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## CHAPTER 15: Quantum cryptography An important new feature of quantum cryptograph

Carter, Noelle, Food Writer has reference to this Academic Journal, PHwiki organized this Journal CHAPTER 15: Quantum cryptography An important new feature of quantum cryptography is that security of cryptographic protocols generation is based on the laws of nature in addition to not on the unproven assumptions of computational complexity theory. Quantum cryptography is the first area in which quantum physics laws are directly exploited to bring an essential advantage in in as long as mation processing. Three main outcomes so far · It has been proven that unconditionally secure quantum generation of classical secret in addition to shared keys is possible (in the sense that any eavesdropping is detectable). · Unconditionally secure basic quantum cryptographic protocols, such as bit commitment in addition to oblivious transfer, are impossible. · Quantum cryptography is already in advanced experimental stage. Be as long as e presenting basic schemes of quantum cryptography basic ideas of quantum in as long as mation processing will be discussed shortly. IV054 Classical versus quantum computing The essense of the difference between classical computers in addition to quantum computers is in the way in as long as mation is stored in addition to processed. In classical computers, in as long as mation is represented on macroscopic level by bits, which can take one of the two values 0 or 1 In quantum computers, in as long as mation is represented on microscopic level using qubits, (quantum bits) which can take on any from the following uncountable many values 0 + b 1 where a, b are arbitrary complex numbers such that a 2 + b 2 = 1. IV054 CLASSICAL EXPERIMENTS Figure 1: Experiment with bullets Figure 2: Experiments with waves IV054

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QUANTUM EXPERIMENTS Figure 3: Two-slit experiment Figure 4: Two-slit experiment with an observation IV054 THREE BASIC PRINCIPLES P1 To each transfer from a quantum state f to a state y a complex number á y f is associated. This number is called the probability amplitude of the transfer in addition to á y f 2 is then the probability of the transfer. IV054 P2 If a transfer from a quantum state f to a quantum state y can be decomposed into two subsequent transfers y ¬ f ¬ f then the resulting amplitude of the transfer is the product of amplitudes of subtransfers: á y f = á y f á f f P3 If a transfer from a state f to a state y has two independent alternatives y j then the resulting amplitude is the sum of amplitudes of two subtransfers. QUANTUM SYSTEMS = HILBERT SPACE Hilbert space Hn is n-dimensional complex vector space with scalar product This allows to define the norm of vectors as Two vectors f in addition to y are called orthogonal if áfy = 0. A basis B of Hn is any set of n vectors b1, b2, , bn of the norm 1 which are mutually orthogonal. Given a basis B, any vector y from Hn can be uniquelly expressed in the as long as m IV054

BRA-KET NOTATION Dirack introduced a very h in addition to y notation, so called bra-ket notation, to deal with amplitudes, quantum states in addition to linear functionals f: H ® C. If y, f Î H, then áyf – scalar product of y in addition to f (an amplitude of going from f to y). f – ket-vector – an equivalent to f áy – bra-vector a linear functional on H such that áy(f) = áyf IV054 QUANTUM EVOLUTION / COMPUTATION EVOLUTION COMPUTATION in in QUANTUM SYSTEM HILBERT SPACE is described by Schrödinger linear equation where h is Planck constant, H(t) is a Hamiltonian (total energy) of the system that can be represented by a Hermitian matrix in addition to (t) is the state of the system in time t. If the Hamiltonian is time independent then the above Shrödinger equation has solution where is the evolution operator that can be represented by a unitary matrix. A step of such an evolution is there as long as e a multiplication of a unitary matrix A with a vector y, i.e. A y IV054 A matrix A is unitary if A · A = A · A = I QUANTUM (PROJECTION) MEASUREMENTS A quantum state is observed (measured) with respect to an observable – a decomposition of a given Hilbert space into orthogonal subspaces (that is each vector can be uniquely represented as a sum of vectors of these subspaces). There are two outcomes of a projection measurement of a state f : 1. Classical in as long as mation into which subspace projection of f was made. 2. Resulting projection (a new state) f into one of subspaces. The subspace into which projection is made is chosen r in addition to omly in addition to the corresponding probability is uniquely determined by the amplitudes at the representation of f as a sum of states of the subspaces. IV054

