Stronger uncertainty relations

Transcription

1 Stronger uncertainty relations Lorenzo Maccone Dip. Fisica, INFN Sez. Pavia, Universita' di Pavia Arun K. Pati Quantum Information and Computation Group, HarishChandra Research Institute, Department of Mathematics, Zhejiang University, Hangzhou

2 What I'm going to talk about Heisenberg uncertainty relations do NOT convey fully the notion of complementarity

3 What I'm going to talk about Heisenberg uncertainty relations do NOT convey fully the notion of complementarity both terms can be zero, even if A and B are incompatible

4 What I'm going to talk about Heisenberg uncertainty relations do NOT convey fully the notion of complementarity both terms can be zero, even if A and B are incompatible e.g. if the state of the system is an eigenstate of A

5 What I'm going to talk about Heisenberg uncertainty relations do NOT convey fully the notion of complementarity both terms can be zero, even if A and B are incompatible e.g. if the state of the system is an eigenstate of A Can we do better?

6 What I'm going to talk about Heisenberg uncertainty relations do NOT convey fully the notion of complementarity both terms can be zero, even if A and B are incompatible e.g. if the state of the system is an eigenstate of A Can we do better? YES!

7 Reminder

8 Reminder Heisenberg-Robertson UR variances and expectations calculated on system state

9 Reminder Heisenberg-Robertson UR variances and expectations calculated on system state Schroedinger UR

10 Reminder Heisenberg-Robertson UR variances and expectations calculated on system state Schroedinger UR both trivial if =eigenstate of A OR B

11 we need to give a bound to the sum (the product is trivially zero if one of the two is)

12 we need to give a bound to the sum (the product simple try: is trivially zero if one of the two is)

13 we need to give a bound to the sum (the product is trivially zero if one of the two is) simple try: Heisenberg

14 we need to give a bound to the sum (the product is trivially zero if one of the two is) simple try: nice, but... Heisenberg

15 we need to give a bound to the sum (the product is trivially zero if one of the two is) simple try: nice, but... Heisenberg...useless!

16 Main result: two lower bounds

17 Main result: two lower bounds nonzero except when is a joint eigenstate of A and B

18 Main result: two lower bounds nonzero except when is a joint eigenstate of A and B

19 Main result: two lower bounds nonzero except when is a joint eigenstate of A and B

20 SUM of variances dimensional regularization needed if A and B have different units

21 SUM of variances dimensional regularization needed if A and B have different units

22 First bound:

23 First bound: choose sign so that this is positive (it's real)

24 First bound: choose sign so that this is positive (it's real) arbitrary state orthogonal to the system state

25 First bound: choose sign so that this is positive (it's real) arbitrary state orthogonal to the system state constructive procedure to choose choose a random one, or just

26 First bound: Green: Heisenb-Robertson UR

27 First bound: Green: Heisenb-Robertson UR Red: new part, always except if is joint eigenstate of A and B

28 First bound: Green: Heisenb-Robertson UR Red: new part, always except if is joint eigenstate of A and B Minimizing over only HR (red) Maximizing over ineq becomes eq

29 First bound: numerical test for random

30 First bound: proof application of Schwartz inequality (like the Robertson)

31 First bound: proof application of Schwartz inequality (like the Robertson)

32 First bound: proof application of Schwartz inequality (like the Robertson) proof by an anonymous referee Masanao Ozawa (original proof more complicated)

33 First bound: minimum uncert. st. togliere questa slide: tutti gli stati sono minimum uncertainty states peri il primo bound: quando ottimizzo su psi^perp>, la disuguaglianza diventa sempre una uguaglianza!

34 First bound: minimum uncert. st. Harmonic osc.

35 First bound: minimum uncert. st. Harmonic osc.

36 First bound: minimum uncert. st. Harmonic osc. Fock states

37 First bound: minimum uncert. st. Harmonic osc. Fock states MUS

38 First bound: minimum uncert. st. Harmonic osc. Fock states Coherent states MUS

39 HR Non-MUS Fock states HR MUS Coherent states They're all MUS (for first, but not second bound) These are just examples, not the ONLY MUS!!

40 Second bound:

41 Second bound:

42 Second bound: this bound is if is not an eigenstate of A+B

43 Second bound: example

44 Second bound: proof application of the parallelogram ineq.

45 Second bound: proof application of the parallelogram ineq.

46 State dependence. both our bounds are state-dependent

47 State dependence. both our bounds are state-dependent

48 State dependence. both our bounds are state-dependent get a state-independent bound from them??

49 State dependence. both our bounds are state-dependent get a state-independent bound from them? the second becomes trivial? minimizing over (just take an eigenstate of A+B)

50 State dependence. both our bounds are state-dependent get a state-independent bound from them? the second becomes trivial? minimizing over (just take an eigenstate of A+B) the first?

