EE641000 Quantum Information and Computation Chung-Chin Lu - PowerPoint PPT Presentation

✫ ✬ Positive Operator-Valued Measure (POVM) Measurements • { M m } : a collection of measurement operators with � M † m M m = I. m • P ( m ) = � ψ | M † m M m | ψ � . 27 • E m ≡ M † m M m : positive operators, called POVM elements � E m = I and P ( m ) = � ψ | E m | ψ � . m • { E m } : a POVM. • Useful when only the measurement statistics matter. ✪ ✩

✫ ✬ For a projective measurement { P m } , all the POVM elements are the same as the measurement operators since 28 E m = P † m P m = P 2 m = P m . ✪ ✩

✫ ✬ What Are POVMs ? • A collection of positive operators { E m } . • Satisfying the completeness relation 29 � E m = I. m The corresponding measurement operators can be chosen as {√ E m } . ✪ ✩

✫ ✬ Postulate 4 – Composite Systems • Q i : i th quantum system. • H i : the Hilbert space associated to the quantum system Q i . 30 • H = ⊗ i H i : the Hilbert space associated to the composite system of Q i ’s. • | ψ i � : a state of quantum system Q i . • | ψ � = ⊗ i | ψ i � : the joint state of the composite system. ✪ ✩

✫ ✬ Entangled States • States in a composite quantum system. • Not a direct product of states of component systems. √ • ( | 00 � + | 01 � ) / 2 is not an entangled state since 31 | 00 � + | 01 � � | 0 � + | 1 � � √ = | 0 � √ . 2 2 • Bell states in a two-qubit system are entangled states | 00 � + | 11 � , | 00 � − | 11 � , | 01 � + | 10 � , | 01 � − | 10 � √ √ √ √ . 2 2 2 2 ✪ ✩

✫ ✬ A Proof Suppose that | 00 � + | 11 � √ = ( a | 0 � + b | 1 � ) ⊗ ( c | 0 � + d | 1 � ) 2 = ac | 00 � + ad | 01 � + bc | 10 � + bd | 11 � , 32 where | a | 2 + | b | 2 = | c | 2 + | d | 2 = 1. Then we have ad = bc = 0 . • a = c = 0 ⇒ | 00 � + | 11 � = e jθ | 11 � , a contradiction. √ 2 • b = d = 0 ⇒ | 00 � + | 11 � = e jθ ′ | 00 � , a contradiction. √ 2 ✪ ✩

✫ ✬ The Density Operator Formulation of Quantum Mechanics • A convenient means for describing quantum systems whose 33 states is not completely known. • A convenient tool for the description of individual subsystems of a composite quantum system. ✪ ✩

✫ ✬ An Ensemble of Quantum Pure States { p i , | ψ i �} • | ψ i � : states of a quantum system, called pure states. • p i : the probability that the quantum system is in pure state | ψ i � , � p i = 1 . 34 i • The density operator or density matrix which represents this ensemble is � ρ = p i | ψ i �� ψ i | . i – Not necessary a spectral decomposition of ρ since {| ψ i �} may not be an orthonormal set. ✪ ✩

✫ ✬ Evolution of a Density Operator • U : a unitary operator, describing the evolution of a closed quantum system during a time interval. • ρ : a density operator, representing an ensemble { p i , | ψ i �} of pure states, which describes the initial state of the system. 35 • UρU † : density operator, describing the final state of the system. U | ψ i � − → U | ψ i � ρ ′ = p i U | ψ i �� ψ i | U † = UρU † . U � � ρ = p i | ψ i �� ψ i | − → i i ✪ ✩

✫ ✬ Measurement Effect on a Density Operator • { M m } : a collection of measurement operators , acting on the Hilbert space associated to the system being measured and satisfying the completeness equation 36 � M † m M m = I. m • m : index which represents possible measurement outcomes. • ρ : a density operator, representing an ensemble { p i , | ψ i �} of pure states. ✪ ✩