QUBITS A qubit is a quantum state in H2 f = a0 + b1 where a, b Î C are such that a2 + b2 = 1 in addition to { 0, 1 } is a (st in addition to ard) basis of H2 IV054 EXAMPLE: Representation of qubits by (a) electron in a Hydrogen atom (b) a spin-1/2 particle Figure 5: Qubit representations by energy levels of an electron in a hydrogen atom in addition to by a spin-1/2 particle. The condition a2 + b2 = 1 is a legal one if a2 in addition to b2 are to be the probabilities of being in one of two basis states (of electrons or photons). HILBERT SPACE H2 STANDARD BASIS DUAL BASIS 0, 1 0, 1 Hadamard matrix H 0 = 0 H 0 = 0 H 1 = 1 H 1 = 1 trans as long as ms one of the basis into another one. General as long as m of a unitary matrix of degree 2 IV054 QUANTUM MEASUREMENT of a qubit state A qubit state can contain unboundly large amount of classical in as long as mation. However, an unknown quantum state cannot be identified. By a measurement of the qubit state a0 + b1 with respect to the basis 0, 1 we can obtain only classical in as long as mation in addition to only in the following r in addition to om way: 0 with probability a2 1 with probability b2 IV054

QUANTUM REGISTERS A general state of a 2-qubit register is: f = a0000 + a0101 + a1010 + a1111 where a00 2 + a01 2 + a10 2 + a11 2 = 1 in addition to 00, 01, 10, 11 are vectors of the st in addition to ard basis of H4, i.e. An important unitary matrix of degree 4, to trans as long as m states of 2-qubit registers: It holds: CNOT : x, yñ Þ x, x Å yñ IV054 QUANTUM MEASUREMENT of the states of 2-qubit registers f = a0000 + a0101 + a1010 + a1111 1. Measurement with respect to the basis { 00, 01, 10, 11 } RESULTS: 00> in addition to 00 with probability a002 01> in addition to 01 with probability a012 10> in addition to 10 with probability a102 11> in addition to 11 with probability a112 IV054 2. Measurement of particular qubits: By measuring the first qubit we get 0 with probability a002 + a012 in addition to f is reduced to the vector 1 with probability a102 + a112 in addition to f is reduced to the vector NO-CLONING THEOREM INFORMAL VERSION: Unknown quantum state cannot be cloned. IV054 FORMAL VERSION: There is no unitary trans as long as mation U such that as long as any qubit state y U (y0) = yy PROOF: Assume U exists in addition to as long as two different states a in addition to b U (a0) = aa U (b0) = bb Let Then However, CNOT can make copies of basis states 0, 1: CNOT (x0) = xx

BELL STATES States as long as m an orthogonal (Bell) basis in H4 in addition to play an important role in quantum computing. Theoretically, there is an observable as long as this basis. However, no one has been able to construct a measuring device as long as Bell measurement using linear elements only. IV054 QUANTUM n-qubit REGISTER A general state of an n-qubit register has the as long as m: in addition to f is a vector in H2^n. Operators on n-qubits registers are unitary matrices of degree 2n. Is it difficult to create a state of an n-qubit register In general yes, in some important special cases not. For example, if n-qubit Hadamard trans as long as mation is used then in addition to , in general, as long as x Î {0,1}n 1The dot product is defined as follows: IV054 QUANTUM PARALLELISM If f : {0, 1, ,2n -1} Þ {0, 1, ,2n -1} then the mapping f :(x, 0) Þ (x, f(x)) is one-to-one in addition to there as long as e there is a unitary trans as long as mation Uf such that. Uf (x0) Þ xf(x) Let us have the state With a single application of the mapping Uf we then get OBSERVE THAT IN A SINGLE COMPUTATIONAL STEP 2n VALUES OF f ARE COMPUTED! IV054

