Entanglement and Complementarity
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1 Entanglement and Complementarity Lorenzo Maccone Dip. Fisica, INFN Sez. Pavia, Universita' di Pavia Chiara Macchiavello Dagmar Bruss
2 What I'm going to talk about We always say that entangled states are more correlated... WHAT DOES IT MEAN exactly?
3 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 Abstract: We show that states that have more correlations among complementary observables must be entangled. The reverse is false: general entangled states do not have more correlations on complementary observables than separable ones. We either prove or conjecture that this is true for different measures of correlation: the mutual information, the sum of conditional probabilities and the Pearson correlation coefficient. We also show that states with nonzero discord typically have less correlation than classically correlated states.
4 Usual approaches to study entanglement Non locality Negative partial transpose Bell inequality violations Enhanced precision in measurements etc.
5 Here: we use correlations among two (or more) COMPLEMENTARY PROPERTIES
6 Remember: Complementary properties.
7 Remember: Complementary properties. Two observables: the knowledge of one gives no knowledge of the other
8 simplest example:
9 simplest example: Maximally entangled state: perfect correlation BOTH on 0/1 and on +/-
10 simplest example: Maximally entangled state: perfect correlation BOTH on 0/1 and on +/- separable state: perfect correlation for 0/1, no correlation for +/-
11 simplest example: Maximally entangled state: perfect correlation BOTH on 0/1 and on +/- separable state: perfect correlation for 0/1, no correlation for +/-
12 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
13 How to measure correlation?
14 How to measure correlation? Mutual information
15 How to measure correlation? Mutual information Pearson correlation coefficient perfect correlation or anticorrelation
16 How to measure correlation? Mutual information Pearson correlation coefficient perfect correlation or anticorrelation Sum of conditional probabilities perfect correlation or anticorrelation
17 Use these to measure correlations among 2 complementary properties complem to of 2 systems
18 Some results...
19 Start with mutual information total correlation given by the sum
20 complem to The system state is maximally entangled iff perfect correlation on both A-B and C-D
21 complem to The system state is maximally entangled iff perfect correlation on both A-B and C-D (for some observ ABCD) maximally entangled
22 complem to The system state is maximally entangled iff perfect correlation on both A-B and C-D (for some observ ABCD) maximally entangled
23 Easy to prove: the state is maximally entangled perfect correlations on all compl observables
24 Easy to prove: the state is maximally entangled perfect correlations on all compl observables Just use simple properties of conditional probabilities, e.g.
25 Easy to prove: the state is maximally entangled perfect correlations on all compl observables Just use simple properties of conditional probabilities, e.g....and write the mutual info as a function of conditional probs.
26 complem to The system state is entangled if correlations on both A-B and C-D are large enough
27 complem to The system state is entangled if correlations on both A-B and C-D are large enough ent
28 complem to The system state is entangled if correlations on both A-B and C-D are large enough ent
29 The system state is entangled if correlations on both A-B and C-D are large enough ent Can the bound be made tighter?
30 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!!
31 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:
32 The systen state is entangled if correlations on both A-B and C-D are large enough is the converse true?
33 The systen state is entangled if correlations on both A-B and C-D are large enough is the converse true? NO!!
34 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
35 Proof: The systen state is entangled if correlations on both A-B and C-D are large enough ent
36 Proof: The systen state is entangled if correlations on both A-B and C-D are large enough ent separable
37 Proof: The systen state is entangled if correlations on both A-B and C-D are large enough ent separable Use the concavity of the entropy:
38 Proof: The systen state is entangled if correlations on both A-B and C-D are large enough ent separable Use the concavity of the entropy: Use Maassen-Uffink's entropic uncertainty relation: (for complem observ)
39 ent What happens at the border with the entangled region?
40 ent What happens at the border with the entangled region? are states more correlated than classically-correlated states?
41 ent What happens at the border with the entangled region? are states more correlated than classically-correlated states? NO!!they're all CC states
42 ent What happens at the border with the entangled region? are states more correlated than classically-correlated states? NO!!they're all CC states CC ZERO DISCORD!