✫ ✬ If the pre-measurement state of the quantum system is | ψ i � , then the probability of getting result m is P ( m | i ) = � ψ i | M † m M m | ψ i � = tr( M † m M m | ψ i �� ψ i | ) , and the post-measurement state of the system is M m | ψ i � | ψ ( m ) � = . i � � ψ i | M † m M m | ψ i � 37 The total probability of getting result m is � � p i tr( M † P ( m ) = p i P ( m | i ) = m M m | ψ i �� ψ i | ) i i � �� �� M † = tr( M † m M m ρ ) = tr( M m ρM † = tr m M m p i | ψ i �� ψ i | m ) . i ✪ ✩

✫ ✬ After a measurement which yields the result m , we have • {P ( i | m ) , | ψ ( m ) �} : an ensemble of pure states i • P ( i | m ) : the probability that the quantum system is in pure state | ψ ( m ) � given that outcome m is measured i P ( i | m ) = p i P ( m | i ) P ( m ) 38 • ρ ( m ) : density operator, describing the state of the quantum system after the outcome m is measured P ( i | m ) M m | ψ i �� ψ i | M † � P ( i | m ) | ψ ( m ) �� ψ ( m ) � ρ ( m ) m = | = i i � ψ i | M † m M m | ψ i � i i i p i M m | ψ i �� ψ i | M † M m ρM † M m ρM † � m m m = = = . tr( M † tr( M m ρM † P ( m ) m M m ρ ) m ) ✪ ✩

✫ ✬ Pure States vs Mixed States • Pure state | ψ � : a quantum system whose state is exactly known as | ψ � and can be described by the density operator ρ = | ψ �� ψ | . 39 • Mixed state ρ : a quantum system whose state is not completely known and is described by the density operator � ρ = p i | ψ i �� ψ i | . i • A pure state can be regarded as a very special mixed state. ✪ ✩

✫ ✬ Characterization of Density Operators ρ is a density operator associated with an ensemble { p i , | ψ i �} if and 40 only if • Unit trace condition : tr( ρ ) = 1. • Positivity condition : ρ is a positive operator. ✪ ✩

✫ ✬ Proof = ⇒ • ρ = � i p i | ψ i �� ψ i | . 41 • tr( ρ ) = � i p i tr( | ψ i �� ψ i | ) = � i p i � ψ i | ψ i � = � i p i = 1. i p i |� ϕ | ψ i �| 2 ≥ 0. • � ϕ | ρ | ϕ � = � i p i � ϕ | ψ i �� ψ i | ϕ � = � ✪ ✩

✫ ✬ Proof ⇐ = • ρ is positive with a spectral decomposition � ρ = λ j | ψ j �� ψ j | . j 42 • λ j : non-negative eigenvalues. • | ψ j � : eigenvectors. • 1 = tr( ρ ) = � j λ j . • { λ j , | ψ j �} : an ensemble of pure states giving rise to the density operator ρ . ✪ ✩

✫ ✬ A Criterion of Pure States A density operator ρ is in a pure state if and only if 43 tr( ρ 2 ) = 1 . • For a mixed (not a pure) state ρ , we have tr( ρ 2 ) < 1. ✪ ✩

✫ ✬ Proof Let ρ be a density operator with spectral decomposition � ρ = λ i | ψ i �� ψ i | , i where λ i ≥ 0 and tr( ρ ) = � i λ i = 1. Since 44 ρ 2 = � λ 2 i | ψ i �� ψ i | , i we have λ i ) 2 = 1 , � � � � tr( ρ 2 ) = λ 2 λ 2 i ≤ i + 2 λ i λ j = ( i i i<j i where equality holds if and only if only one λ i is non-zero and is equal to one, i.e., ρ = | ψ i �� ψ i | , a pure state. ✪ ✩

✫ ✬ Mixture of Mixed States � ρ = p i ρ i . i • ρ i : density operator corresponding to an ensemble { p ij , | ψ ij �} � ρ i = p ij | ψ ij �� ψ ij | . j 45 • p i : probability that the state of the quantum system is prepared in ρ i . The probability of being in the pure state | ψ ij �} is p i p ij and the overall density operator to describe the state of the quantum system is � � � � ρ = p i p ij | ψ ij �� ψ ij | = p i p ij | ψ ij �� ψ ij | = p i ρ i . ij i j i ✪ ✩

