A New Logical Measure for Quantum Information

Figures & Tables

	Logic ℘(U) of subsets of U	Logic of partitions Π(U) on U
Its/Dits	Elements of subsets	Distinctions of partitions
P.O.	Inclusion of subsets	Inclusion of ditsets
Join	Union of subsets	Union of ditsets
Meet	Subset of common elements	Ditset of common dits
Impl.	S ⊃ T = U iff S ⊆ T	σ ⇒ π = 1_U iff σ ≾ π
Top	Subset U with all elements	Partition 1_U with all distinctions
Bottom	Subset ∅ with no elements	Partition 0_U with no distinctions

Set concept	Vector-space concept
Subset S ⊆ U	Subspace [S] ⊆ V
Partition {f⁻¹(r)}_r∈f(U)	DSD {V_r}_r∈f(U)
Disjoint union U = ⊎_r∈f(U) f⁻¹(r)	Direct sum V = ⊕_r∈f(U)V_r
Numerical attribute f : U → ℝ	Observable F u_i = f(u_i)u_i
f ↾ S = rS	F u_i = ru_i
Constant set S of f	Eigenvector u_i of F
Value r on constant set S	Eigenvalue r of eigenvector u_i
Characteristic fcn. χ_S : U → {0, 1}	Projection operator P_{[S]u_i} = χ_S (u_i)u_i
∑_r∈f(U) χ_{f⁻¹ (r)} = χ_U	∑_r∈f(U) P_r = I : V → V
Spectral Decomp. f = ∑_r∈f(U) rχf⁻¹(r)	Spectral Decomp. F = ∑_r∈f(U) rP_r
Set of r-constant sets ℘(f⁻¹(r))	Eigenspace V_r of r-eigenvectors
Direct product U × U	Tensor product V ⊗ V

Logical entropy	Quantum logical entropy
U = {u₁, ..., u_n}	ON basis U for Hilbert space V
f, g :U → ℝ	Commuting F , G : V → V
{r}_r∈f(U), {S}_s∈g(U)	Eigenvalues of F and G
π = {f⁻¹(r)}_r∈f(U), σ = {^g−1(S)}_s∈g(U)	DSDs of eigenspaces of F , G
Dits of π : (u_i, u_k), f(u_i) ≠ f(u_k)	Qudits F : u_i ⊗ u_k, f(u_i) ≠ f(u_k)
Dits of σ : (u_i, u_k), g(u_i) ≠ g(u_k)	Qudits G: u_i ⊗ u_k, g(u_i) ≠ g(u_k)
Ditset of π: dit(π)	[qudit(F)]: Subspace generated in V ⊗ V
Ditset of σ: dit(σ)	[qudit(G)]: Subspace generated in V ⊗ V
Join: dit(π) ∪ dit(σ) ⊆ U × U	[qudit(F) ∪ qudit(G)] ⊆ V ⊗ V
Difference: dit(π) − dit(σ) ⊆ U × U	[qudit(F) − qudit(G)] ⊆ V ⊗ V
Mutual: dit(π) ∩ dit(σ) ⊆ U × U	[qudit(F) ∩ qudit(G)] ⊆ V ⊗ V

	Logical Probability Theory	Logical Information Theory
Outcomes	Elements of S	Distinctions of π
Events	Subsets S ⊆ U	Ditsets dit(π) ⊆ U × U
pi=1n p_i = {1 \over n}	Pr⁡(S)=\|S\|[U] \Pr \left( S \right) = {{\left\| S \right\|} \over {\left[ U \right]}}	h(π)=\|dit(π)\|\|U×U\| h\left( \pi \right) = {{\left\| {{\rm{dit}}\left( \pi \right)} \right\|} \over {\left\| {U \times U} \right\|}}
Probs. p	Pr(S) = ∑_{u_i∈S} p_i	h(π) = ∑_{(u_i,} _{u_k)∈dit(π)} p_ip_k
Interpretation	1-draw prob. of S-element	2-draw prob. of π-distinction

Set concept with probabilities	Set level density matrix concept
Partition π with point probs. p	Density matrix ρ(π)=∑j=1mPr⁡(Bj)\|bj〉〈bj\| \rho (\pi ) = \sum\nolimits_{j = 1}^m \Pr (B_j )\|b_j \rangle \langle b_j \|
Point probabilities {p₁, ..., p_n}	Value of diagonal entries of ρ(π)
Trivial indits (u_i, u_i) of π	Diagonal entries of ρ(π)
Non-trivial indits of π	Non-zero off-diagonal entries of ρ(π)
Dits of π	Zero entries of ρ(π)
Sum Pr(B_j) = ∑_{u_i∈ B_j} p_i	Trace tr[P_{B_j}ρ(π)]
Block probabilities Pr(B_j) in π	Eigenvalues ≠ 0 of ρ(π)
Block prob. 1 of U in 0_U = {U}	Non-zero eigenvalue of 1 for ρ(0_U)

	The Dit-Bit Transform: 1−pi⇝log⁡(1pi) 1 - p_i \rightsquigarrow \log \left( {{1 \over {p_i }}} \right)
h(p) =	∑_ip_i(1 − p_i)
H(p) =	∑_ip_i log(1/p_i)
h(X, Y) =	∑_x,y p (x, y) [1 − p(x, y)]
H(X, Y) =	∑x,yp(x,y)log⁡(1p(x,y)) \sum\nolimits_{x,y} p\left( {x,y} \right)\log \left( {{1 \over {p\left( {x,y} \right)}}} \right)
h(X\|Y) =	∑_x,y p (x, y) [(1 − p(x, y) − (1 − p(y))]
H(X\|Y) =	∑x,yp(x,y)[log⁡(1p(x,y))−log⁡(1p(y))] \sum\nolimits_{x,y} p\left( {x,y} \right)\left[ {\log \left( {{1 \over {p\left( {x,y} \right)}}} \right) - \log \left( {{1 \over {p\left( y \right)}}} \right)} \right]
m(X, Y)	∑_x,y p(x,y) [[1 − p(x)] + [1 − p(y)] − [1 − p(x,y)]]
I(X, Y)	∑x,yp(x,y)[log⁡(1p(x))+log⁡(1p(y))−log⁡(1p(x,y))] \sum\nolimits_{x,y} p\left( {x,y} \right)\left[ {\log \left( {{1 \over {p\left( x \right)}}} \right) + \log \left( {{1 \over {p\left( y \right)}}} \right) - \log \left( {{1 \over {p\left( {x,y} \right)}}} \right)} \right]