43 Examples Threshold for entanglement
44 Examples Threshold for entanglement
45 Examples Threshold for entanglement Werner states
46 Examples Threshold for entanglement Werner states
47 Examples Nonzero discord except for p=0,1
48 Example Randomly generated 2 qubit states (uniform in Haar measure) distribution of separable states distribution of entangled states
49 Example Randomly generated 2 qubit states (uniform in Haar measure) distribution of separable states distribution of entangled states A large overlap between the two curves (but still distinguishable).
50 Example Randomly generated 2 qubit states (uniform in Haar measure) distribution of separable states distribution of entangled states A large overlap between the two curves (but still distinguishable). Can we do better with other correlation measures?
51 Example Randomly generated 2 qubit states (uniform in Haar measure) distribution of separable states distribution of entangled states A large overlap between the two curves (but still distinguishable). Can we do better with other correlation measures? YES!
52 Another measure of correlation...
53 Pearsoncorrelation correlationcoefficient coefficient Pearson perfect correlation or anticorrelation
54 Pearsoncorrelation correlationcoefficient coefficient Pearson perfect correlation or anticorrelation it can be complex for quantum expectation values
55 Pearsoncorrelation correlationcoefficient coefficient Pearson perfect correlation or anticorrelation it can be complex for quantum expectation values... but its modulus is still :
56 Pearsoncorrelation correlationcoefficient coefficient Pearson perfect correlation or anticorrelation it can be complex for quantum expectation values... but its modulus is still :
57 Pearsoncorrelation correlationcoefficient coefficient Pearson perfect correlation or anticorrelation it can be complex for quantum expectation values... but its modulus is still : Using Schroedinger's uncertainty relation:
58 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:
59 Total correlation: again use the sum complem to
60 complem to The system state is maximally entangled iff perfect correlation on both A-B and C-D
61 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)
62 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) linear = linear in the eigenvalues e.g. and not
63 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
64 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
65 complem to The system state is entangled if correlations on both A-B and C-D are large enough?
66 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!
67 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)
68 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)
69 Conjecture: state is ent. Again, the inequality is tight:
70 Conjecture: state is ent. Again, the inequality is tight: separable state
71 Conjecture: state is ent. Again, the inequality is tight: separable state (perfect correl on one basis, no correl on the complem)
72 States on the border are zero-discord?
73 States on the border are zero-discord? NOT with
74 States on the border are zero-discord? NOT with
75 States on the border are zero-discord? NOT with It's always at the boundary (and has nonzero discord for p=0,1)
76 Conjecture: numerical evidence: state is ent. distribution of separable states distribution of entangled states threshold
77 Is the Pearson correlation weaker than the mutual info? only linear correlations all correlations
78 Is the Pearson correlation weaker than the mutual info? NO!! only linear correlations all correlations
79 Is the Pearson correlation weaker than the mutual info? NO!! Has negligible mutual info for only linear correlations all correlations
80 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
81 Still another measure of correlation...
82 Sum of conditional probabilities [similar approach (but joint probabilities):pra 86,022311]
83 Sum of conditional probabilities [similar approach (but joint probabilities):pra 86,022311] conditional probability of finding result i for A when I found result i for B:
84 Sum of conditional probabilities [similar approach (but joint probabilities):pra 86,022311] conditional probability of finding result i for A when I found result i for B: results are always the same
85 Sum of conditional probabilities [similar approach (but joint probabilities):pra 86,022311] conditional probability of finding result i for A when I found result i for B: results are always the same results are always different
![Sum of conditional probabilities [similar approach (but joint probabilities):pra 86,022311] conditional probability of](/docs-images/86/93503943/images/86-0.jpg "finding result i for A when I found result i for B: results are always the same results are always different perfect")
86 Sum of conditional probabilities [similar approach (but joint probabilities):pra 86,022311] conditional probability of finding result i for A when I found result i for B: results are always the same results are always different perfect correlation d
87 Total correlation: again use the sum complem to
88 complem to The system state is maximally entangled iff perfect correlation on both A-B and C-D
89 complem to The system state is maximally entangled iff perfect correlation on both A-B and C-D True also using
90 complem to The system state is maximally entangled iff perfect correlation on both A-B and C-D True also using (for some observ ABCD) maximally entangled
91 complem to The system state is maximally entangled iff perfect correlation on both A-B and C-D complem complem to to True also using (for some observ ABCD) maximally entangled
92 Again, simple proof using properties of the conditional probabilities
93 complem to The system state is entangled if correlations on both A-B and C-D are large enough?