✫ ✬ Density Operator After Unspecified Measurement { M m } 46 M m ρM † ρ ′ = P ( m ) ρ ( m ) = � � � tr( M m ρM † M m ρM † m m ) = m . tr( M m ρM † m ) m m m ✪ ✩

✫ ✬ Average for Projective Measurement • ρ : density operator for a quantum system • M : an observable for the quantum system with spectral decomposition � M = mP m 47 m • P ( m ) = tr( P m ρP m ) = tr( P 2 m ρ ) = tr( P m ρ ) : the probability that outcome m occurs • � M � : the average measurement value � M � = � m m P ( m ) = � m m tr( P m ρ ) = tr( Mρ ). ✪ ✩

✫ ✬ What Class of Ensembles Gives Rise to a Particular ρ ? • ρ = 3 4 | 0 �� 0 | + 1 4 | 1 �� 1 | (spectral decomposition). � � � � 3 1 3 1 • | a � = 4 | 0 � + 4 | 1 � , | b � = 4 | 0 � − 4 | 1 � . 48 1 2 | a �� a | + 1 2 | b �� b | = 3 4 | 0 �� 0 | + 1 4 | 1 �� 1 | = ρ. • A lesson : the collection of eigenstates of a density operator is not an especially privileged ensemble. ✪ ✩

✫ ✬ Unitary Freedom in the Ensemble for Density Operators Two ensembles { p i , | ψ i �} and { q i , | ϕ j �} give rise to the same density operator ρ , i.e., � � ρ = p i | ψ i �� ψ i | , ρ = q j | ϕ j �� ϕ j | 49 i j if and only if √ p i | ψ i � = z ij √ q j | ϕ j � � j where z ij is a unitary matrix of complex numbers and pure states with zero probability are padded to the smaller ensemble to have the same size as the larger one. ✪ ✩

✫ ✬ Proof ⇐ = • | v i � ≡ √ p i | ψ i � , | w j � ≡ √ q j | ϕ j � . Since � | v i � = z ij | w j � , j we have � � � � z ij z ∗ 50 p i | ψ i �� ψ i | = | v i �� v i | = ik | w j �� w k | i i i jk �� � � z ij z ∗ = | w j �� w k | ik jk i � = | w j �� w j | j � = q j | ϕ j �� ϕ j | . j ✪ ✩

✫ ✬ Proof = ⇒ By spectral decomposition of ρ , we have � � | k ′ �� k ′ | , ρ = λ k | k �� k | = k k where λ k are positive, | k � are orthonormal and | k ′ � = √ λ k | k � . • | u � : a vector in the orthogonal complement Span {| k ′ �} ⊥ of 51 Span {| k ′ �} . Then � � � � u | k ′ �� k ′ | u � = � u | ρ | u � = |� u | v i �| 2 0 = � u | v i �� v i | u � = k i i which implies that | u � ∈ Span {| v i �} ⊥ . ✪ ✩

✫ ✬ Thus Span {| k ′ �} ⊥ ⊆ Span {| v i �} ⊥ and then Span {| v i �} ⊆ Span {| k ′ �} . For each | v i � , we have � c ik | k ′ � | v i � = k Then 52 �� � � � � | k ′ �� k ′ | = c ik c ∗ | k ′ �� l ′ | ρ = | v i �� v i | = il i i k kl Since the operators | k ′ �� l ′ | are linearly independent, we have � c ik c ∗ il = δ kl i By appending more columns to the matrix C = [ c ik ], we obtain a ✪ ✩

✫ ✬ unitary matrix T = [ t ik ] such that � t ik | k ′ � | v i � = k where some zero vectors are padded into the list of | k ′ � . Similarly, there is a unitary matrix S = [ jk ] such that 53 � s jk | k ′ � | w j � = k Then with Z = TS † a unitary matrix and Z = [ z ij ], we have � | v i � = z ij | w j � j since ✪ ✩

✫ ✬ � � � � t ik s ∗ s jl | l ′ � z ij | w j � = jk j j k l � � t ik | l ′ � s ∗ = jk s jl 54 kl j � t ik | k ′ � = k = | v i � ✪ ✩

