Minimum Uncertainty for Entangled States

Transcription

1 Minimum Uncertainty for Entangled States Tabish Qureshi Centre for Theoretical Physics Jamia Millia Islamia New Delhi Collaborators: N.D. Hari Dass, Aditi Sheel Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 1 / 23

2 Outline 1 Uncertainty and Entanglement 2 Examples 3 General Analysis & Results 4 Conclusions Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 2 / 23

3 Outline Uncertainty and Entanglement 1 Uncertainty and Entanglement 2 Examples 3 General Analysis & Results 4 Conclusions Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 3 / 23

4 Uncertainty and Entanglement Heisenberg s Microscope Optical resolution of the microscope x = λ sin ɛ omentum recoil on the electron = h/λ. the recoil cannot be exactly known, since the direction of he scattered photon is undetermined within the bundle of ays entering the microscope" x p x p x h λ sin ɛ ( ) ( ) λ h sin ɛ λ sin ɛ = h. Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 4 / 23

5 Uncertainty and Entanglement Heisenberg s Microscope Optical resolution of the microscope x = λ sin ɛ omentum recoil on the electron = h/λ. the recoil cannot be exactly known, since the direction of he scattered photon is undetermined within the bundle of ays entering the microscope" x p x p x h λ sin ɛ ( ) ( ) λ h sin ɛ λ sin ɛ = h. Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 4 / 23

6 Uncertainty and Entanglement Uncertainty inherent in Quantum Mechanics The uncertanties in two observables A and B, are defined as A = ψ A 2 ψ ψ A ψ 2 B = ψ B 2 ψ ψ B ψ 2 Uncertainty is an inherent property which depends on the state of the system. It is not a shortcoming of the measurement process. If ψ is an eigenstate of A, A = 0 Noncommuting observables cannot have common eigenstates. Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 5 / 23

7 Uncertainty and Entanglement Uncertainty inherent in Quantum Mechanics The uncertanties in two observables A and B, are defined as A = ψ A 2 ψ ψ A ψ 2 B = ψ B 2 ψ ψ B ψ 2 Uncertainty is an inherent property which depends on the state of the system. It is not a shortcoming of the measurement process. If ψ is an eigenstate of A, A = 0 Noncommuting observables cannot have common eigenstates. Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 5 / 23

8 Uncertainty and Entanglement Uncertainty Relations Heisenberg Uncertainty Relation (HUR) 1 ( X) 2 ( Y ) [X, Y] 2, (1) Schrödinger-Robertson inequality (SR) 2 ( X) 2 ( Y ) [X, Y] { X, Ỹ} 2 (2) where, X = X X, Ỹ = Y Y 1 W. Heisenberg, Zeitschrift für Physik 43, (1927). H. P. Robertson, Phys. Rev. 34, (1929). 2 E. Schrödinger, Ber. Kgl. Akad. Wiss. No. 296 (1930); H. P. Robertson, Phys. Rev. 35, 667A (1930). Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 6 / 23

9 Uncertainty and Entanglement Uncertainty and Entanglement Question: Does entanglement put a bound on the uncertainty relations? System A entangled with System B X A Y A? Let it be Simplest bipartite entangled state Ψ = c 1 ψ 1 A α 1 B + c 2 ψ 2 A α 2 B ψ i A two states of system A, α j B are two orthonormal states of system B. Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 7 / 23

10 Uncertainty and Entanglement Uncertainty and Entanglement Question: Does entanglement put a bound on the uncertainty relations? System A entangled with System B X A Y A? Let it be Simplest bipartite entangled state Ψ = c 1 ψ 1 A α 1 B + c 2 ψ 2 A α 2 B ψ i A two states of system A, α j B are two orthonormal states of system B. Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 7 / 23

11 Uncertainty and Entanglement Uncertainty and Entanglement Question: Does entanglement put a bound on the uncertainty relations? System A entangled with System B X A Y A? Let it be Simplest bipartite entangled state Ψ = c 1 ψ 1 A α 1 B + c 2 ψ 2 A α 2 B ψ i A two states of system A, α j B are two orthonormal states of system B. Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 7 / 23

12 Uncertainty and Entanglement Uncertainty and Entanglement The uncertainties in two observables X A 1 B and Y A 1 B, in the entangled state Ψ, are defined as For a generic observable O ( X) 2 Ψ = Ψ X2 Ψ Ψ X Ψ 2 ( Y ) 2 Ψ = Ψ Y2 Ψ Ψ Y Ψ 2 ( O) 2 Ψ = c 1 2 ( O) c 2 2 ( O) c 1 2 c 2 2 ( O 1 O 2 ) 2 where ( O) 2 1 = ψ 1 O 2 ψ 1 ψ 1 O ψ 1 2 ( O) 2 2 = ψ 2 O 2 ψ 2 ψ 2 O ψ 2 2 Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 8 / 23

