1. Mutual uncertainty, conditional uncertainty and ...
Sk Sazim
HRI
27th Feb, 2017
arXiv:

2. Statements of UR
HUP: One cannot measure position and momentum simultanuously with infinite precision.
RUR: It is impossible to prepare a quantum ensamble where one will be able to measure two incompatible observables with infinite precision.
EDR: Measurement of one dof in a quantum state will necessarily disturb its complementary one.
SUR: Sum of uncertainties of two observables is always >= the uncertainty of sum of them.

3. Confusions
Heisenberg never derived the UR. Its Lenard who derived the position momentum uncertainty relation.
Roberson generalized it for any observables.
Robertson’s relation has nothing to do with EDR.
We are dealing with two uncertainty principles:
a static uncertainty principle, namelely, RUR.
a dynamical uncertainty principle, a statement that should establish a tradeoff between the accuracy with which A is measured and the disturbance caused on a non commuting B.

4. SUR and MU The SUR: or The MU:
It is positive and symmetric in observables.
Expression resembles with relative entropy mutual information.
δ𝐴+δ𝐵≥δ 𝐴+𝐵
δ𝐴+δ𝐵−δ 𝐴+𝐵 ≥0
𝑀 𝐴:𝐵 =δ𝐴+δ𝐵−δ 𝐴+𝐵
𝑀 𝐴 1 : 𝐴 2 :...: 𝐴 𝑛 = δ 𝐴 𝑖 −δ 𝐴 𝑖
δ 𝐴 2 =𝑇𝑟 ρ 𝐴 2 − 𝑇𝑟 ρ𝐴 2

5. CU and ... CU: Chain rules: CMU: δ 𝐴∣𝐵 =δ 𝐴+𝐵 −δ𝐵
δ 𝐴 𝑖 = δ 𝐴 𝑖 ∣ 𝐴 𝑖− 𝐴 1
δ 𝐴 𝑖 ∣𝑋 = δ 𝐴 𝑖 ∣𝑋+ 𝐴 𝐴 𝑖−1
𝑀 𝐴:𝐵∣𝐶 =δ 𝐵∣𝐶 −δ 𝐵∣𝐶+𝐴

6. SSA-U If then Conditioning on more observables reduces uncertainty.
𝑀 𝐵:𝐶 =0
δ 𝐴∣𝐵+𝐶 ≤δ 𝐴∣𝐵
𝑀 𝐴:𝐵 ≤𝑀 𝐴:𝐵+𝐶
𝑀 𝐴:𝐵 =0
𝑀 𝐴+𝐵:𝐶 ≥𝑀 𝐴:𝐵+𝐶

7. MU and Entanglement If ; separable ( ) If then
For product, classical, and classical quantum states
𝑀 𝐴:𝐵 ≤2− 2
𝐴=𝐼⊗σ. 𝑎 ,𝐵=σ. 𝑏 ⊗𝐼
𝑇𝑟 2 𝐴ρ = 𝑇𝑟 1 𝐵ρ =0; 𝑇𝑟 2 𝐴𝐴ρ = 𝑇𝑟 1 𝐴𝐴ρ =1
𝐶𝑜𝑛𝑐𝑢𝑟𝑟𝑒𝑛𝑐𝑒= 1 2𝑡 2+𝑀 𝑀−4
𝑀=2− 2

8. MU and steering Unsteerable: If ; steering.
Werner state shows steering for
ρ 12 = 𝑝 μ ρ 1 μ ρ 2 𝑄
η=𝑀 σ 1 : σ 2 +𝑀 σ 2 : σ 3 +𝑀 σ 3 : σ 1
η≥2− 2
η<2− 2
𝑝> −8

9. Thanks
Questions!!!