51 State dependence. both our bounds are state-dependent get a state-independent bound from them? the second becomes trivial? minimizing over (just take an eigenstate of A+B) the first?

52 Using the same techniques, we can tighten the Heis-Rob bound!

53 Using the same techniques, we can tighten the Heis-Rob bound!

54 Heisenberg uncertainty relation vs. principle two very different notions!!!!!!!!!!!!!!!!!!!

55 Heisenberg uncertainty relation vs. principle two very different notions!!!!!!!!!!!!!!!!!!! much confusion in the literature, originating from Heisenberg's paper where he uses them interchangeably (criticized by Bohr)

56 Heisenberg uncertainty relation vs. principle two very different notions!!!!!!!!!!!!!!!!!!! much confusion in the literature, originating from Heisenberg's paper where he uses them interchangeably (criticized by Bohr) u principle: measurement-disturbance tradeoff. A precise measure induces a disturbance

57 Heisenberg uncertainty relation vs. principle two very different notions!!!!!!!!!!!!!!!!!!! much confusion in the literature, originating from Heisenberg's paper where he uses them interchangeably (criticized by Bohr) u principle: measurement-disturbance tradeoff. A precise measure induces a disturbance WRONG interpetation!

58 Heisenberg uncertainty relation vs. principle two very different notions!!!!!!!!!!!!!!!!!!! much confusion in the literature, originating from Heisenberg's paper where he uses them interchangeably (criticized by Bohr) u principle: measurement-disturbance tradeoff. A precise measure induces a disturbance WRONG interpetation! u relation: refers only to the preparation of the state. Can't prepare a state with sharp values for incompatible observables.

59 Heisenberg uncertainty relation vs. principle two veryin different notions!!!!!!!!!!!!!!!!!!! this talk: unc much confusion in the literature, originating from Heisenberg's paper where he uses them interchangeably (criticized by Bohr) RELATIONS! RELATIONS u principle: measurement-disturbance tradeoff. A precise measure induces a disturbance WRONG interpetation! u relation: refers only to the preparation of the state. Can't prepare a state with sharp values for incompatible observables.

60 Asher Peres introduced this notation uncertainty relation: The only correct interpretation of [the uncertainty relations for x and p] is the following: If the same preparation procedure is repeated many times, and is followed either by a measurement of x, or by a measurement of p, the various results obtained for x and for p have standard deviations, Δx and Δp, whose product cannot be less than ħ/2. There never is any question here that a measurement of x disturbs the value of p and vice versa, as sometimes claimed. These measurements are indeed incompatible, but they are performed on different particles (all of which were identically prepared) and therefore these measurements cannot disturb each other in any way. The uncertainty relation [...] only reflects the intrinsic randomness of the outcomes of quantum tests. (he didn't want to talk about unc principle) try looking up uncertain principle in his book.

61 What did I say?!? Lorenzo Maccone maccone@unipv.it stronger uncertainty relations for variances with nontrivial lower bound. PRL Nature India:

62 Entanglement and Complementarity Lorenzo Maccone Dip. Fisica, INFN Sez. Pavia, Universita' di Pavia Chiara Macchiavello Dagmar Bruss

63 What I'm going to talk about We always say that entangled states are more correlated... WHAT DOES IT MEAN exactly?

64 What I'm going to talk about We always say that entangled states are more correlated... WHAT DOES IT MEAN exactly? they have more correlations among complementary observables than separable ones

65 Usual approaches to study entanglement Non locality Negative partial transpose Bell inequality violations Enhanced precision in measurements etc.

66 Here: we use correlations among two (or more) COMPLEMENTARY PROPERTIES different way to think about entanglement, as correlations among complementary properties

67 Remember: Complementary properties.

68 Remember: Complementary properties. Two observables: the knowledge of one gives no knowledge of the other

69 simplest example:

70 simplest example: Maximally entangled state: perfect correlation BOTH on 0/1 and on +/-

71 simplest example: Maximally entangled state: perfect correlation BOTH on 0/1 and on +/- separable state: perfect correlation for 0/1, no correlation for +/-

72 simplest example: Maximally entangled state: perfect correlation BOTH on 0/1 and on +/- separable state: perfect correlation for 0/1, no correlation for +/-

73 Simple experiment On system 1 measure either A or C On system 2 measure either B or D Calculate correlations A-B and C-D

