QUANTUM MECHANICS AND QUANTUM INFORMATION SCIENCE
WHAT IS Ψ?
IS Ψ ONTIC OR EPISTEMIC?
WHAT IS A QUANTUM MEASUREMENT?
QUANTUM PARADOXES
THE DOUBLE-SLIT: WAVE-PARTICLE DUALITY (1905-1927-)
SCHRöDINGER CAT (1035)
EPR PARADOX AND QUANTUM ENTANGLEMENT (1935)
INTERPRETATIONS
BOHR VS VON NEUMANN/DIRAC
EVERETT (MANY WORLDS?) (1957-)
HIDDEN VARIABLES (1935-)
OTHERS (MODAL, CONSISTENT HISTORIES, ETC)
NO-GO THEOREMS FOR HIDDEN VARIABLES
VON NEUMANN THEOREM (1932)
BELL’S THEOREM (1964)
KOCHEN-SPECKER THEOREM (1967) CONTEXTUAL HVT ARE INCOMPATIBLE WITH QM
HOWEVER, BOHM’S HIDDEN VARIABLE THEORY IS NOT RULED OUT BY THESE THEOREMS
BELL’S THEOREM
STANDARD OR ORTHODOX QUANTUM MECHANICS IS INCOMPATIBLE WITH
LOCAL REALISM
(USES LOCAL HIDDEN VARIABLES)
EINSTEIN’S 1927 ARGUMENT USES A SINGLE PARTICLE
EINSTEIN’S 1927 ARGUMENT
Ψ = (1/√2) [ψa + ψb]
p(1a Λ 1b |ψ) = p (1a|ψ) p(1b|1a,ψ)
= p(1a|ψ) p(1b|ψ) locality = ¼THIS CONTRADICTS THE STANDARD QMPREDICTION
p(1a Λ 1b |ψ) = 0
BIRTH OF QUANTUM INFORMATION AGE
1982 FEYNMAN SHOWED THAT A CLASSICAL TURING MACHINE WOULD EXPERIENCE EXPONENTIAL SLOW DOWN WHEN SIMULATING QUANTUM PROCESSES BUT HIS HYPOTHETICAL UNIVERSAL QUANTUM SIMULATOR WOULD NOT.
1985 DAVID DEUTSCH DEFINED A UNIVERSAL QUANTUM COMPUTER
1996 SETH LLOYD SHOWED THAT A QUANTUM COMPUTER CAN BE PROGRAMMED TO SIMULATE ANY LOCAL QUANTUM SYSTEM EFFICIENTLY.
BITS AND QUBITS
IN QUANTUM COMPUTING THE ANALOGUE OF THE CLASSICAL UNIT OF INFORMATION, THE BIT, IS A QUBIT WHICH IS A TWO-LEVEL QUANTUM SYSTEM LIKE THE TWO STATES OF POLARIZATION OF A SINGLE PHOTON WHICH CAN BE IN A SUPERPOSITION OF STATES:
|ψ> = α|0> + β|1>
with
| α|2 + | β |2 = 1
BREAKTHROUGHQUANTUM ALGORITHMS
1992 DEUTSCH-JOZSA: exponentially faster than any deterministic classical algorithm
1998 improved by CLEVE, EKERT, MACCHIAVELLO and MOSCA
1994 SHOR: integer factorization
1996 GROVER: quantum search
OTHER ALGORITHMS FOR
QUANTUM FOURIER TRANSFORM
QUANTUM GATES
QUANTUM ADIABATIC
QUANTUM ERROR CORRECTION
NO-CLONING THEOREM
WOOTERS, ZUREK, DIEKS (1982)
QUANTUM MECHANICS FORBIDS THE CREATION OF IDENTICAL COPIES OF AN UNKNOWN QUANTUM STATE
NO-DELETING THEOREM
A K PATI & S L BRAUNSTEIN, NATURE 2000
GIVEN TWO COPIES OF SOME UNKNOWN AND ARBITRARY QUANTUM STATE, IT IS IMPOSSIBLE TO DELETE ONE OF THE COPIES
IT IS A TIME REVERSED DUAL TO THE NO-CLONING THEOREM
IN SOME INSTANCES QUANTUM STATES CAN BE ROBUST
QUANTUM INFORMATION PROCESSING SCIENCE
QUANTUM COMPUTING QUANTUM COMPLEXITY THEORY QUANTUM CRYPTOGRAPHY QUANTUM ERROR CORRECTION QUANTUM COMMUNICATION
COMPLEXITY QUANTUM ENTANGLEMENT QUANTUM DENSE CODING
QUANTUM ENTANGLEMENT: CHIEF RESOURCE IN QI SCIENCE
NON-SEPARABLE STATES
COMPLETE KNOWLEDGE OF THE STATE DOES NOT IMPLY COMPLETE KNOWLEDGE OF THE PARTS
STRONG MEASUREMENT RESULTS IN CONDITIONAL DISJUNCTION OF THE STATE
POVMs
IN QI PROCESSING CONVENTIONAL PROJECTIVE MEASUREMENT IS REPLACED BY MORE GENERAL
POVMs: CHOICE OF NON-ORTHOGONAL BASIS FOR MEASUREMENTS WITH THE NEW PROJECTORS STILL SUMMING TO UNITY
REASON:
PROJECTIVE MEASUREMENTS ON A LARGER SYSTEM, DESCRIBED BY A PROJECTION-VALUED MEASURE (PVM), WILL ACT ON A SUB-SYSTEM IN WAYS THAT CANNOT BE DESCRIBED BY A PVM ON THE SUB-SYSTEM ALONE
ENTANGLEMENT MEASURES
BELL INEQUALITY VIOLATION IS A MEASURE OF ENTANGLEMENT
BUT NOT ALL ENTANGLED STATES VIOLATE BIs.