IN WHAT LIES POWER OF QUANTUM COMPUTING In quantum interference or in quantum parallelism NOT, in QUANTUM ENTANGLEMENT! Let be a (global) state of two very distant particles, as long as example on two planets Measurement of one of the particels, with respect to the st in addition to ard Basis, make collapse of the above state to one of the states 00> Or 11>. This means that subsequent measurement of other particle (on another planet) provides the same result as the measurement of the first particle. This indicate that in quantum world non-local influences, correlations exist. IV054 POWER of ENTANGLEMENT Quantum state > of a bipartite quantum state A x B is called entangled if it cannot be decomposed into tensor product of the states from A in addition to B. Quantum entanglement is an important quantum resource that alllows To create phenomena that are impossible in the classical world ( as long as example teleportation) To create quantum algorithms that are asymptotically more efficient than any classical algorithm as long as the same problem. To cretae communication protocols that are asymptotically more efficient than classical communication protocols as long as the same task To create, as long as two parties, shared secret binary keys To increase capacity of quantum channels CLASSICAL versus QUANTUM CRYPTOGRAPHY Security of classical cryptography is based on unproven assumptions of computational complexity ( in addition to it can be jeopardize by progress in algorithms in addition to /or technology). Security of quantum cryptography is based on laws of quantum physics that allow to build systems where undetectable eavesdropping is impossible. IV054 Since classical cryptography is volnurable to technological improvements it has to be designed in such a way that a secret is secure with respect to future technology, during the whole period in which the secrecy is required. Quantum key generation, on the other h in addition to , needs to be designed only to be secure against technology available at the moment of key generation.

QUANTUM KEY GENERATION Quantum protocols as long as using quantum systems to achieve unconditionally secure generation of secret (classical) keys by two parties are one of the main theoretical achievements of quantum in as long as mation processing in addition to communication research. Moreover, experimental systems as long as implementing such protocols are one of the main achievements of experimental quantum in as long as mation processing research. It is believed in addition to hoped that it will be quantum key generation (QKG) another term is quantum key distribution (QKD) where one can expect the first transfer from the experimental to the development stage. IV054 QUANTUM KEY GENERATION – EPR METHOD Let Alice in addition to Bob share n pairs of particles in the entangled state IV054 POLARIZATION of PHOTONS Polarized photons are currently mainly used as long as experimental quantum key generation. Photon, or light quantum, is a particle composing light in addition to other as long as ms of electromagnetic radiation. Photons are electromagnetic waves in addition to their electric in addition to magnetic fields are perpendicular to the direction of propagation in addition to also to each other. An important property of photons is polarization – it refers to the bias of the electric field in the electromagnetic field of the photon. Figure 6: Electric in addition to magnetic fields of a linearly polarized photon IV054

POLARIZATION of PHOTONS Figure 6: Electric in addition to magnetic fields of a linearly polarized photon If the electric field vector is always parallel to a fixed line we have linear polarization (see Figure). IV054 POLARIZATION of PHOTONS There is no way to determine exactly polarization of a single photon. However, as long as any angle q there are q-polarizers filters – that produce q-polarized photons from an incoming stream of photons in addition to they let q1-polarized photons to get through with probability cos2(q – q1). Figure 6: Photon polarizers in addition to measuring devices-80% Photons whose electronic fields oscillate in a plane at either 0O or 90O to some reference line are called usually rectilinearly polarized in addition to those whose electric field oscillates in a plane at 45O or 135O as diagonally polarized. Polarizers that produce only vertically or horizontally polarized photons are depicted in Figure 6 a, b. IV054 POLARIZATION of PHOTONS Generation of orthogonally polarized photons. Figure 6: Photon polarizers in addition to measuring devices-80% For any two orthogonal polarizations there are generators that produce photons of two given orthogonal polarizations. For example, a calcite crystal, properly oriented, can do the job. Fig. c – a calcite crystal that makes q-polarized photons to be horizontally (vertically) polarized with probability cos2 q (sin2 q). Fig. d – a calcite crystal can be used to separate horizontally in addition to vertically polarized photons. IV054

