diff --git a/src/QuantumInterface.jl b/src/QuantumInterface.jl
index c1bc651..80c7b1a 100644
--- a/src/QuantumInterface.jl
+++ b/src/QuantumInterface.jl
@@ -7,23 +7,88 @@ module QuantumInterface
 """
     basis(a)
 
-Return the basis of an object.
+Return the basis of a quantum object.
 
-If it's ambiguous, e.g. if an operator has a different left and right basis,
-an [`IncompatibleBases`](@ref) error is thrown.
+If it's ambiguous, e.g. if an operator has a different
+left and right basis, an [`IncompatibleBases`](@ref) error is thrown.
+
+See [`StateVector`](@ref) and [`AbstractOperator`](@ref)
 """
 function basis end
 
+"""
+    basis_l(a)
+
+Return the left basis of an operator.
+"""
+function basis_l end
+
+"""
+    basis_r(a)
+
+Return the right basis of an operator.
+"""
+function basis_r end
+
 """
 Exception that should be raised for an illegal algebraic operation.
 """
 mutable struct IncompatibleBases <: Exception end
 
+#function bases end
+
+function spinnumber end
+
+function cutoff end
+
+function offset end
 
 ##
 # Standard methods
 ##
 
+"""
+    multiplicable(a, b)
+
+Check if any two subtypes of `StateVector` or `AbstractOperator`,
+can be multiplied in the given order.
+"""
+function multiplicable end
+
+"""
+    check_multiplicable(a, b)
+
+Throw an [`IncompatibleBases`](@ref) error if the objects are
+not multiplicable as determined by `multiplicable(a, b)`.
+
+If the macro `@compatiblebases` is used anywhere up the call stack,
+this check is disabled.
+"""
+function check_multiplicable end
+
+"""
+    addible(a, b)
+
+Check if any two subtypes of `StateVector` or `AbstractOperator`
+ can be added together.
+
+Spcefically this checks whether the left basis of a is equal
+to the left basis of b and whether the right basis of a is equal
+to the right basis of b.
+"""
+function addible end
+
+"""
+    check_addible(a, b)
+
+Throw an [`IncompatibleBases`](@ref) error if the objects are
+not addible as determined by `addible(a, b)`.
+
+If the macro `@compatiblebases` is used anywhere up the call stack,
+this check is disabled.
+"""
+function check_addible end
+
 function apply! end
 
 function dagger end
diff --git a/src/abstract_types.jl b/src/abstract_types.jl
index 0650290..7e63a53 100644
--- a/src/abstract_types.jl
+++ b/src/abstract_types.jl
@@ -1,35 +1,40 @@
 """
-Abstract base class for all specialized bases.
+Abstract type for all specialized bases of a Hilbert space.
 
-The Basis class is meant to specify a basis of the Hilbert space of the
-studied system. Besides basis specific information all subclasses must
-implement a shape variable which indicates the dimension of the used
-Hilbert space. For a spin-1/2 Hilbert space this would be the
-vector `[2]`. A system composed of two spins would then have a
-shape vector `[2 2]`.
+The `Basis` type specifies an orthonormal basis for the Hilbert
+space of the studied system. All subtypes must implement `Base.:(==)`,
+and `Base.size`. `size` should return a tuple representing the total dimension
+of the Hilbert space with any tensor product structure the basis has such that 
+`length(b::Basis) = prod(size(b))` gives the total Hilbert dimension
 
-Composite systems can be defined with help of the [`CompositeBasis`](@ref)
-class.
+Composite systems can be defined with help of [`CompositeBasis`](@ref).
+
+All relevant properties of subtypes of `Basis` defined in `QuantumInterface`
+should be accessed using their documented functions and should not
+assume anything about the internal representation of instances of these  
+types (i.e. don't access the struct's fields directly).
 """
 abstract type Basis end
 
 """
-Abstract base class for `Bra` and `Ket` states.
+Abstract type for `Bra` and `Ket` states.
 
-The state vector class stores the coefficients of an abstract state
-in respect to a certain basis. These coefficients are stored in the
-`data` field and the basis is defined in the `basis`
-field.
+The state vector class stores an abstract state with respect
+to a certain basis. All subtypes must implement the `basis`
+method which should this basis as a subtype of `Basis`.
 """
 abstract type StateVector{B,T} end
 abstract type AbstractKet{B,T} <: StateVector{B,T} end
 abstract type AbstractBra{B,T} <: StateVector{B,T} end
 
 """
-Abstract base class for all operators.
+Abstract type for all operators and super operators.
 
-All deriving operator classes have to define the fields
-`basis_l` and `basis_r` defining the left and right side bases.
+All subtypes must implement the methods `basis_l` and
+`basis_r` which return subtypes of `Basis` and
+represent the left and right bases that the operator
+maps between and thus is compatible with a `Bra` defined
+in the left basis and a `Ket` defined in the right basis.
 