13 Uncertainty and Entanglement Uncertainty and Entanglement The product of uncertainties can be worked out to be ( X) 2 Ψ ( Y )2 Ψ = c 1 4 ( X) 2 1 ( Y )2 1 + c 2 4 ( X) 2 2 ( Y ) c 1 2 c 2 2 ( X) 1 ( Y ) 1 ( X) 2 ( Y ) 2 + c 1 4 c 2 4 ( Y 1 Y 2 ) 2 ( X 1 X 2 ) 2 + c 1 2 c 2 2 {( X) 1 ( Y ) 2 ( X) 2 ( Y ) 1 } 2 + c 1 2 c 2 2 { i c i 2 ( X) 2 i ( Y 1 Y 2 ) 2 + i c i 2 ( Y ) 2 i ( X 1 X 2 ) 2 } Necessary conditions for this to reach its minimum value are 1 X 1 = X 2, Y 1 = Y 2 2 ψ 1, ψ 2 be minimum uncertainty states themselves 3 ( X) 1 /( Y ) 1 = ( X) 2 /( Y ) 2. Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 9 / 23

14 Outline Examples 1 Uncertainty and Entanglement 2 Examples 3 General Analysis & Results 4 Conclusions Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 10 / 23

15 Examples Angular Momentum Operators HUR for J x and J y ( J x ) 2 ( J y ) i J z 2 State: Ψ = c 1 m 1 α 1 + c 2 m 2 α 2 The uncertainties for m are given by ( J x ) 2 = ( J y ) 2 = 2 2 (j(j + 1) m2 ) minimum for m = ±j So Ψ can be entangled only if m 1 = +j and m 2 = j, or vice-versa. But, ( J x ) 2 Ψ ( J y) 2 Ψ = j2 2 4 whereas J z Ψ 2 = j 2 2 ( c 1 2 c 2 2 ) 2. ( J x ) 2 ( J y ) 2 > 1 4 i J z 2 Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 11 / 23

16 Examples Angular Momentum Operators HUR for J x and J y ( J x ) 2 ( J y ) i J z 2 State: Ψ = c 1 m 1 α 1 + c 2 m 2 α 2 The uncertainties for m are given by ( J x ) 2 = ( J y ) 2 = 2 2 (j(j + 1) m2 ) minimum for m = ±j So Ψ can be entangled only if m 1 = +j and m 2 = j, or vice-versa. But, ( J x ) 2 Ψ ( J y) 2 Ψ = j2 2 4 whereas J z Ψ 2 = j 2 2 ( c 1 2 c 2 2 ) 2. ( J x ) 2 ( J y ) 2 > 1 4 i J z 2 Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 11 / 23

17 Position-Momentum Examples The HUR has the form ( X) 2 ( P) Entangled state made up of two Gaussian states entangled with two orthogonal states of another system. ( ) 1 x ψ i = (2πσi 2 ) 1/4 eip i x/ exp (x x i) 2 4σi 2 (3) X i = x i, P i = p i, ( X) i = σ i, ( p) i = 2σ i. The minimum uncertainty conditions (1-3) yield: x 1 = x 2, p 1 = p 2 and σ 1 = σ 2. the Gaussians are identical the state is disentangled. Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 12 / 23

18 EPR-like state Examples Two entangled particles A and B: Ψ(x A, x B ) = C dp e ipxb/ e ipxa/ e p2 /4 2 σ 2 e (x A +x B ) 2 16Ω 2 In the limit σ, Ω, the state reduces to the EPR state. The uncertainties in position and momentum of particle A (say) X A = Ω 2 + 1/16σ 2, P A = σ Ω 2. X A P A = /2 is obtained only if Ω = 1/4σ. In that case, the state becomes 1 Ψ(x A, x B ) = e (x A 2+x B 2)2σ2 π/4σ 2 Disentangled! Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 13 / 23

19 EPR-like state Examples Two entangled particles A and B: Ψ(x A, x B ) = C dp e ipxb/ e ipxa/ e p2 /4 2 σ 2 e (x A +x B ) 2 16Ω 2 In the limit σ, Ω, the state reduces to the EPR state. The uncertainties in position and momentum of particle A (say) X A = Ω 2 + 1/16σ 2, P A = σ Ω 2. X A P A = /2 is obtained only if Ω = 1/4σ. In that case, the state becomes 1 Ψ(x A, x B ) = e (x A 2+x B 2)2σ2 π/4σ 2 Disentangled! Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 13 / 23