A WERNER STATE, A MIXTURE OF THE MAXIMALLY ENTANGLED STATE AND THE MAXIMALLY MIXED STATE, CAN BE ENTANGLED AND YET NOT VIOLATE THE CONVENTIONAL BELL INEQUALITY.
OTHER MEASURES
CONCURRENCE
TANGLE
ENTROPY
ENTROPY
ENTROPY OF ENTANGLEMENT IS A GOOD ENTANGLEMENT MEASURE FOR BIPARTITE PURE STATES.
FOR A PURE STATE ρ(ab) = |ψ>< ψ|
Ε(ρ(ab) ) = S(ρ(a)) = S (ρ(b))
WHERE ρ(a) = Trb ρ(ab) ρ(b) = Tra ρ(ab)
AND S IS THE VON NEUMANN ENTROPY
S = - Tr (ρ ln ρ)
MONOGAMY OF ENTANGLEMENT
IF TWO QUBITS A AND B ARE MAXIMALLY QUANTUMLY CORRELATED, THEY CANNOT BE CORRELATED AT ALL WITH A THIRD QUBIT C
FOR ANY TRIPARTITE SYSTEM
E(A|B1) + E(A|B2) ≤ E(A|B1B2)
QUANTUM TELEPORTATION
C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres and W. K. Wootters (1993)
BIRTH OF ALICE AND BOB
ALICE CAN SEND BOB ‘QUANTUM INFORMATION’ (i.e. THE EXACT STATE OF A QUBIT) BY SHARING AN ENTANGLED STATE BETWEEN THEM AND EXCHANGING 2 BITS CLASSICAL INFORMATION.
QUANTUM CRYPTOGRAPHY
QM GUARANTEES THAT MEASURING QUANTUM DATA DISTURBS THAT DATA, AND THIS CAN BE
USED TO DETECT EAVESDROPPING IN QUANTUM KEYDISTRIBUTIONS. THIS IS DONE BY ENCODING THEINFORMATION IN NON-ORTHOGONAL STATES WHICHCANNOT BE MEASURED WITHOUT DISTURBING THEORIGINAL STATE.
PROTOCOLS
C H BENNETT, G BRASSARD (BB84)DEVELOPED A NEW METHOD OF SECURE QUANTUM KEY DISTRIBUTION BASED ON ‘CONJUGATE VARIABLES’
A EKERT (1990) DEVELOPED ANOTHER METHOD BY USING ENTANGLED PHOTON PAIRS
VARIOUS OTHER PROTOCOLS HAVE BEEN DESIGNED AND ARE BEING PUT TO COMMERCIAL USE
QUANTUM KEY DISTRIBUTION NETWORKS
DARPA
SECOQC
SWISSQUANTUM
TOKYO QKD
LOS ALAMOS NATIONAL LABS
ENTANGLEMENT IN CLASSICAL POLARIZATION OPTICS
AZIMUTHAL PLARIZATION
ENTANGLEMENT IS SOMETIMES ENOUGH
NATURAL UNPOLARIZED THERMAL LIGHT IS A BELL STATE |e> = (1/√2) [ |u1> |f1> + |u2> |f2> ] :
BI VIOLATION WITHOUT NONLOCALITY
PARTIALLY POLARIZED LIGHT IS NOT MAXIMALLY ENTANGLED
|e> = κ1 |u1> |f1> + κ2 |u2> |f2> ]
FULLY POLARIZED LIGHT IS A PRODUCT STATE
BI VIOLATION IS NOT A UNIQUE INDICATOR OF ENTANGLEMENT, QUANTUMNESS OR NONLOCALITY
BELL-LIKE INEQUALITIES ARE VIOLATED BY SUCH LIGHT
R J C SPREEUW (1998) P GHOSE & M K SAMAL (2001) B N SIMON et al (2010), BORGES et al (2010), G S AGARWAL et al (2013), X-F Qian and J. H. Eberly (2013), K H KAGALWALA et al (2013)P GHOSE AND A MUKHERJEE, Rev of Theoret Sc vol. 2, pp 1-14, 2014.
