Lecture from Quantum Mechanics

Marek Zrałek Field Theory and Particle Physics Department. Silesian University Lecture 11

The entanglement of quantum states can be used to kind of "communication" Classically  BITS, e.g. Morse code Morse Code A. – B –... C –. –. D –.. E. F.. –. G –.. H.... I.. J. – – – K –. – L. –.. M – – N –. O – – – P. – –. R. –. S... T – U.. – W. – – V... – X –.. – Y –. – Z – –.. 1. – – – – 2.. – – – 3... – – – – – –... 8 – – –.. 9 – – – –. 0 – – – – – In Quantum World at first glance there is no difference ALICE BOB  − One classic BIT = with each QUBIT When sending single QUBITS no other option But let's change the rules and use entangled states entangled qubits Capacitor charged == 1 Capacitor non-charged == electrons Interaction of separate particles Classical computer Quantum computer

Classically - unit of information is the bit that takes two values (1.0). The corresponding unit in quantum computing is qubit(2) Qubit - it is information contained in the simplest possible quantum system. The simplest nontrivial state space is two-dimensional: Common phase has no physical meaning. The measurement of observable does not disturb the state only when a = 0 or b = 0. If the values of a and b are not known there is no way of determining them by one measurement. After measuring, the state of qubit will be described by the state or, and is different from the state before the measurement. In the case of the classical bit we can measured without any disorder and decipher all the information that it contains. Qubit(2), Qutrit(3), Qudit(>3) QUBIT

Classically, the bit can take only two values (0.1), qubit is characterized by two real numbers, for example we can take: To thoroughly examine the differences between bit and qubit let us consider the situation: Bit 0 1 p 0 p 1 Qubit The properties of qubits we can study by considering spin ½ : With respect to given axis (θ,φ) determine a spin orientation.

In particular, we have: If in any of these states the spin in z-direction is measured, then with probability of ½ the projection is obtained and the same for Consider another combination: By measuring a spin in the x direction again we can get two values. What will be if we measure a spin in the z-direction as before? In a classical situation the answer would be as follows: The system may be in the state + x or -x, in each with a probability of ½. If it is in the state + x by measuring the spin in a direction z with a probability of ½ we receive two results -1/2 or 1/2. Similarly, when the system is in a – x state. In that case, from theory of probability it follows, that measuring component of spin in z direction, with probability ½ we should get two results, +1/2 and -1/2. What is the answer i Quantum Mechanics? By measuring in that state the z component of the spin we get with probability 1 only one value /2. We never get value -1/2. The answer is simple. In quantum physics we can add states, there is quantum interference, the relative phases are important.

In quantum computing (in operation on qubits) all further possibilities of Quantum Mechanics appear, and so:  We can consider two qubits (and any number), then you can see all the subtle and wonderful properties of QM.  Two qubits can be in a pure or mixed state.  Two qubits can be entangled. With state We create statistical operator and statistical operators for individual qubits: As we know, two qubits in the state are entangled, if and only if the statistical operators and describe mixed states.

Alice can send Bob a classical information using bits Alice can also transfer qubits eg. photons in any state of polarization. This will be the transmission of information in the spin 0 = up, 1 = spin down. What do we gain by sending quantum information in comparison with sending a message in the classic way. At first glance, not much: Alice can prepare the state or and send it to Bob. Bob receives a spin down or up. In this way Alice can transmit 1 bit of information to each qubit. But let's change the rules and let's assume that Alice and Bob share a pair of entangled particles, eg. in state Alice can make on the particle from the entangled pair, one of the four unitary transformation, and she knows that Bob will get his particle in the entangled states: Bob can now make the unitary transformation defined by the matrix: Check that the indicated unitary transformations really give the states

And he obtain: If Alice send to Bob one bit of information, for example. 0 = {(+)} == I received states 1 or 3 1 = {(-)} == I received states of 2 or 4 Bob measure his particle and obtain the state 0 = {(+)} or {1 = (-)}. In the same time he has a two-bits information {00, 01, 10, 11} {++, + -, - +, -} With one bit Bob receives two Dense coding Entanglement of more QUBITS - can provide more information

Two entangled QUBITS

System of binary numbers: AND OR XOR NAND NOR Logical operations:

Quantum „communication”: Quantum teleportation, Dense coding, Quantum cryptography. Quantum cryptography What do we mean by classical cryptography ? Transferring information so that they are not possible to decipher by unauthorized persons Caesar cipher - an ordinary alphabet where the letters are shifted by a predetermined number of steps (= j), for example. Ola => sod for j = 5. The text goes in the cipher text when we know the key ALICEBOB Private key Transmitted in the normal way encrypted

Vernam cipher(1917) Sent text - the text written in the form (0.1) Key - a random string (0.1) of the same length as the text output Encrypted text - arises from the combination of the transmitted text with the key Example: Sent text { } Key { } encrypted text-- { } Bob must decode: { } { } { } The key can only be used only once, even two times used can be broken The use of a new key every time - troublesome Currently - important diplomatic messages Currently popular are cryptosystems with a public key:  Alice and Bob do not exchange a key,  Bob to the public provides information that can be used by anyone, Alice uses this information to encode,  but a sent message can not be decoded using information supplied by Bob,  only Bob knows the key to decoding. One such cryptosystems was introduced by Rivest, Shamir and Adleman (RSA protocol) in XOR