QUANTUM KEY GENERATION – PROLOGUE Very basic setting Alice tries to send a quantum system to Bob in addition to an eavesdropper tries to learn, or to change, as much as possible, without being detected. Eavesdroppers have this time especially hard time, because quantum states cannot be copied in addition to cannot be measured without causing, in general, a disturbance. Key problem: Alice prepares a quantum system in a specific way, unknown to the eavesdropper, Eve, in addition to sends it to Bob. The question is how much in as long as mation can Eve extract of that quantum system in addition to how much it costs in terms of the disturbance of the system. Three special cases Eve has no in as long as mation about the state y Alice sends. Eve knows that y is one of the states of an orthonormal basis {fi}ni=1. Eve knows that y is one of the states f1, , fn that are not mutually orthonormal in addition to that pi is the probability that y = fi. IV054 TRANSMISSION ERRORS If Alice sends r in addition to omly chosen bit 0 encoded r in addition to omly as 0 or 0′ or 1 encoded as r in addition to omly as 1 or $1′ in addition to Bob measures the encoded bit by choosing r in addition to omly the st in addition to ard or the dual basis, then the probability of error is ¼=2/8 If Eve measures the encoded bit, sent by Alice, according to the r in addition to omly chosen basis, st in addition to ard or dual, then she can learn the bit sent with the probability 75% . If she then sends the state obtained after the measurement to Bob in addition to he measures it with respect to the st in addition to ard or dual basis, r in addition to omly chosen, then the probability of error as long as his measurement is 3/8 – a 50% increase with respect to the case there was no eavesdropping. Indeed the error is IV054 BB84 QUANTUM KEY GENERATION PROTOCOL Quantum key generation protocol BB84 (due to Bennett in addition to Brassard), as long as generation of a key of length n, has several phases: Preparation phase Alice generates two private r in addition to om binary sequences of bits of length m >> n bits in addition to Bob generates one such private r in addition to om sequence. IV054 Quantum transmission Alice is assumed to have four transmitters of photons in one of the following four polarizations 0, 45, 90 in addition to 135 degrees Figure 8: Polarizations of photons as long as BB84 in addition to B92 protocols Expressed in a more general as long as m, Alice uses as long as encoding states from the set {0, 1,0′, 1′}. Bob has a detector that can be set up to distinguish between rectilinear polarizations (0 in addition to 90 degrees) or can be quickly reset to distinguish between diagonal polarizations (45 in addition to 135 degrees).

QUANTUM TELEPORTATION I Total state of three particles: can be expressed as follows: in addition to there as long as e the measurement of the first two particles projects the state of the Bob’s particle into a small modification y1ñ of the unknown state yñ = 1/sqrt 2 (a0ñ + b1ñ). The unknown state yñ can there as long as e be obtained from y1ñ by applying one of the four operations sx, sy, sz, I in addition to the result of the Bell measurement provides two bits specifying which of the above four operations should be applied. These four bits Alice needs to send to Bob using a classical channel (by email, as long as example). IV054 QUANTUM TELEPORTATION II If the first two particles of the state are measured with respect to the Bell basis then Bob’s particle gets into the mixed state to which corresponds the density matrix The resulting density matrix is identical to the density matrix as long as the mixed state Indeed, the density matrix as long as the last mixed state has the as long as m IV054 QUANTUM TELEPORTATION – COMMENTS Alice can be seen as dividing in as long as mation contained in yñ into quantum in as long as mation – transmitted through EPR channel classical in as long as mation – transmitted through a classical cahnnel IV054 In a quantum teleportation an unknown quantum state fñ can be disambled into, in addition to later reconstructed from, two classical bit-states in addition to an maximally entangled pure quantum state. Using quantum teleportation an unknown quantum state can be teleported from one place to another by a sender who does not need to know – as long as teleportation itself – neither the state to be teleported nor the location of the intended receiver. The teleportation procedure can not be used to transmit in as long as mation faster than light but it can be argued that quantum in as long as mation presented in unknown state is transmitted instanteneously (except two r in addition to om bits to be transmitted at the speed of light at most). EPR channel is irreversibly destroyed during the teleportation process.

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