20 EPR-like state Examples Two entangled particles A and B: Ψ(x A, x B ) = C dp e ipxb/ e ipxa/ e p2 /4 2 σ 2 e (x A +x B ) 2 16Ω 2 In the limit σ, Ω, the state reduces to the EPR state. The uncertainties in position and momentum of particle A (say) X A = Ω 2 + 1/16σ 2, P A = σ Ω 2. X A P A = /2 is obtained only if Ω = 1/4σ. In that case, the state becomes 1 Ψ(x A, x B ) = e (x A 2+x B 2)2σ2 π/4σ 2 Disentangled! Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 13 / 23

21 Examples Question Can the equality in the uncertainty relations be acheived for entangled states? B.G. Englert s counter-example 3 Consider the hermitian observables ( = 1) A = , B = together with the mixed-state density operator ρ = i 0 i For these we have ( A ) 2 ( ) 2 = B = 1 i [ A, B ] = 1 3 arxiv: [quant-ph] Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 14 /

22 Examples Question Can the equality in the uncertainty relations be acheived for entangled states? B.G. Englert s counter-example 3 Consider the hermitian observables ( = 1) A = , B = together with the mixed-state density operator ρ = i 0 i For these we have ( A ) 2 ( ) 2 = B = 1 i [ A, B ] = 1 3 arxiv: [quant-ph] Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 14 /

23 Examples Question Can the equality in the uncertainty relations be acheived for entangled states? B.G. Englert s counter-example 3 Consider the hermitian observables ( = 1) A = , B = together with the mixed-state density operator ρ = i 0 i For these we have ( A ) 2 ( ) 2 = B = 1 i [ A, B ] = 1 3 arxiv: [quant-ph] Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 14 /

24 Outline General Analysis & Results 1 Uncertainty and Entanglement 2 Examples 3 General Analysis & Results 4 Conclusions Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 15 / 23

25 General Analysis & Results General Analysis Finite-dimensional Hilbert spaces Entangled state Ψ admits a Schmidt decomposition Ψ = s i=1 c i a i A b i B a i A, b i B orthonormal basis vectors in H A, H B respectively. s d A Schmidt rank. Consider the operators and the states X A = X A X A Ψ, Ỹ A = Y A Y A Ψ (X A X A Ψ ) Ψ (Y A Y A Ψ ) Ψ Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 16 / 23

26 General Analysis & Results General Analysis Finite-dimensional Hilbert spaces Entangled state Ψ admits a Schmidt decomposition Ψ = s i=1 c i a i A b i B a i A, b i B orthonormal basis vectors in H A, H B respectively. s d A Schmidt rank. Consider the operators and the states X A = X A X A Ψ, Ỹ A = Y A Y A Ψ (X A X A Ψ ) Ψ (Y A Y A Ψ ) Ψ Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 16 / 23

27 General Analysis General Analysis & Results Schwarz inequality of the states (X A X A Ψ ) Ψ, (Y A Y A Ψ ) Ψ gives X 2 A Ψ Ỹ2 A Ψ 1 4 [X A, Y A ] Ψ { X A, ỸA} Ψ 2 Note: X 2 A Ψ Ỹ2 A Ψ = ( X A ) 2 Ψ ( Y A) 2 Ψ. This is nothing but Schrödinger-Robertson inequality! Minimum uncertainty The equality holds if the two state-vectors X A Ψ and ỸA Ψ are parallel: (X A X A Ψ ) Ψ + Γ (Y A Y A Ψ ) Ψ = 0 Γ imaginary HUR equality Γ complex SR equality Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 17 / 23

28 General Analysis General Analysis & Results Schwarz inequality of the states (X A X A Ψ ) Ψ, (Y A Y A Ψ ) Ψ gives X 2 A Ψ Ỹ2 A Ψ 1 4 [X A, Y A ] Ψ { X A, ỸA} Ψ 2 Note: X 2 A Ψ Ỹ2 A Ψ = ( X A ) 2 Ψ ( Y A) 2 Ψ. This is nothing but Schrödinger-Robertson inequality! Minimum uncertainty The equality holds if the two state-vectors X A Ψ and ỸA Ψ are parallel: (X A X A Ψ ) Ψ + Γ (Y A Y A Ψ ) Ψ = 0 Γ imaginary HUR equality Γ complex SR equality Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 17 / 23

29 General Analysis & Results Minimum uncertainty for the entangled state s i=1 c i {(X A X A Ψ ) + Γ (Y A Y A Ψ )} a i A b i B = 0 This can be satisfied only if {(X A X A Ψ ) + Γ (Y A Y A Ψ )} a i A = 0 for every i. Following conclusions can be drawn from the above: 1 X A i = X A Ψ, Y A i = Y A Ψ 2 ( X A ) 2 i ( Y A) 2 i = 1 4 [X A, Y A ] i 2 3 ( X A ) 2 i /( Y A) 2 i = 2Γ 2 These constitute a generalization of conditions (1-3) spelt out earlier. Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 18 / 23

