Einstein-Podolsky-Rosen-like correlation on a coherent-state basis and Continuous-Variable entanglement

Transcription

1 12/02/13 Einstein-Podolsky-Rosen-like correlation on a coherent-state basis and Continuous-Variable entanglement Ryo Namiki Quantum optics group, Kyoto University 京大理 並木亮 求職中 arxiv:

2 Quantum Entanglement Foundation of Quantum mechanics: Einstein-Podolsky-Rosen paradox; Bell's theorem. Experimental goal: Generation of entangled states and Observation of Quantum correlation Resource of communication and computation in quantum information processing B Quantum key distribution Measurement-based one way quantum computation Entanglement + Local measurement Universal quantum computer = Simulation of any physical system

3 Continuous-variable systems in Quantum Optics Bosonic fields described by a set of canonical variables [ x, p ]=[ x B, p B ]= =i 2 x 2 p 1/4 Gaussian states -Minimum uncertainty states Coherent states/ Squeezed states /... Wave function is Gaussian -ny property is determined by the second moments: 2 x, 2 p, 2 x B, 2 x p B, p p B, x x B, p p B, x B, Measurable by optical homodyne detection B

4 EPR correlation Einstein-Podolsky-Rosen (EPR) state B := dx x x B / 2 Positions are correlated and Momentums are anti-correlated 2 p p B ~0 ; 2 x x B ~0 B x 1... x 1 x 2... x 2 x 3... x 3 p 1... p 1 p 2... p 2 p 3... p 3 B Stronger Simultaneous correlation can be a signature of entanglement

5 EPR-like uncertainties and entanglement Product criterion for entanglement 2 u x B 2 u p C 2 C := [x, p] /2 the state is entangled. V. Giovannetti et al., Phys. Rev. 67, (2003) EPR correlation 2 x x B ~0 ; 2 p p B ~0 EPR-like operators u x B ; u p u 2 v 2 =1

6 EPR-like uncertainties and entanglement Product criterion for entanglement 2 u x B 2 u p C 2 2 u x B 2 u p v p B C 2 C := [x, p] /2 Uncertainty relation! V. Giovannetti et al., Phys. Rev. 67, (2003) 2 x x B ~0 ; 2 p p B ~0 EPR-like operators u x B ; u p u 2 v 2 =1

7 EPR-like uncertainties and entanglement Product criterion for entanglement 2 u x B 2 u p C 2 2 u x B 2 u p v p B C 2 C := [x, p] /2 Uncertainty relation! V. Giovannetti et al., Phys. Rev. 67, (2003) 2 x x B ~0 ; 2 p p B ~0 EPR-like operators u x B ; u p u 2 v 2 =1

8 EPR-like uncertainties and entanglement Product criterion for entanglement 2 u x B 2 u p C 2 2 u x B 2 u p v p B C 2 C := [x, p] /2 Uncertainty relation! V. Giovannetti et al., Phys. Rev. 67, (2003) EPR p 2 paradox 3 Uncertainty relation! 4 V x 2 p 2 C 2 4 V 3 EPR paradox! UV U = 2 u x v x B /C V = 2 u p /C U x U

9 EPR-like uncertainties and entanglement Product criterion for entanglement 2 u x B 2 u p C 2 C := [x, p] /2 the state is entangled. Outline of outline of proof. V. Giovannetti et al., Phys. Rev. 67, (2003) 1. ssume separable form 2. Prove the inequality 2 U 2 V 1 B = i p i i i B U = 2 u x v x B /C V = 2 u p /C

10 EPR-like uncertainties and entanglement Product criterion for entanglement 2 u x B 2 u p C 2 C := [x, p] /2 the state is entangled. Outline of outline of proof. V. Giovannetti et al., Phys. Rev. 67, (2003) 1. ssume separable form 2. Prove the inequality B = i p i i i B 2 U 2 V 1 U = 2 u x v x B /C V = 2 u p /C

11 EPR-like uncertainties and entanglement Product criterion for entanglement 2 u x B 2 u p C 2 C := [x, p] /2 Outline of outline of proof. V. Giovannetti et al., Phys. Rev. 67, (2003) 1. ssume separable form 2. Prove the inequality B = i p i i i B the state is entangled. 2 U 2 V 1 Violation of the inequality U = 2 u x v x B /C V = 2 u p /C

