Quantum Information 2015.7.22 and Communication quantum mechanics uncertainty principle wave function superposition state information/communication cryptography signal processing Quantum Information/Communication quantum cryptography quantum computation
Quantum mechanical superposition state - Schrödinger s cat -? or determined Q: What is the cat s state in the box? A1:The cat is either alive or dead. It is determinative though not observed A2:The cat may be alive and dead. We do not know which, thus it can be alive and dead. Both is possible. classical quantum
Quantum mechanics employs A2; the cat in the box is alive and dead. Quantum mechanical superposition state A superposition state is mathematically expressed as: alive (cat in the box)= a + b dead Parameters a and b are coefficients representing the probabilities. When the box is opened, the cat state is instantly determined to be dead or alive. a + b opening = observation or
To be a superposition state, whether dead or alive is unknown in principle. with a timer (The cat necessarily dies when the time comes) Dead or alive is known in principle, even when the box is sealed. before the set time after the set time not superposition state
by the way Anything is composed of minimum units that cannot be further divided. A material is composed of nuclei and electrons electron nucleus Light energy is composed of photons attenuator detector signal Detection signal is discrete photon time
Superposition state of a photon Ex.1) A beam-split photon half mirror attenuation Ex.2)A photon split and recombined output photon=a (transmitted photon) +b (reflected photon) output photon=a (photon@{a, fast}) +b (photon@{b, slow}) attenuation half mirror B A +c (photon@{a, fast}) +d (photon@{b, slow})
Coefficients in superposition are regarded as wave Ex.3)A photon split and recombined (2) B The output state should be different from that in Ex.2. glass plate A output photon=a (photon@{a, fast}) +b (photon@{b, fast}) +c (photon@{a, slow}) +d (photon@{b, slow}) change due to the glass plate Here are postulates; - A coefficient has a fine structure like wave. The change due to the glass plate is expressed by a shift in the wave-like structure - The probability is given by its height. The probability is the same for c and c. c c prob.
Coefficients in superposition behave like wave Ex.3)A photon split and recombined (2) L1 L2 indistinguishable B A output photon @A = c 1 (photon via L1) + c 2 (photon via L2) = (c 1 + c 2 ) (photon@a) c 1 c 2 +
Total of coefficients depends on relative undulation + + ==enhanced canceled
photon probability @A thickness of glass plate interference fringe attenuation A B The coefficient of photon via A and that via B are recombined on the screen.
Which slit a photon passes through must be unknown for interference to occur A photon@slit=a (photon@a) + b (photon@b) B observation A B photon@slit= (photon@a) or (photon@b)
Applications to information/communication technologies Utilizing superposition states for cryptography Quantum cryptography (quantum key distribution) Utilizing superposition states for computation Quantum computer
Quantum cryptography Alice Bob encrypt decrypt
present cryptosystem Public key cryptosystem Alice Bob encrypting key (public) decrypting key (closed) public key 367 521 =? :easy(answer is191207 )?? = 191207 :difficult Deciphering is possible in principle
Secret key cryptosystem Alice Bob secret key (random bit sequence) secret key (random bit sequence) Quantum key distribution system (providing secure keys) Quantum Cryptography (Quantum Key Distribution) purpose To provide secret keys to distant parties selling point The security of the keys is guaranteed by quantum mechanics
Quantum Cryptography (Quantum Key Distribution) Cryptosystem utilizing quantum superposition states normal condition superposition state correct signal Key data are transmitted utilizing interference of a photon. eavesdropping?! state change due to observation Eavesdropping is revealed from the state change
Configuration of QKD system (example) Superposition states on temporal position are transmitted light source Alice modulator attenuation Bob detector B detector A long path short path time A photon is detected by detector A or B, according to the relative undulation of coefficients
Quantum Computing -computing utilizing superposition states- Plural data are simultaneously expressed by a superposition state Ultra parallel processing utilizing quantum interference
Quantum Bit photon s polarization state a (horizontal state) + b (vertical state) electron s spin state - Fundamental element for quantum processinga (up-spin) + b (down-spin) atom s energy state a (upper state) + b (lower state) E 2 superconducting current state a (right direction) + b (left direction) E 1
Plural data are simultaneously expressed by a superposition Two quantum bits a b 1 2 3 4 a b a b a b a b Each Q bit is a superposition of and, thus the whole state is; > = c 1 > + c 2 > + c 3 > + c 4 > a b a b a b a b a b = c 1 1 + c 2 2 + c 3 3 + c 4 4 4 digits are expressed N quantum bits 2 n digits can be simultaneously expressed
Parallel processing utilizing a superposition Problem: To find x satisfying the following equation f(x)=c classical way ex) prime decomposition 367 521 = 191207 191207 = x y quantum way x 1 x 2 x 3 x n φ> = x 1 > + x 2 > + x 3 > + + x n > f(x 1 ) f(x 2 ) f(x 3 ) f(x n ) U φ> = U x 1 > + U x 2 > + U x 3 > + + U x n > a b c z φ out > = 0 x 1 > + 0 x 2 > + 1 x 3 > + + 0 x n > Examining all candidates one by one Conducting one operation
Image of parallel processing Simultaneous operation based on interference prepared state 1 mod N m 1 mod N m 2 1 mod N m q -1 mod N interference
Unfortunately, implementation is quit difficult Quantum bits are very fragile. They easily collapse due to disturbances. 2 準位系 E 2 E 1
quantum mechanics uncertainty principle wave function superposition state information/communication cryptography signal processing Quantum Information/Communication quantum cryptography quantum computation