✫ ✬ Postulates of Quantum Mechanics 55 – Density Operator Version ✪ ✩

✫ ✬ Postulate 1 – States Associated to an isolated physical system is a Hilbert space H (eg, a finite-dimensional complex inner product space). The state of the system is completely described by its density operator , which is a 56 positive operator with trace one acting on the Hilbert space H . If the quantum system is in the state ρ i with probability p i , then the density operator for this system is � ρ = p i ρ i . i ✪ ✩

✫ ✬ Postulate 2 - Time Evolution The evolution of a closed quantum system is described by a unitary operator . That is, the state ρ of the system at time t 1 is related to 57 the state ρ ′ of the system at time t 2 by a unitary operator U which depends only on the times t 1 and t 2 , ρ ′ = UρU † . ✪ ✩

✫ ✬ Postulate 3 – Quantum Measurements • { M m } : a collection of measurement operators , acting on the Hilbert space associated to the system being measured and 58 satisfying the completeness equation � M † m M m = I. m • m : measurement outcomes that may occur in the experiment. ✪ ✩

✫ ✬ If the pre-measurement state of the quantum system is ρ , then the probability that result m occurs is given by P ( m ) = tr( M m ρM † m ) , and the post-measurement state of the system is M m ρM † m . tr( M m ρM † m ) 59 The completeness euqation expresses the fact that probabilities sum to one � � � tr( M m ρM † tr( M † P ( m ) = m ) = m M m ρ ) m m m ��� � � M † = tr m M m ρ = tr( ρ ) = 1 . m ✪ ✩

✫ ✬ Postulate 4 – Composite Systems • Q i : i th quantum system. • H i : the Hilbert space associated to the quantum system Q i . 60 • H = ⊗ i H i : the Hilbert space associated to the composite system of Q i ’s. • ρ i : the state in which the quantum system Q i is prepared. • ρ = ⊗ i ρ i : the joint state of the composite system. ✪ ✩

✫ ✬ Reduced Density Operator 61 ✪ ✩

✫ ✬ Definition ρ A △ = tr B ( ρ AB ) . 62 • ρ AB : density operators for composite quantum system AB . • ρ A △ = tr B ( ρ AB ) : reduced density operator for subsystem A . – A description for the state of subsystem A : justification needed. ✪ ✩

✫ ✬ A Simple Justification • ρ AB = ρ ⊗ σ : a direct product density operator for composite 63 quantum system AB . • ρ A = tr B ( ρ AB ) = ρ tr( σ ) = ρ : correct description of system A . • ρ B = tr A ( ρ AB ) = tr( ρ ) σ = σ : correct description of system B . ✪ ✩

✫ ✬ A Further Justification 64 ✪ ✩

✫ ✬ Local and Global Observables • M : the observable on subsystem A for a measurement carrying out on subsystem A , a Hermitian operator with spectral decomposition � M = mP m . m 65 • M ⊗ I : the corresponding observable on the composite system AB for the same measurement carrying out on subsystem A , a Hermitian operator with spectral decomposition � M ⊗ I = m ( P m ⊗ I ) . m • | m � is an eigenstate of the observable M and | ψ � is any state of subsystem B ⇐ ⇒ | m � ⊗ | ψ � is an eigenstate of M ⊗ I . ✪ ✩

✫ ✬ When System AB Is Prepared With State | m � ⊗ | ψ � • m : the outcome which occurs with probability one by the observable M on subsystem A . 66 • m : the outcome which occurs with probability one by the observable M ⊗ I on the composite system AB . • Consistency. ✪ ✩

✫ ✬ When System AB Is in a Mixed State ρ AB • f ( ρ AB ) : a density operator on subsystem A as a function of the density operator on system AB , serving as an appropriate description of the state of subsystem A . 67 • Measurement statistics must be consistent between the local observable M on subsystem A and the global observable M ⊗ I on system AB tr( Mf ( ρ AB )) = � M � = � M ⊗ I � = tr(( M ⊗ I ) ρ AB ) . ✪ ✩

✫ ✬ Existence : f ( ρ AB ) = tr B ( ρ AB ) • ρ AB = � i α i T A i ⊗ T B : a linear operator on the state space of i the composite system AB . tr(( M ⊗ I ) ρ AB ) 68 � � α i T A i ⊗ T B α i ( MT A i ) ⊗ T B = tr(( M ⊗ I )( i )) = tr( i ) i i � � α i ( MT A i ) ⊗ T B α i ( MT A i ) tr( T B = tr(tr B ( i )) = tr( i )) i i � � α i T A tr( T B α i T A i ⊗ T B = tr( M ( i ))) = tr( M tr B ( i )) i i i tr( M tr B ( ρ AB )) . = ✪ ✩

✫ ✬ Uniqueness • H : the Hilbert space associated to the quantum system A . • L H ( H ) : the real inner product space of all Hermitian operators on H with trace inner product. • { M i } : an orthonormal basis of L H ( H ). • f ( ρ AB ) = � i M i tr( M i f ( ρ AB )) : the expansion of f ( ρ AB ) by 69 the orthonormal basis { M i } . Since tr( M i f ( ρ AB )) = tr(( M i ⊗ I ) ρ AB ) ∀ i, we have � f ( ρ AB ) = M i tr(( M i ⊗ I ) ρ AB ) i which uniquely specifies the function f . ✪ ✩

✫ ✬ An Example • Suppose a two-qubit system is in a pure Bell state | 00 � + | 11 � √ 2 with density operator 70 � | 00 � + | 11 � � � � 00 | + � 11 | � ρ 12 = √ √ 2 2 | 00 �� 00 | + | 11 �� 00 | + | 00 �� 11 | + | 11 �� 11 | = . 2 ✪ ✩

✫ ✬ • ρ 1 : the reduced density operator of the first qubit ρ 1 tr 2 ( ρ 12 ) = tr 2 ( | 00 �� 00 | ) + tr 2 ( | 11 �� 00 | ) + tr 2 ( | 00 �� 11 | ) + tr 2 ( | 11 �� 11 | ) = 2 | 0 �� 0 |� 0 | 0 � + | 1 �� 0 |� 0 | 1 � + | 0 �� 1 |� 1 | 0 � + | 1 �� 1 |� 1 | 1 � 71 = 2 | 0 �� 0 | + | 1 �� 1 | = I = 2 . 2 • Reduced density operator ρ 1 for the first qubit is in a mixed state while the two-qubit system is in a pure state. ✪ ✩

✫ ✬ Schmidt Decomposition and Purification 72 ✪ ✩

✫ ✬ Schmidt Decomposition For each pure state | ψ � in a composite quantum system AB , there exist a set {| i A �} of orthonormal states for subsystem A and a set {| i B �} of orthonormal states for subsystem B of the same size such that � | ψ � = λ i | i A �| i B � i 73 where λ i are non-negative real numbers with � λ 2 i = 1 . i • λ i : Schmidt coefficients. • {| i A �} and {| i B �} : Schmidt “bases” for A and B respectively. – Dependent on | ψ � . • # of non-zero values λ i : Schmidt number for | ψ � . ✪ ✩

✫ ✬ Proof • {| j �} , {| k �} : given orthonormal bases of the Hilbert spaces of subsystems A and B respectively � | ψ � = c jk | j �| k � . jk 74 • C = UDV : singular value decomposition C = [ c jk ] , U = [ u ji ] , D = diag( d ii ) , V = [ v ik ] , � c jk = u ji d ii v ik . i – U and V : unitary matrices. – D : a diagonal matrix, not necessarily square. ✪ ✩

✫ ✬ � � | ψ � = u ji d ii v ik | j �| k � jk i ⎛ ⎞ �� � � ⎝� � = d ii u ji | j � v ik | k � = λ i | i A �| i B � . ⎠ i j i k • | i A � = � j u ji | j � : orthonormal states of subsystem A � � � i A | i ′ u ∗ ji u j ′ i ′ � j | j ′ � = u ∗ ji u ji ′ = δ ii ′ . A � = 75 jj ′ j • | i B � = � k v ik | k � : orthonormal states of subsystem B � � � i B | i ′ v ∗ ik v i ′ k ′ � k | k ′ � = v ∗ B � = ik v i ′ k = δ ii ′ . kk ′ k • λ i = d ii : non-negative real numbers � � λ i λ i ′ � i A | i ′ A �� i B | i ′ λ 2 1 = � ψ | ψ � = B � = i . ii ′ i ✪ ✩