Bob --- chooses two large prime numbers p and q, and creates N = p q, who publicly announces, Alice ----knowing N in a special way encode their messages, to decode them we must know p and q (But only Bob knows it). Eve ---- to decode must distribute the N into prime factors: time  When N has 130 digits, 100 workstations require for about 1 month of working,  When N has 400 digits - the time required is the age of the universe, years,  In 1994, Peter Shor showed that a quantum computer could do it in three years,  Until the invention of quantum computers, the security of data is guaranteed. The only safe way of information transfer will be quantum cryptography. Classically ---- we do not know whether the information transferred were monitored  any security is mathematically difficult algorithms, Quantum ---- any interference in the system changes its quantum state  we will know if someone intercepts the messages. The current record for distribution of the N number of 176 digital (several months of calculations on a cluster of PC) “Eavesdropper "- eavesdropping under the door

When Heisenberg's principle is working Quantum Cryptography. Cryptosystem based on two non-commuting observables proposed by S.Wiesner (1970), and later by C.H.Bennett and G.Brassard (1984) A.K. Ekert, Phys. Rev. Lett. 67, 661 (1991); A.K. Ekert, J.G. Rarity, P.R. Tapster, and G.M. Palma, Phys. Rev. Lett. 69, 1293(1992) C.H. Bennett, Phys. Rev. Lett. 68, 3121 (1992). 1984)S. Wiesner, SIGACT News 15, 78 (1983); original manuscript written circa C.H. Bennett and G. Brassard, in "Proc. IEEE Int. Conference on Computers, Systems and Signal Processing", IEEE, New York (1984). C.H. Bennett, F. Bessette, G. Brassard, L. Salvail, and J.Smolin, "Experimental quantum cryptography," J. Cryptology 5, 3 (1992). A Alice could send photons in the 4 polarization states, 0, 45, 90 and 135 degrees. Bob measures the polarizations of the incoming photons. May set its device so as to distinguish between polarizations (0, 90) or (45, 135). However he does not distinguish between these two types of polarization.  Alice sends photons with random polarizations.  Bob sets his instruments to measure the polarization randomly, remember the ways of measurement and the results, and then publicly announce the measuring method (black, red).  Then Alice decide which measurements were correct and so those that were measured by the same alphabet (c, c). Alice knows what results Bob will get (x = 0, z = 1), so it can send the key by system of bits Protocol BB84 Protocol E91 Cryptosystem based on entanglement of quantum states and based on Bell theorem, proposed by A.K. Ekert (1991) The system is based on two non-orthogonal states of vectors according to C.H. Bennett (1992)

 If Eve wants to detect the key, she must measure each photon polarizations. Even if she know the directions (c, c) does not know what photon hits, of necessity so it must randomly select them. By setting in one direction your polarizer, Eve did not measure two non- commuting observables. By measuring using mismatched apparatus will introduce the disturbance in the polarization states of photons.  Alice and Bob communicate with each other about obtained results, so they easy detect if Eve, as a result of mismatched measurements, changes the polarization states of photons. They can not prevent "spying", but they know when they occurred, and these results will not be used to send key. REQUEST Show that when Eve place her polarizer in any direction, and after measurement of photon polarization in this direction, will send it to Bob, in 25% of cases Bob and Alice receive different results of their photons polarization. B ---As before, a pair of particles are produced in the entangled state (e.g. polarization entangled photons). In this case, Alice and Bob measure the polarization of a particle in three directions Alice  Bob  They decide on measurements along the axis selected at random. Way. --- Choosing two pairs from three - the correlation functions may be found. Offset by 45 degrees with respect to the Alice settings

They publicly communicate about the selected measurement axis for spin projection Measurements are divided into two categories depending on what kind of axis to measurements they choose: - they chose different axes (measure correlations S) - they have chosen the same axes (then they can convey a secret message)

Correlations in the spin singlet state The entire discussion can be repeated for particles with spin ½. Source of spin ½ particles a b z z x x Alice Bob Experimental results:

--- MK provides that --- Alice and Bob inform the public, which axes they chosen for measurement. If the axes do not coincide, the correlation coefficient should be minimal ie. --- When Eve will "spy" correlation will not be minimal --- Alice and Bob can inform each other when they have chosen to measure the same axe Quantum security systems for information transmission already entered the commercial phase The company from Geneva Id Quantique introduced on market a device for encoding Świat Nauki, February, Then, due to the existing correlation, can send safely a secret key

MagieQ Technologies, from New York A system based on light pipe to 100km NEC from Tokyo Up to 150 km. QinetiQ, Farnborough, Greit Britain In air up to 10 km DARPA – Defense Advanced Research Project Agency POSSIBILITIES 1000 km to communicate with satellites Census data, Various provisions, Technology Control of commercial satellites The first network cryptographic

For example, let's take a Protocol BB84 It is not possible to obtain information that will help distinguish between two non- orthogonal states. Alice could send to Bob one of four states: z x Alice and Bob can realize that their states are disturbed by Eve. Let’s take more generally, let will be two non-orthogonal states in the state space H.

Let H E be a state space of Eve. We perform unitary transformation in the state space of Eve H E : Unitarity of transformation implies:

This means that the states e and f are identical: Thus, Eve has not possibility to distinguish between both states: So Eve has not possibility to answer the question, which of the four states has been sent by Alice. However, in the case of transfer of individual bits the situation is no different from the classic. Let U operates as follows: Qubit E is copied to the identical qubit A

Whereas, when the first qubit is in the state: Entangled state of two qubits More generally, taken into accout the Bob:

Unitarity of U matrix imply: And hence when: So with the precision to phase Unitary transformation is not able to make a copy of states when they are different and orthogonal

There is no possibility of acquiring information that would allow to distinguish between two non-orthogonal quantum states without disturbing those states Or else in other way: