 |0i + e sin 2 |1i = iΦ θ ➲ Single qubit states e sin 2 ➲ Generation of single-qubit " # states

Diagrams of States: Practical Applications

Entanglement Purification Cartesian coordinates in the

Conclusions and Bibliography three-dimensional space em- bedding the Bloch sphere:

Sara Felloni, September 17th 2009 Quantum Information Processing and Diagrams of States - p. 9/31 Generation of single-qubit states

...is obtained by applying a θ−rotation about the y axis of the Quantum Information Processing Bloch sphere and a phase-shift gate with initial state |0i: ➲ Quantum Computing ➲ Quantum Information Processing θ θ θ ➲ Quantum Algorithms 1 0 cos 2 − sin 2 1 cos 2 ➲ Quantum Information |Ψi = iδ θ θ = iδ θ Protocols 0 e sin 2 cos 2 0 e sin 2 ➲ Possibilities of Quantum " # " # " # " # Computing ➲ Quantum Information Representations ➲ Single qubit states ➲ Generation of single-qubit states

Diagrams of States: Practical Applications

Entanglement Purification

Conclusions and Bibliography Quantum circuit (left) and Diagram of states (right) ▲ The initial state |0i determines the active information ▲ From left to right, active information flows on the thick lines, while thin lines correspond to absence of information

Sara Felloni, September 17th 2009 Quantum Information Processing and Diagrams of States - p. 10/31 Quantum Information Processing

Diagrams of States: Practical Applications ➲ Quantum Diagrams of States ➲ Synthesis of Multi-qubit Diagrams of States: Practical Gates ➲ Bell States Applications ➲ Bell States: Diagrams of States ➲ Quantum Teleportation ➲ Teleportation: Diagram of States ➲ Quantum Dense Coding ➲ Dense Coding: Diagrams of States ➲ Grover’s Search Algorithm ➲ Grover’s Algorithm: Diagram of States

Entanglement Purification

Conclusions and Bibliography

Sara Felloni, September 17th 2009 Quantum Information Processing and Diagrams of States - p. 11/31 Quantum Diagrams of States

... graphically represent and analyze how the information Quantum Information Processing encoded in quantum states is processed during

Diagrams of States: Practical computations performed by quantum circuits. Applications ➲ Quantum Diagrams of States ➲ Synthesis of Multi-qubit To analyze given quantum processes: Gates ➲ Bell States ▲ ➲ Bell States: Diagrams of Derive complete diagrams directly from implementations States ➲ Quantum Teleportation by quantum circuits ➲ Teleportation: Diagram of States ➲ Quantum Dense Coding ▲ Rearrange into simplified diagrams to visualize the overall ➲ Dense Coding: Diagrams of States effects of computations ➲ Grover’s Search Algorithm ➲ Grover’s Algorithm: Diagram of States To conceive new quantum computations: Entanglement Purification

Conclusions and Bibliography ▲ Describe schematically the desired manipulations of information by simple diagrams ▲ Expand into the equivalent complete diagrams to obtain implementations by quantum circuits

Sara Felloni, September 17th 2009 Quantum Information Processing and Diagrams of States - p. 12/31 Synthesis of Multi-qubit Gates

A general three-qubit controlled U−gate, with control from the two most significant qubits, is obtained by applying two CNOT gates and three controlled V −gates with control from a single-qubit. ▲ The diagram replaces a five 8 × 8-dimensional matrix multiplication ▲ The controlled unitary matrix V is such that V 2 = U

Sara Felloni, September 17th 2009 Quantum Information Processing and Diagrams of States - p. 13/31 Bell States

Bell states are defined as maximally entangled states of two qubits: 1 1 |Ψ±i = √2 {|01i±|10i} |Φ±i = √2 {|00i±|11i} ▲ Perfectly correlated even when spatially separated – EPR phenomenon ▲ Fundamental resources for several main algorithms and experiments: quantum teleportation, dense coding, entanglement-based cryptography, entanglement-based protocols...

▲ Bell states generation from the computational basis states (left) ▲ Bell measurements – measurements in respect to Bell basis (right)

Sara Felloni, September 17th 2009 Quantum Information Processing and Diagrams of States - p. 14/31 Bell States: Diagrams of States

Bell states generation (left) and Bell measurements (right) ▲ The initial state determines the active information lines ▲ Output states are determined by constructive and destructive interference in the active lines caused by Hadamard gates

Sara Felloni, September 17th 2009 Quantum Information Processing and Diagrams of States - p. 15/31 Quantum Teleportation

▲ An unknown quantum state |ψi is transferred by classical communication and by sharing a Bell state ▲ A direct measurement of the quantum system would perturb its state, offering too little information to reconstruct the state |ψi

+ (1) definition of initial states; (2) Hadamard and CNOT gates to generate the Bell state Φ ; (3) CNOT and | i Hadamard gates to perform Bell measurements; (4) measurement in the computational basis of the two most significant qubits (sender); (5) action on the least significant qubit (receiver) determined by the sender’s measurement results (double lines denote classical information).

Sara Felloni, September 17th 2009 Quantum Information Processing and Diagrams of States - p. 16/31 Teleportation: Diagram of States

▲ The initial state parameters a, b determine the active information lines ▲ The active information is spread by Hadamard gates ▲ The action on the least significant qubit (receiver) is determined by the sender’s measurement results ▲ The state |ψi is perfectly reconstructed in output

Sara Felloni, September 17th 2009 Quantum Information Processing and Diagrams of States - p. 17/31 Quantum Dense Coding

▲ Two classical bits of information are communicated by actually transmitting only one quantum bit of information ▲ A direct measurement of one quantum bit would give only a single classical bit of information

+ (1) definition of initial states; (2) Hadamard and CNOT gates to generate the Bell state Φ ; (3) unitary operation | i (sender) on half entangled pair, according to the two classical bits to communicate; (4) CNOT and Hadamard gates to perform Bell measurements; (5) measurement in the computational basis.

Sara Felloni, September 17th 2009 Quantum Information Processing and Diagrams of States - p. 18/31 Dense Coding: Diagrams of States

▲ Four possible unitary operations are performed by the sender according to the two classical bits to communicate ▲ Output states are determined by constructive and destructive interference caused by Hadamard gates and by the sender’s operation ▲ The two desired classical bits are communicated with unit probability

Sara Felloni, September 17th 2009 Quantum Information Processing and Diagrams of States - p. 19/31 Grover’s Search Algorithm

▲ Main representative of search heuristics for unstructured problems ▲ It addresses the problem of searching for one marked item inside an unstructured database of N = 2n items ▲ Quadratic speed up in resolution of general search problems

A quantum circuit implementing Grover’s algorithm to find one item out of N = 4, represented by two qubits: “A” - preparation of the state of the register and ancillary qubits; “B” - oracle query; “C” - main instruction.

Sara Felloni, September 17th 2009 Quantum Information Processing and Diagrams of States - p. 20/31 Grover’s Algorithm: Diagram of States

▲ Information in thick lines is spread by the Hadamard gates and subsequently manipulated by the oracle function ▲ The output state is determined by constructive and destructive interference caused by Hadamard gates ▲ The final active output lines correspond to basis states |100i and |101i: Measurement gives as outcome the marked item “10” with certainty ▲ The search problem is solved with a single query of the oracle function f

Sara Felloni, September 17th 2009 Quantum Information Processing and Diagrams of States - p. 21/31 Quantum Information Processing

Diagrams of States: Practical Applications

Entanglement Purification ➲ Reliable Quantum Entanglement Purification Transmissions ➲ Entanglement Purification ➲ Quantum Privacy Amplification ➲ QPA Protocol: Diagram of States ➲ QPA Iterations ➲ Fidelity and Survival Probability

Conclusions and Bibliography

Sara Felloni, September 17th 2009 Quantum Information Processing and Diagrams of States - p. 22/31 Reliable Quantum Transmissions

▲ A fundamental problem in quantum communication is how to Quantum Information Processing reliably transmit information through noisy channels:

Diagrams of States: Practical ● Applications decoherence – undesired interactions between qubits

Entanglement Purification carrying information and the environment ➲ Reliable Quantum Transmissions ● ➲ Entanglement Purification imperfections in the quantum components implementing ➲ Quantum Privacy the communication apparatus Amplification ➲ QPA Protocol: Diagram of ● States eavesdropping operations performed on the qubits ➲ QPA Iterations ➲ Fidelity and Survival carrying information in a cryptographic scenario Probability Conclusions and Bibliography ▲ In communication and cryptographic protocols the communicating parties resources are spatially separated ▲ Any information reconciliation or error-correcting procedure must be LOCC – local operations and classical communication

Sara Felloni, September 17th 2009 Quantum Information Processing and Diagrams of States - p. 23/31 Entanglement Purification

When considering entanglement-based communication Quantum Information Processing protocols, special LOCC protocols – also entanglement-based

Diagrams of States: Practical – can be used to: Applications ▲ Entanglement Purification improve the quality and the amount of entanglement in the ➲ Reliable Quantum Transmissions initially imperfect shared pairs ➲ Entanglement Purification ➲ Quantum Privacy ● in protocols requiring high-quality entanglement: Amplification ➲ QPA Protocol: Diagram of teleportation, quantum repeaters... States ➲ QPA Iterations ➲ Fidelity and Survival ▲ reduce the entanglement with any outside system to Probability

Conclusions and Bibliography arbitrarily low values ● eliminating eavesdropping in cryptography...