✫ ✬ Schmidt Number for State | ψ � = � i λ i | i A �| i B � • ”Amount” of entanglement between systems A and B when the composite system AB is in state | ψ � . • Invariance under unitary transformations on subsystem A or 76 subsystem B alone. – U : a unitary operator on subsystem A . – U | i A � : orthonormal states of subsystem A . � � ( U ⊗ I ) | ψ � = λ i ( U ⊗ I )( | i A � ⊗ | i B � ) = λ i U | i A �| i B � . i i ✪ ✩

✫ ✬ Purification • ρ A : a density operator for system A with ensemble { p i , | i A �} � ρ A = p i | i A �� i A | . i • R : a reference system. 77 • {| i R �} : an orthonormal basis of the Hilbert space associated to system R , having the same cardinality as that of {| i A �} . • | AR � : a pure state of the composite system AR with √ p i | i A �| i R � . △ � | AR � = i ✪ ✩

✫ ✬ √ p i p j tr R ( | i A �� j A | ⊗ | i R �� j R | ) � tr R ( | AR �� AR | ) = ij √ p i p j | i A �� j A | tr( | i R �� j R | ) � = ij 78 � = p i | i A �� i A | = ρ A . i • A mixed state of a local system is a local view of a pure state in a global composite system. ✪ ✩

✫ ✬ Applications 79 ✪ ✩

✫ ✬ Non-orthogonal States Cannot Be Distinguished • { M j } : measurement operators • | ψ 1 � and | ψ 2 � : two non-orthogonal states to be distinguished and 80 | ψ 2 � = α | ψ 1 � + β | ψ � , where | ψ 1 � and | ψ � are orthonormal. Note that | α | 2 + | β | 2 = 1 and then | β | < 1. • f ( · ) : a rule to guess which state vector is observed based on the outcome of the measurement, i.e., either f ( j ) = 1 or f ( j ) = 2. ✪ ✩

✫ ✬ Suppose that | ψ 1 � and | ψ 2 � can be distinguished reliably, i.e., ⎛ ⎞ � � ψ 1 | M † � M † ⎠ | ψ 1 � = � ψ 1 | G 1 | ψ 1 � = 1 j M j | ψ 1 � = � ψ 1 | j M j ⎝ j : f ( j )=1 j : f ( j )=1 ⎛ ⎞ � � ψ 2 | M † � M † ⎠ | ψ 2 � = � ψ 2 | G 2 | ψ 2 � = 1 j M j | ψ 2 � = � ψ 2 | j M j ⎝ j : f ( j )=2 j : f ( j )=2 j : f ( j )= i M † 81 where G i = � j M j , for i = 1 , 2. Since G 1 + G 2 = I , we have � ψ 1 | ( G 1 + G 2 ) | ψ 1 � = 1 and then � � � � ψ 1 | G 2 | ψ 1 � = 0 ⇒ G 2 | ψ 1 � = 0 ⇒ G 2 | ψ 2 � = β G 2 | ψ � Thus a contradiction is obtained as follows � ψ 2 | G 2 | ψ 2 � = | β | 2 � ψ | G 2 | ψ � ≤ | β | 2 < 1 since � ψ | G 2 | ψ � ≤ � ψ | ( G 1 + G 2 ) | ψ � = � ψ | ψ � = 1. ✪ ✩

✫ ✬ Superdense Coding • Goal : Alice wants to send two classical bits of information to Bob by transmitting only one qubit to Bob • Initialization : preparing a pair of qubits in a Bell State | ψ � = | 00 � + | 11 � √ 2 82 Alice held the first qubit and Bob held the second qubit before apart (may send by a third party) • Alice takes action on her qubit according the two bits of information she wants to send ( I ⊗ I ) | ψ � = | 00 � + | 11 � 00 : | ψ � → √ 2 ( Z ⊗ I ) | ψ � = | 00 � − | 11 � √ 01 : | ψ � → 2 ✪ ✩

✫ ✬ ( X ⊗ I ) | ψ � = | 10 � + | 01 � √ 10 : | ψ � → 2 ( iY ⊗ I ) | ψ � = | 10 � − | 01 � √ 11 : | ψ � → 2 83 • Alice sends her qubit to Bob • The four Bell states form an orthonormal basis of the two-qubit system and can form a projective measrement • With two qubits together, Bob makes the projective measurement ✪ ✩

