The Quantum Measurement Process and the Quantum Eraser Christos Karapoulitidis Supervisor: Anastasios Petkou Aristotle University of Thessaloniki Department of Physics October 3, 2019 Christos Karapoulitidis October 3, 2019 1 / 49
Contents Density Matrix 1 Entanglement - Non Locality 2 Quantum Measurement Problem 3 Quantum Eraser and Delayed-Choice Experiments 4 Christos Karapoulitidis October 3, 2019 2 / 49
Plan Density Matrix 1 Projection Operator Statistical Ensemble Definition, Mean Value and Probabilities Pure and Mixed States Reduced density matrix Entanglement - Non Locality 2 Quantum Measurement Problem 3 Quantum Eraser and Delayed-Choice Experiments 4 Christos Karapoulitidis October 3, 2019 3 / 49
Projection Operator Definition: ˆ P j = | j � � j | (1) the operator which performs a projection of the state | ψ � = � n c n | n � on the state c j | j � Properties ˆ P j is hermitian P 2 ˆ j = ˆ P j P j ˆ ˆ P j ′ = 0 � | j � � j | = 1 j Christos Karapoulitidis October 3, 2019 4 / 49
Statistical Ensemble Definition Statistical ensemble is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents a possible state that the real system might be in. In other words, a statistical ensemble is a probability distribution for the state of the system. Ensembles are usefull for calculation of statistical predictions of theory like the mean value � ˆ O � = � ψ | O | ψ � Ensembles can be not only pure. There is the possibility to composed from ensembles which one corresponds in another quantum state | ψ i � .This mixed ensemble described from the statistical weight of its parts. Christos Karapoulitidis October 3, 2019 5 / 49
Definition, Mean Value and Probabilities We assume the statistical mixture quantum mechanical states: { p 1 , | ψ 1 �} , { p 2 , | ψ 2 �} , ... , { p n , | ψ n �} and the observables of the system ˆ O . � ˆ � p i � ψ i | ˆ � p i � ˆ O � = O | ψ i � ≡ O � i i i We assume that the eigenstates of the operator ˆ O are the | j � and the eigenequation will be: ˆ O | j � = o j | j � So, � ˆ p i |� j | ψ i �| 2 o j � O � = i , j Through the term |� j | ψ i �| 2 = � j | ψ i � � j | ψ i � ∗ we can expand the above mean value: � ˆ � ˆ � p i � j | ψ i � � ψ i | ˆ � � � O � = O | j � = � j | p i | ψ i � � ψ i | O | j � i , j j i Christos Karapoulitidis October 3, 2019 6 / 49
Definition, Mean Value and Probabilities Definition For a finite-dimensional function space, the most general density operator is of the form: � ̺ ≡ ˆ p i | ψ i � � ψ i | (2) i Mean Value � ˆ ̺ ˆ ̺ ˆ O � = � � � j � j | ˆ O | j � = tr ˆ O Probability p ψ ( o j ) = | � j | ψ � | 2 = � j | ˆ ̺ ˆ � � ̺ | j � = tr ˆ P j Christos Karapoulitidis October 3, 2019 7 / 49
Pure and Mixed States Pure State The state of the system described by a ket vector in Hilbert space. ̺ = | ψ � � ψ | ˆ ̺ 2 = ˆ ˆ ̺ 1 tr (ˆ ̺ ) = 1 2 ̺ 2 ) = 1 tr (ˆ 3 Mixed State Cannot be described with a single ket vector, but with its associated density matrix. ˆ ̺ = � ˜ i p i | ψ i � � ψ i | tr (ˆ ̺ ) = 1 ˜ 1 tr (ˆ ̺ 2 ) < 1 ˜ 2 Christos Karapoulitidis October 3, 2019 8 / 49
̺ and ˆ Difference between ˆ ˜ ̺ Two-level polarization ̺ = c h | 2 | h � � h | + | c v | 2 | v � � v | + c h c ∗ v | h � � v | + c ∗ | ψ � = c h | h � + c v | v � → ˆ h c v | v � � h | ̺ = | c h | 2 | h � � h | + | c v | 2 | v � � v | Mixture : ˆ ˜ � 1 � � 0 � In matrices representation: | h � = , | v � = 0 1 Pure State Mixed State � | c h | 2 � | c h | 2 c h c ∗ � 0 � ˆ v ̺ = ˆ (3) ̺ = ˜ (4) c ∗ | c v | 2 | c v | 2 h c v 0 Calculate probabilities p h and p v of detection horizontal and vertical polarization, respectively will give us that are in both cases equal | c h | 2 and | c v | 2 respectively. Christos Karapoulitidis October 3, 2019 9 / 49
Pure and mixed states We consider the polarization in a rotated basis, for example in the basis |ր� , |տ� where � 1 1 1 � 1 � 1 1 � � � � � √ √ √ √ |ր� = | h � + | v � = , |տ� = | h �−| v � = 1 − 1 2 2 2 2 1 1 √ � � √ � � | h � = |ր� + |տ� , | v � = |ր� − |տ� 2 2 The state vector in the basis rotated by 45 ◦ will be | ψ � = c h + c v |ր� + c h − c v √ √ |տ� (5) 2 2 which corresponds to a density matrix | c h + c v | 2 ( c h + c v )( c h − c v ) ∗ ̺ տր = 1 � � ˆ (6) ( c h + c v ) ∗ ( c h − c v ) | c h − c v | 2 2 Christos Karapoulitidis October 3, 2019 10 / 49