94 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
95 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 ent (for some observ ABCD)
96 complem to results are always statetheissame entangled if The system correlations on both A-B and C-D results are 1/d are large enough? uncorrelated CONJECTURE: we don't know if it's true also using ent (for some observ ABCD)
97 Conjecture: ent (for some observ ABCD) again, inequality is tight:
98 Conjecture: ent (for some observ ABCD) again, inequality is tight:
99 Conjecture: ent (for some observ ABCD) again, inequality is tight: d+1 (separable)
100 Conjecture: ent (for some observ ABCD) again, inequality is tight: d+1 (separable) Perfect correlation on the 0>, 1> basis No correlation on the +>, - > basis
101 Conjecture: ent Histogram of random states (for some observ ABCD) numerical evidence
102 Role of discord?
103 Role of discord? zero discord states can be correlated only on one of the complem properties.
104 Role of discord? zero discord states can be correlated only on one of the complem properties. perfect correlation only on 0/1
105 Role of discord? zero discord states can be correlated only on one of the complem properties. perfect correlation only on 0/1 Perfect correlation on the 0>, 1> basis /d No correlation on the +>, -> basis
106 CC, CQ, QC states can be correlated only on one of the complem properties. perfect correlation only on 0/1 Only CC states can have perfect correlation on one obs
107 CC, CQ, QC states can be correlated only on one of the complem properties. perfect correlation only on 0/1 Only CC states can have perfect correlation on one obs CQ/QC states can have only partial correl
108 CC, CQ, QC states can be correlated only on one of the complem properties. perfect correlation only on 0/1 Only CC states can have perfect correlation on one obs CQ/QC states can have only partial correl CC CQ QC bases
109 What about QQ states? QQ
110 What about QQ states? QQ nonzero discord states can be partially correlated on more properties.
111 What about QQ states? QQ nonzero discord states can be partially correlated on more properties.
112 DISCORD: CC states can have maximal correlation only on one property CQ states cannot have maximal correlation in any property QQ states can have partial correlation on multiple properties
113 DISCORD: CC states can have maximal correlation only on one property CQ states cannot have maximal correlation in any property QQ states can have partial correlation on multiple properties Only pure, maximally entangled states have max correlations on more properties
114 What are these results good for, practically?
115 Simple criterion for entanglement detection!!
116 Simple criterion for entanglement detection!! Just measure two complementary properties. Are the correlations greater than perfect correlation on one?
117 Simple criterion for entanglement detection!! Just measure two complementary properties. Are the correlations greater than perfect correlation on one? The state is entangled!
118 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.
119 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
120 Simple and effective criterion for maximal entanglement detection!!
121 Simple and effective criterion for maximal entanglement detection!! Just measure two complementary properties. Are the correlations maximal on both properties?
122 Simple and effective criterion for maximal entanglement detection!! Just measure two complementary properties. Are the correlations maximal on both properties? The state is maximally entangled!
123 Conclusions
124 What did I say?!? Entanglement as correlation among complementary observables Using different measures of correlation: Mutual info Pearson correlation Sum of conditional prob Some theorems and some conjectures Role of discord
125 What did I say?!? Results: necessary and sufficient conditions for maximal entanglement necessary conditions for entanglement discord: mutual info: states on the boundary have no discord! correlation properties of CC, CQ, QC, and QQ states.
126 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 arxiv:
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