✫ ✬ Quantum Teleportation M 1 ψ H 84 M 2 { β 00 M ψ M X Z 2 1 ψ ψ ψ ψ ψ 1 2 3 0 4 ✪ ✩

✫ ✬ Quantum Teleportation • | ψ 2 � : state of the three-qubit system before Alice makes her measurement 1 | ψ 2 � = 2 ( | 00 � ( α | 0 � + β | 1 � ) + | 01 � ( α | 1 � + β | 0 � ) + | 10 � ( α | 0 � − β | 1 � ) + | 11 � ( α | 1 � − β | 0 � )) • {| 00 �� 00 | , | 01 �� 01 | , | 10 �� 10 | , | 11 �� 11 |} : a POVM measurement 85 made by Alice on her two qubits • ρ = | ψ 2 �� ψ 2 | : density operator for the three-qubit system before the measurement • ρ ′ : density operator for the three-qubit system after the unspecified (from Bob’s point of view) measurement ρ ′ = � � M m ρM † M m | ψ 2 �� ψ 2 | M † m = m m m ✪ ✩

✫ ✬ • | 00 �� 00 | ψ 2 � = (1 / 2) | 00 � ( α | 0 � + β | 1 � ) • | 01 �� 01 | ψ 2 � = (1 / 2) | 01 � ( α | 1 � + β | 0 � ) • | 10 �� 10 | ψ 2 � = (1 / 2) | 10 � ( α | 0 � − β | 1 � ) • | 11 �� 11 | ψ 2 � = (1 / 2) | 11 � ( α | 1 � − β | 0 � ) • ρ B : the reduced density operator of Bob’s qubit ρ B = tr A ( ρ ′ ) = � tr A ( M m | ψ 2 �� ψ 2 | M † m ) 86 m 1 4(( α | 0 � + β | 1 � )( α | 0 � + β | 1 � ) † + ( α | 1 � + β | 0 � )( α | 1 � + β | 0 � ) † = +( α | 0 � − β | 1 � )( α | 0 � − β | 1 � ) † + ( α | 1 � − β | 0 � )( α | 1 � − β | 0 � ) † ) 2( | α | 2 + | β | 2 ) | 0 �� 0 | + 2( | α | 2 + | β | 2 ) | 1 �� 1 | = 4 | 0 �� 0 | + | 1 �� 1 | = I = 2 2 ✪ ✩

✫ ✬ • Bob does not have any information about the state | ψ � if Alice does not send him her measurement result, preventing Alice 87 from using teleportation to transmit information to Bob faster than light ✪ ✩

✫ ✬ Anti-correlations in the EPR Experiment • V : the state space of a qubit • B = {| 0 � , | 1 �} : an orthonormal basis of V • M = ασ x + βσ y + γσ z : an observable on V 88 | α | 2 + | β | 2 + | γ | 2 = 1 , X = [ σ x ] B , Y = [ σ y ] B , Z = [ σ z ] B • ± 1 : eigenvalues of M • B ′ = {| a � , | b �} : unit eigenvectors of M | 0 � = α | a � + β | b � | 1 � = γ | a � + δ | b � ✪ ✩

✫ ✬ with coordinate transformation matrix U , which is unitary ⎡ ⎤ ⎣ α β U = [ B ′ → B ] = ⎦ , | det( U ) | = | αδ − βγ | = 1 γ δ √ • | ψ � = ( | 01 � − | 10 � ) / 2 : a Bell state prepared on a two-qubit quantum system | 01 � − | 10 � = ( αδ − βγ ) | ab � − | ba � √ √ 89 2 2 • M ⊗ M : observable on the two-qubit system M ⊗ M = ( I ⊗ M )( M ⊗ I ) with spectral decomposition M ⊗ I = ( | a �� a | ⊗ I ) − ( | b �� b | ⊗ I ) I ⊗ M = ( I ⊗ | a �� a | ) − ( I ⊗ | b �� b | ) ✪ ✩