D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu and A. Sanpera Quantum Privacy Amplification and the Security of Quantum Cryptography over Noisy Channels – Phys. Rev. Lett. 77, 2818 (1996)

Sara Felloni, September 17th 2009 Quantum Information Processing and Diagrams of States - p. 24/31 Quantum Privacy Amplification

The QPA protocol purifies entanglement by creating nearly Quantum Information Processing perfect EPR states out of exponentially many partially

Diagrams of States: Practical entangled states: Applications

Entanglement Purification ➲ Reliable Quantum Transmissions ➲ Entanglement Purification ➲ Quantum Privacy Amplification ➲ QPA Protocol: Diagram of States ➲ QPA Iterations ➲ Fidelity and Survival Probability

Conclusions and Bibliography

The initial mixed pairs are described by the density matrices ρAB • The two communicating parties locally perform rotations and CNOT gates • The two most significant qubits are measured • ′ The final output is a purified state ρAB , when the detectors D0 and D1 give the same • outcomes

Sara Felloni, September 17th 2009 Quantum Information Processing and Diagrams of States - p. 25/31 QPA Protocol: Diagram of States

▲ + 1 Purification of imperfect state |Φ i = √2 (|00i + |11i) ▲ |Φ+i components determine the active information lines ▲ The purified output state is determined by constructive and destructive interference caused by Hadamard gates (arrows at the bottom)

Sara Felloni, September 17th 2009 Quantum Information Processing and Diagrams of States - p. 26/31 QPA Iterations

▲ Quantum circuit implementing two iterations of the QPA protocol ▲ At least 2n imperfect pairs are needed in input to obtain in output one purified pair after n iterations ▲ This number can be significantly larger – pairs must be discarded whenever Alice and Bob obtain different measurement outcomes

Sara Felloni, September 17th 2009 Quantum Information Processing and Diagrams of States - p. 27/31 Fidelity and Survival Probability

Deviation 1 − F of the fidelity Survival probability + + n F = hΦ | ρAB′ |Φ i P (n)= i=1 pi from its optimal value F = 1 on a with pi = probabilityQ of coinciding logarithmic scale outcomes at step i ...both as a function of the number of iterations n (horizontal axis) ▲ The protocol is successful for weak (solid line), middle (dashed line) and strong (dot-dashed line) eavesdropping intrusion or noise perturbation

Sara Felloni, September 17th 2009 Quantum Information Processing and Diagrams of States - p. 28/31 Quantum Information Processing

Diagrams of States: Practical Applications Entanglement Purification Conclusions and Bibliography Conclusions and Bibliography ➲ Conclusive Remarks ➲ References

Sara Felloni, September 17th 2009 Quantum Information Processing and Diagrams of States - p. 29/31 Conclusive Remarks

▲ For any given quantum computation, diagrams of states offer both: ● a detailed description of each gate action (complete diagram) ● an overall visualization from input to output (simplified diagram) ▲ The diagram dimension grows exponentially in respect to the dimension of the examined quantum system: ● Clearer visualization in respect to traditional descriptions ▲ Diagrams of states are most useful whenever quantum operations are described by sparse matrices: ● Only non-null entries of matrices are associated with diagram lines ● Only significant information flow and processing are shown ● This requirement is indeed satisfied by most quantum computations

Sara Felloni, September 17th 2009 Quantum Information Processing and Diagrams of States - p. 30/31 References

[*] Quantum Hacking Group: http://www.iet.ntnu.no/groups/optics/qcr/, Department of Quantum Information Electronics and Telecommunications, NTNU, and UNIK – University Graduate Center. Processing [*] This work was carried out during the tenure of an ERCIM “Alain Bensoussan” Diagrams of States: Practical Applications Fellowship Programme.

Entanglement Purification [1] G. Benenti, G. Casati, G. Strini, Principles of Quantum Computation and Information, Conclusions and Bibliography Volume I: Basic Concepts, World Scientific, 2004. ➲ Conclusive Remarks ➲ References [2] G. Benenti, G. Casati, G. Strini, Principles of Quantum Computation and Information, Volume II: Basic Tools And Special Topics, World Scientific, 2007. [3] M. A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000. [4] S. Felloni, A. Leporati, G. Strini, Diagrams of states in quantum information: An illustrative tutorial, International Journal of Unconventional Computing, to appear. [5] S. Felloni, A. Leporati, G. Strini, Evolution of quantum systems by diagrams of states, manuscript. [6] S. Felloni, G. Strini, Entanglement-based computations by diagrams of states, manuscript. [7] S. Felloni, G. Strini, Quantum algorithms by diagrams of states: Deutsch’s and Grover’s algorithms, submitted for publication. [8] G. Benenti, S. Felloni, G. Strini, Effects of single-qubit quantum noise on entanglement purification, Eur. Phys. J. D 38, p. 389, 2006.

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