QUANTUMNESS OTHER THAN ENTANGLEMENT?
THE LEGGETT-GARG INEQUALITY(1985)
MACROREALISM:
A) A MACROSCOPIC OBJECT WHICH HAS AVAILABLE TO IT TWO OR MORE MACROSCOPICALLY DISTINCT STATES IS AT ANY GIVEN TIME IN A DEFINITE ONE OF THOSE STATES
B) NON-INVASIVE MEASUREABILITY: IT IS POSSIBLE IN PRINCIPLE TO DETERMINE WHICH OF THESE STATES THE SYSTEM IS IN WITHOUT ANY EFFECT ON THE SYSTEM ITSELF OR ON THE SUBSEQUENT SYSTEM DYNAMICS
QUANTUM SYSTEMS, NO MATTER HOW MACROSCOPIC, VIOLATE THESE POSTULATES
ONTOLOGICAL MODELS OF Ψ
HARRIGAN AND SPEKKENS (2010)
DOES THE QUANTUM STATE REPRESENT REALITY OR MERELY OUR KNOWLEDGE OF REALITY?
IS REALITY LOCAL OR NONLOCAL?
WHAT IS AN ONTOLOGICAL MODEL?
THEORY MUST BE FORMULATED OPERATIONALLY, i.e. THE PRIMITIVES OF DESCRIPTION ARE PREPARATIONS AND MEASUREMENTS
IN AN ONTOLOGICAL MODEL OF AN OPERATIONAL THEORY THE PRIMITIVES ARE PROPERTIES OF THE MICROSCOPIC SYSTEMS
A PREPARATION P PREPARES A SYTEM WITH CERTAIN PROPERTIES AND A MEASUREMENT M REVEALS THOSE PROPERTIES
A COMPLETE SPECIFICATION OF THE PROPERTIES OF A SYSTEM IS CALLED AN ‘ONTIC STATE’ AND IS DENOTED BY λ
THE ONTIC STATE SPACE IS DENOTED BY Λ
EVEN WHEN AN OBSERVER KNOWS THE PREPARATION PROCEDURE P, SHE MAY NOT KNOW THE EXACT ONTIC STATE THAT IS PRODUCED, AND ASSIGNS OVER Λ A PROBABILITY DISTRIBUTION μ(ψ|λ) >0 AND AN ‘INDICATOR FUNCTION’ ξ (ψ|λ) TO EACH STATE ψ SUCH THAT THE BORN RULE IS REPRODUCED:
BORN RULE
∫ d λ ξ (φ|λ) μ(ψ|λ) = |< φ| ψ>|2
∫ d λ μ(ψ|λ) = 1
AN INDICATOR/RESPONSE FUNCTION IS DEFINED BY
ξ (ψ|λ) = 1 FOR ALL λ IN Λψ
= 0 ELSEWHERE
SCHEMATIC VIEWS OF THE ONTIC STATE SPACE FOR 3 MODELS
SCHEMATIC REPRESENTATIONS OF PROBABILITY DISTRIBUTIONS ASSOCIATED WITH ψ IN 3 MODELS
TWO DISTINCTIONS AND THREE CLASSES OF ONTOLOGICAL MODELS
THE PBR THEOREM
PUSEY, BARRETT AND RUDOLPH (2012)
UNDER THE REASONABLE ASSUMPTION OF PREPARATION INDEPENDENCE
Ψ-EPISTEMIC MODELS ARE INCOMPATIBLE WITH STANDARD ORTHODOX QUANTUM MECHANICS
INFORMATION AGE
IN THIS AGE OF QUANTUM INFORMATION SCIENCE Ψ IS REGARDED PRIMARILY AS MERE KNOWLEDGE. THE PBR THEOREM IS A SHOCK IN THIS RESPECT.
EINSTEIN PREFERRED THE EPISTEMIC INTERPRETATION OF Ψ
QUANTUM BAYESIANISM (QBISM) ADVOCATES AN EPISTEMIC INTERPRETATIONFuchs, Mermin and Schack
SOME INDIAN RESEARCH GROUPS IN QIP
IISc BANGALORE AND IISER PUNE (NMR)
HARISH-CHANDRA RESEARCH INSTITUTE, ALLAHABAD
S N BOSE NATIONAL CENTRE FOR BASIC SCIENCES & BOSE INSTITUTE, KOLKATA
INDIAN INSTITUTE OF MATHEMATICAL SCIENCES, CHENNAI
IIT KANPUR (EXP QUANTUM OPTICS)
IOP, BHUBANESWAR
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