✫ ✬ I ⊗ M +1 : | ab � with prob. 1 − → − 1 : | ab � with prob. 1 | ab � − | ba � M ⊗ I 2 √ 90 − → I ⊗ M 2 − 1 : | ba � with prob. 1 − → +1 : | ba � with prob. 1 2 ✪ ✩

✫ ✬ The Argument of Einstein, Podolsky and Rosen • Any ”element of reality” must be represented in any complete physical theory • It is sufficient to say a physical property to be an element of reality if it is possible to predict with cartainty the value that 91 property will have, immediately before measurement • As in the anti-correlation experiment on a Bell state, once Alice gets her measurement result +1 (-1), she can predict with certainly that Bob will measure -1 (+1) on his qubit • The physical property revealed by various observables M on Bob’s qubit is an element of reality of Bob’s qubit ✪ ✩

✫ ✬ • Quantum mechanics only tell one how to calculate the probability of the respective measurement outcomes if M is measured, it does not include any fundamental element 92 intended to represent such a physical property • Quantum mechanics is not a complete physical theory ✪ ✩

✫ ✬ Bell’s Inequality 93 • A compelling example which illustrates an essential difference between quantum and classical physics ✪ ✩

✫ ✬ The Setup • Charlie prepares two particles 94 – He is capable of repeating the experimental procedure • Charlie sends one particle to Alice and another particle to Bob ✪ ✩

✫ ✬ Derivation of Bell’s Inequality in Classical Scenario • P Q , P R : two physical properties of Alice’s particle • Q, R : values of P Q , P R respectively – Assumed to exist independent of measurement, i.e., assumed to be objective properties of Alice’s particle – Merely revealed by measurement apparatuses 95 – Variables each taking +1 or -1 • P S , P T : two physical properties of Bob’s particle • S, T : values of P S , P T respectively – Assumed to exist independent of measurement, i.e., assumed to be objective properties of Bob’s particle – Merely revealed by measurement apparatuses – Variables taking +1 or -1 ✪ ✩

✫ ✬ • Alice and Bob do their measurements at the same time, which are assumed uncorrelatedly – Alice performing her measurement does not disturb the result of Bob’s measurement – Bob performing his measurement does not disturb the result of Alice’s measurement 96 • QS + RS + RT − QT : a derived quantity QS + RS + RT − QT = ( Q + R ) S + ( R − Q ) T = ± 2 , since Q, R = ± 1 implies that either ( Q + R ) S = 0 or ( R − Q ) T = 0 ✪ ✩

✫ ✬ • P ( q, r, s, t ) : probability that, before the measurements are performed, the system is in a state where Q = q, R = r , S = s, T = t – Dependent on how Charlie prepares the two particles – Dependent on experimental noise • Bell’s inequality : E ( QS ) + E ( RS ) + E ( RT ) − E ( QT ) 97 = E ( QS + RS + RT − QT ) � = P ( q, r, s, t )( qs + rs + rt − qt ) q,r,s,t, � ≤ P ( q, r, s, t ) · 2 q,r,s,t, = 2 • It doesn’t matter how Charlie prepares the particles ✪ ✩

✫ ✬ A Quantum Mechnical Scenario • Charlie prepares a quantum system of two qubits in the Bell state | ψ � = | 01 � − | 10 � √ 2 98 • Q, R : observables performed by Alice on her qubit Q = Z 1 , R = X 1 • S, T : observables performed by Bob on his qubit S = − Z 2 − X 2 , R = Z 2 − X 2 √ √ 2 2 ✪ ✩

✫ ✬ A Quantum Mechnical Scenario (Cont’) • � QS � , � RS � , � RT � , � QT � : average values of measurements � ψ | ( Z 1 ⊗ − Z 2 − X 2 1 � QS � = √ ) | ψ � = √ 2 2 � ψ | ( X 1 ⊗ − Z 2 − X 2 1 √ √ � RS � = ) | ψ � = 2 2 � ψ | ( X 1 ⊗ Z 2 − X 2 1 99 √ √ � RT � = ) | ψ � = 2 2 � ψ | ( Z 1 ⊗ Z 2 − X 2 ) | ψ � = − 1 √ √ � QT � = 2 2 √ � QS � + � RS � + � RT � − � QT � = 2 2 • But the Bell’s inequality says that the above quantity cannot exceed two ✪ ✩