74 How to measure correlation?

75 How to measure correlation? Mutual information

76 How to measure correlation? Mutual information Pearson correlation coefficient perfect correlation or anticorrelation

77 Use these to measure correlations among 2 complementary properties complem to of 2 systems

78 Some results...

79 Start with mutual information total correlation given by the sum

80 complem to The system state is maximally entangled iff perfect correlation on both A-B and C-D

81 complem to The system state is maximally entangled iff perfect correlation on both A-B and C-D (for some observ ABCD) maximally entangled

82 complem to The system state is maximally entangled iff perfect correlation on both A-B and C-D (for some observ ABCD) maximally entangled

83 complem to The system state is entangled if correlations on both A-B and C-D are large enough

84 complem to The system state is entangled if correlations on both A-B and C-D are large enough ent

85 complem to The system state is entangled if correlations on both A-B and C-D are large enough ent

86 The system state is entangled if correlations on both A-B and C-D are large enough ent Can the bound be made tighter?

87 The system state is entangled if correlations on both A-B and C-D are large enough ent Can the bound be made tighter? NO!!

88 The system state is entangled if correlations on both A-B and C-D are large enough ent Can the bound be made tighter? NO!! the separable state saturates it:

89 The systen state is entangled if correlations on both A-B and C-D are large enough is the converse true?

90 The systen state is entangled if correlations on both A-B and C-D are large enough is the converse true? NO!!

91 The systen state is entangled if correlations on both A-B and C-D are large enough is the converse true? NO!! is entangled but has negligible mutual info for

92 Another measure of correlation...

93 Pearsoncorrelation correlationcoefficient coefficient Pearson perfect correlation or anticorrelation

94 Pearsoncorrelation correlationcoefficient coefficient Pearson perfect correlation or anticorrelation it can be complex for quantum expectation values

95 Pearsoncorrelation correlationcoefficient coefficient Pearson perfect correlation or anticorrelation it can be complex for quantum expectation values... but its modulus is still :

96 Pearsoncorrelation correlationcoefficient coefficient Pearson perfect correlation or anticorrelation it can be complex for quantum expectation values... but its modulus is still : not a problem for us: A and B commute, so it's REAL Using Schroedinger's uncertainty relation:

97 Total correlation: again use the sum complem to

98 complem to The system state is maximally entangled iff perfect correlation on both A-B and C-D

99 complem to The system state is maximally entangled iff perfect correlation on both A-B and C-D True also using Pearson! (for linear observables: Pearson measures only linear correl)

100 complem to The system state is maximally entangled iff perfect correlation on both A-B and C-D True also using Pearson! (for linear observables: Pearson measures only linear correl) (for some observ ABCD) maximally entangled

101 The system state is maximally entangled iff perfect correlation on both A-B and C-D True also using Pearson! (for linear complem complem to to observables: Pearson measures only linear correl) (for some observ ABCD) maximally entangled

102 complem to The system state is entangled if correlations on both A-B and C-D are large enough?

103 complem to The system state is entangled if correlations on both A-B and C-D are large enough? CONJECTURE: we don't know if it's true also using Pearson!

104 complem to The system state is entangled if correlations on both A-B and C-D are large enough? CONJECTURE: we don't know if it's true also using Pearson! ent (for some observ ABCD)

105 complem to The system state is entangled if correlations on both A-B and C-D are large enough? CONJECTURE: we don't know if it's true also using Pearson! complem complem to to ent (for some observ ABCD)

106 Conjecture: state is ent. Again, the inequality is tight:

107 Conjecture: state is ent. Again, the inequality is tight: separable state

108 Conjecture: state is ent. Again, the inequality is tight: separable state (perfect correl on one basis, no correl on the complem)

109 Is the Pearson correlation weaker than the mutual info? only linear correlations all correlations

110 Is the Pearson correlation weaker than the mutual info? NO!! only linear correlations all correlations

111 Is the Pearson correlation weaker than the mutual info? NO!! Has negligible mutual info for only linear correlations all correlations

112 Is the Pearson correlation weaker than the mutual info? NO!! Has negligible mutual info for but Pearson correlation always >1! only linear correlations all correlations

113 Simple criterion for entanglement detection!! Just measure two complementary properties. Are the correlations greater than perfect correlation on one? The state is entangled! Simple to measure and simple to optimize. Unfortunately: not very effective in finding entanglement in random states

114 What did I say?!? Entanglement as correlation among complementary observables Using different measures of correlation: Mutual info Pearson correlation Some theorems and some conjectures

115 Take home message The most correlated states are entangled but ent states are not the most correlated Correlations on complementary prop. help understanding entanglement Lorenzo Maccone PRL