̺ and ˆ Difference between ˆ ˜ ̺ For the mixed state will be | c h | 2 + | c v | 2 | c h | 2 − | c v | 2 ̺ տր = 1 ˆ ˜ (7) 2 | c h | 2 − | c v | 2 | c h | 2 + | c v | 2 Pure State Mixed State ̺ տր |ր� = | c h | 2 + | c v | 2 p ψ (45 ◦ ) = �ր| ̺ տր |ր� = | c h + c v | 2 ′ ψ (45 ◦ ) = �ր| ˆ ˜ p 2 2 ̺ տր |տ� = | c h | 2 − | c v | 2 p ψ (135 ◦ ) = �տ| ̺ տր |տ� = | c h − c v | 2 ′ ψ (135 ◦ ) = �տ| ˆ p ˜ 2 2 (8) (9) So the difference between a mixed state and a pure state lies in the ability of quantum amplitudes to interfere. A mixture of states describes a situation in which a system really is in one of these two states, and we merely do not know which state this is. On the contrary, when a system is in a superposition of states, it is definitely not in either of these states. Christos Karapoulitidis October 3, 2019 11 / 49
Reduced density matrix The density matrix obtained by tracing out partial degrees of freedom of a compound system is called reduced density matrix. A couple of two-level systems in factorized form state 1 1 √ � � √ � � | ψ � 1 = |↑� 1 + |↓� 1 , | ψ � 2 = |↑� 2 + |↓� 2 2 2 Whole system state vector: | Ψ � 12 = | ψ � 1 ⊗ | ψ � 2 (10) = 1 � � |↑� 1 |↑� 2 + |↓� 1 |↓� 2 + |↑� 1 |↓� 2 + |↓� 1 |↑� 2 2 we note here that the state vector is in factorized form . Christos Karapoulitidis October 3, 2019 12 / 49
Reduced density matrix The corresponding density matrix � � ̺ 12 = ˆ | ψ � 1 � ψ | ⊗ | ψ � 2 � ψ | = 1 � � |↑� 1 �↑| + |↓� 1 �↓| + |↑� 1 �↓| + |↓� 1 �↑| (11) 2 ⊗ 1 � � |↑� 2 �↑| + |↓� 2 �↓| + |↑� 2 �↓| + |↓� 2 �↑| 2 � ̺ 1 = Tr 2 (ˆ ˆ ̺ 12 ) = 2 � j | ˆ ̺ 12 | j � 2 j = ↑ , ↓ (12) = 1 � � |↑� 1 �↑| + |↓� 1 �↓| + |↑� 1 �↓| + |↓� 1 �↑| = | ψ � 1 � ψ | 2 ̺ 2 = 1 � � ˆ |↑� 2 �↑| + |↓� 2 �↓| + |↑� 2 �↓| + |↓� 2 �↑| = | ψ � 2 � ψ | (13) 2 Christos Karapoulitidis October 3, 2019 13 / 49
Plan Density Matrix 1 Entanglement - Non Locality 2 Quantum Measurement Problem 3 Quantum Eraser and Delayed-Choice Experiments 4 Christos Karapoulitidis October 3, 2019 14 / 49
Description of Entanglement via Density Matrix Definition Two system S 1 and S 2 are said to be entangled with respect to a certain de- gree of freedom if their total state | Ψ � 12 , relative to that degree of freedom, cannot be written in a factorized form as a product | ψ � 1 ⊗ | ψ � 2 We are coming back to the example of electrons spin, A couple of two-level systems in a non-factorized form state 1 � � √ | Ψ � 12 = |↑� 1 |↓� 2 + |↓� 1 |↑� 2 2 ̺ 12 = | Ψ � 12 � Ψ | ˆ = 1 � � (14) |↑� 1 �↑| ⊗ |↓� 2 �↓| + |↓� 1 �↓| ⊗ |↑� 2 �↑| 2 � + |↑� 1 �↓| ⊗ |↓� 2 �↑| + |↓� 1 �↑| ⊗ |↑� 2 �↓| Christos Karapoulitidis October 3, 2019 15 / 49
Reduced density matrix ̺ 21 | j � 2 = 1 � � � ̺ 1 = Tr 2 (ˆ ˆ ̺ 21 ) = 2 � j | ˆ |↑� 1 �↑| + |↓� 1 �↓| (15) 2 j = ↑ , ↓ In similar way, ̺ 21 | j � 1 = 1 � � � ̺ 2 = Tr 1 (ˆ ˆ ̺ 21 ) = 1 � j | ˆ |↑� 2 �↑| + |↓� 2 �↓| (16) 2 j = ↑ , ↓ It is clear that the above density matrices are mixed states. In this case there is a loss of information, when we want to describe the subsystems of the whole. This is a totally different outcome from the case of factorized state vector. So in conclusion the separation of the subsystems is only virtual in the case of entangled states. Christos Karapoulitidis October 3, 2019 16 / 49
Plan Density Matrix 1 Entanglement - Non Locality 2 Quantum Measurement Problem 3 Statement of the problem Von Neumann - Projective Measurement The role of apparatus Decoherence - Zurek’s model Quantum Eraser and Delayed-Choice Experiments 4 Christos Karapoulitidis October 3, 2019 17 / 49
Statement of the problem Suppose a system in a superposition pure state state before measuring � | ψ � = c n | ψ n � (17) n As a result of measurement we obtain the component | ψ k � of the initial state. Measurement Problem measurement | ψ � − − − − − − − − → | ψ k � (18) There in no way to obtain this result from a superposition state through a unitary evolution. Christos Karapoulitidis October 3, 2019 18 / 49
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