12 EPR-like uncertainties and entanglement Product criterion for entanglement 2 u x B 2 u p C 2 C := [x, p] /2 V. Giovannetti et al., Phys. Rev. 67, (2003) Outline of outline of proof. 1. ssume separable form 2. Prove the inequality B = i p i i i B the state is entangled. 2 U 2 V 1 B i p i i i B Violation of the inequality Definition of entangled states. U = 2 u x v x B /C V = 2 u p /C

13 Product criterion and Sum criterion Product criterion Entanglement is signified by the EPR paradox. Sum criterion Sufficient for the detection of Gaussian entanglement (with an optimization of local Gaussian unitary operation) 4 V 3 UV U U = 2 u x v x B /C V = 2 u p /C

14 Product criterion Entanglement is signified by the EPR paradox. Sum criterion Sufficient for the detection of Gaussian entanglement (with an optimization of local Gaussian unitary operation) 4 V Product criterion and Sum criterion Duan et al, Phys. Rev. Lett (2000) 3 UV 1 2 U V U U = 2 u x v x B /C V = 2 u p /C

15 Quantum noise & entanglement We can say basically, Entanglement = EPR-like correlation Entanglement detection = Observation of EPR-paradox Homodyne detection in Experiments 2 x 2 p 1/4 4 V 3 UV U

16 nother measure of the correlation.. Variances (Uncertainties) 2 x x B. Intensities of the distributions P x, x B

17 nother measure of the correlation.. Variances (Uncertainties) 2 x x B. Intensities of the distributions P x, x B =x i p B =x B i p B Conjugate-amplitude pair of coherent states g *

18 nother measure of the correlation.. Variances (Uncertainties) 2 x x B. Intensities of the distributions P x, x B =x i p B =x B i p B Conjugate-amplitude pair of coherent states Coherent states a = C Husimi -Q function Q = / g * Heterodyne measurement

19 New criterion with the EPR-like correlation on a coherent-state basis Strength of the correlation for entanglement u 2 v 2 =1 v v u * B u * Projection to the pair of coherent states uncertainty relation x p [ x, p] /2 EPR paradox! sort of uncertainty relation th 1 th = e 2 d 2 = 1 n=0 1 1 n n n R. Namiki, arxiv:

20 New criterion with the EPR-like correlation on a coherent-state basis Strength of the correlation for entanglement d 2 p v v u * B u * 1 u 2 v 2 =1 p := e 2 uncertainty relation x p [ x, p] /2 EPR paradox! the state is entangled. R. Namiki, PR 83, (2011) sort of uncertainty relation th 1 th = e 2 d 2 = 1 n=0 1 1 n n n R. Namiki, arxiv:

21 New criterion with the EPR-like correlation on a coherent-state basis Strength of the correlation for entanglement d 2 p v v u * B u * 1 =x i p d 2 p v v u B u 1 p := e 2 Universal relation uncertainty relation x p [ x, p] /2 EPR paradox! sort of uncertainty relation th 1 th = e 2 d 2 = 1 n=0 1 1 n n n R. Namiki, arxiv:

22 Performance of new criterion Strength of the correlation for entanglement d 2 p v v u * B u * 1 For a standard form of the Gaussian state & λ U 1 V 1 1 U := 2 u x B /C V := 2 u p /C EPR-like uncertainties!

23 Performance of new criterion Strength of the correlation for entanglement d 2 p v v u * B u * 1 For a standard form of the Gaussian state & λ U 1 V 1 1 U := 2 u x B /C V := 2 u p /C EPR-like uncertainties! UV 1 U V 2 New criterion covers Gaussian entanglement!

24 Summary EPR-like correlation EPR-paradox signifies the entanglement Detection of Gaussian entanglement 2 x 2 p 1/4 Coherent-state-based EPR-like correlation New criterion for entanglement Detection of Gaussian entanglement Simple interrelation to the standard method Probability distribution (c.f. Quantum noise) Heterodyne measurements Novel basis to comprehend quantum entanglement R. Namiki, arxiv: Benchmark for genuine quantum memories and gates R. Namiki and Y. Tokunaga, PR 85, (R) (2012) Poster #80