30 General Analysis & Results More conclusions Operators (X A X A Ψ ) and (Y A Y A Ψ ) are zero in the subspace spanned by a 1, a 2... a s. If s = d A (maximal Schmidt rank) it implies that all the states a i are simultaneous degenerate eigenstates of X A, Y A. (Impossible for non-commuting X A, Y A ) For entangled states with maximal Schmidt rank, equality in HUR and SR cannot be reached! For s < d A, equality in HUR and SR can be reached for some entangled states. Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 19 / 23

31 General Analysis & Results More conclusions Operators (X A X A Ψ ) and (Y A Y A Ψ ) are zero in the subspace spanned by a 1, a 2... a s. If s = d A (maximal Schmidt rank) it implies that all the states a i are simultaneous degenerate eigenstates of X A, Y A. (Impossible for non-commuting X A, Y A ) For entangled states with maximal Schmidt rank, equality in HUR and SR cannot be reached! For s < d A, equality in HUR and SR can be reached for some entangled states. Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 19 / 23

32 General Analysis & Results More conclusions Operators (X A X A Ψ ) and (Y A Y A Ψ ) are zero in the subspace spanned by a 1, a 2... a s. If s = d A (maximal Schmidt rank) it implies that all the states a i are simultaneous degenerate eigenstates of X A, Y A. (Impossible for non-commuting X A, Y A ) For entangled states with maximal Schmidt rank, equality in HUR and SR cannot be reached! For s < d A, equality in HUR and SR can be reached for some entangled states. Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 19 / 23

33 General Analysis & Results Infinite-d Hilbert space: Position-Momentum Condition for equality in Heisenberg Uncertainty Relation {(P A P A Ψ ) + Γ (Q A Q A Ψ )} a i A = 0 In the position representation: { i d } dq + iγ Iq ( P Ψ + iγ I Q Ψ ) ψ ai (q) = 0 The solution is ψ ai (q) = Ce i P Ψq/ exp ( Γ (q Q Ψ) 2 ) 2 HUR equality can be acheived only when all the ψ ai (q) are Gaussian states with same centre and same average momentum. Such a state is not entangled For position and momentum, equality in Heisenberg uncertainty relation can never be reached for any entangled state! Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 20 / 23

34 General Analysis & Results Infinite-d Hilbert space: Position-Momentum Condition for equality in Heisenberg Uncertainty Relation {(P A P A Ψ ) + Γ (Q A Q A Ψ )} a i A = 0 In the position representation: { i d } dq + iγ Iq ( P Ψ + iγ I Q Ψ ) ψ ai (q) = 0 The solution is ψ ai (q) = Ce i P Ψq/ exp ( Γ (q Q Ψ) 2 ) 2 HUR equality can be acheived only when all the ψ ai (q) are Gaussian states with same centre and same average momentum. Such a state is not entangled For position and momentum, equality in Heisenberg uncertainty relation can never be reached for any entangled state! Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 20 / 23

35 General Analysis & Results Infinite-d Hilbert space: Position-Momentum Condition for equality in Heisenberg Uncertainty Relation {(P A P A Ψ ) + Γ (Q A Q A Ψ )} a i A = 0 In the position representation: { i d } dq + iγ Iq ( P Ψ + iγ I Q Ψ ) ψ ai (q) = 0 The solution is ψ ai (q) = Ce i P Ψq/ exp ( Γ (q Q Ψ) 2 ) 2 HUR equality can be acheived only when all the ψ ai (q) are Gaussian states with same centre and same average momentum. Such a state is not entangled For position and momentum, equality in Heisenberg uncertainty relation can never be reached for any entangled state! Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 20 / 23

36 Outline Conclusions 1 Uncertainty and Entanglement 2 Examples 3 General Analysis & Results 4 Conclusions Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 21 / 23

37 Conclusions Summary Equality in the uncertainty relations cannot be reached for entangled states with maximal Schmidt rank for any two observables. For some entangled states with non-maximal Schmidt rank, the minimum uncertainty equality can be reached. For position and momentum, no entangled state can acheive minimum uncertainty equality. Entanglement does put some restrictions on the uncertainty relations. N.D. Hari Dass, Tabish Qureshi, Aditi Sheel Minimum Uncertainty and Entanglement arxiv: [quant-ph] Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 22 / 23

38 Thank You! Conclusions Tabish Qureshi (CTP, JMI) Minimum Uncertainty for Entangled States IWQI12 23 / 23