 For fast time evolution also at least the function
 `mul!(result::Ket,op::AbstractOperator,x::Ket,alpha,beta)` should be
@@ -53,3 +58,5 @@ A_{br_1,br_2} = B_{bl_1,bl_2} S_{(bl_1,bl_2) ↔ (br_1,br_2)}
 ```
 """
 abstract type AbstractSuperOperator{B1,B2} end
+
+const AbstractQObjType = Union{<:StateVector,<:AbstractOperator}
diff --git a/src/bases.jl b/src/bases.jl
index e178a66..9f583e6 100644
--- a/src/bases.jl
+++ b/src/bases.jl
@@ -7,7 +7,7 @@
 
 Total dimension of the Hilbert space.
 """
-Base.length(b::Basis) = prod(b.shape)
+Base.length(b::Basis) = prod(b.shape) # change to prod(size(b)) when downstream Bases are updated
 
 """
     GenericBasis(N)
@@ -24,7 +24,7 @@ end
 GenericBasis(N::Integer) = GenericBasis([N])
 
 Base.:(==)(b1::GenericBasis, b2::GenericBasis) = equal_shape(b1.shape, b2.shape)
-
+Base.size(b::GenericBasis) = b.shape
 
 """
     CompositeBasis(b1, b2...)
@@ -42,8 +42,11 @@ end
 CompositeBasis(bases) = CompositeBasis([length(b) for b ∈ bases], bases)
 CompositeBasis(bases::Basis...) = CompositeBasis((bases...,))
 CompositeBasis(bases::Vector) = CompositeBasis((bases...,))
+#bases(b::CompositeBasis) = b.bases
 
 Base.:(==)(b1::T, b2::T) where T<:CompositeBasis = equal_shape(b1.shape, b2.shape)
+Base.size(b::CompositeBasis) = length.(b.bases)
+Base.getindex(b::CompositeBasis, i) = getindex(b.bases, i)
 
 ##
 # Common bases
@@ -69,6 +72,9 @@ struct FockBasis{T} <: Basis
 end
 
 Base.:(==)(b1::FockBasis, b2::FockBasis) = (b1.N==b2.N && b1.offset==b2.offset)
+Base.size(b::FockBasis) = (b.N - b.offset + 1,)
+cutoff(b::FockBasis) = b.N
+offset(b::FockBasis) = b.offset
 
 
 """
@@ -88,6 +94,7 @@ struct NLevelBasis{T} <: Basis
 end
 
 Base.:(==)(b1::NLevelBasis, b2::NLevelBasis) = b1.N == b2.N
+Base.size(b::NLevelBasis) = (b.N,)
 
 """
     NQubitBasis(num_qubits::Int)
@@ -106,6 +113,7 @@ struct NQubitBasis{S,B} <: Basis
 end
 
 Base.:(==)(pb1::NQubitBasis, pb2::NQubitBasis) = length(pb1.bases) == length(pb2.bases)
+Base.size(b::NQubitBasis) = b.shape
 
 """
     SpinBasis(n)
@@ -132,7 +140,8 @@ SpinBasis(spinnumber::Rational) = SpinBasis{spinnumber}(spinnumber)
 SpinBasis(spinnumber) = SpinBasis(convert(Rational{Int}, spinnumber))
 
 Base.:(==)(b1::SpinBasis, b2::SpinBasis) = b1.spinnumber==b2.spinnumber
-
+Base.size(b::SpinBasis) = (numerator(b.spinnumber*2 + 1),)
+spinnumber(b::SpinBasis) = b.spinnumber
 
 """
     SumBasis(b1, b2...)
@@ -151,3 +160,4 @@ SumBasis(bases::Basis...) = SumBasis((bases...,))
 Base.:(==)(b1::T, b2::T) where T<:SumBasis = equal_shape(b1.shape, b2.shape)
 Base.:(==)(b1::SumBasis, b2::SumBasis) = false
 Base.length(b::SumBasis) = sum(b.shape)
+# TODO how should `.bases` be accessed? `getindex` or a `sumbases` method?
diff --git a/src/deprecated.jl b/src/deprecated.jl
index 680b8f3..dd439eb 100644
--- a/src/deprecated.jl
+++ b/src/deprecated.jl
@@ -43,12 +43,6 @@ function check_samebases(b1, b2)
     end
 end
 
-function check_multiplicable(b1, b2)
-    if BASES_CHECK[] && !multiplicable(b1, b2)
-        throw(IncompatibleBases())
-    end
-end
-
 samebases(b1::Basis, b2::Basis) = b1==b2
 samebases(b1::Tuple{Basis, Basis}, b2::Tuple{Basis, Basis}) = b1==b2 # for checking superoperators
 samebases(a::AbstractOperator) = samebases(a.basis_l, a.basis_r)::Bool # FIXME issue #12
@@ -68,5 +62,3 @@ function multiplicable(b1::CompositeBasis, b2::CompositeBasis)
     end
     return true
 end
-
-multiplicable(a::AbstractOperator, b::AbstractOperator) = multiplicable(a.basis_r, b.basis_l) # FIXME issue #12
diff --git a/src/expect_variance.jl b/src/expect_variance.jl
index 86f5e64..678fb5f 100644
--- a/src/expect_variance.jl
+++ b/src/expect_variance.jl
@@ -3,33 +3,35 @@
 
 If an `index` is given, it assumes that `op` is defined in the subsystem specified by this number.
 """
-function expect(indices, op::AbstractOperator{B1,B2}, state::AbstractOperator{B3,B3}) where {B1,B2,B3<:CompositeBasis}
-    N = length(state.basis_l.shape)
-    indices_ = complement(N, indices)
-    expect(op, ptrace(state, indices_))
-end
+expect(indices, op::AbstractOperator, state::AbstractOperator) =
+    expect(op, ptrace(state, complement(nsubsystems(state), indices)))
+
+expect(index::Integer, op::AbstractOperator, state::AbstractOperator) = expect([index], op, state)
 
-expect(index::Integer, op::AbstractOperator{B1,B2}, state::AbstractOperator{B3,B3}) where {B1,B2,B3<:CompositeBasis} = expect([index], op, state)
 expect(op::AbstractOperator, states::Vector) = [expect(op, state) for state=states]
+
 expect(indices, op::AbstractOperator, states::Vector) = [expect(indices, op, state) for state=states]
 
-expect(op::AbstractOperator{B1,B2}, state::AbstractOperator{B2,B2}) where {B1,B2} = tr(op*state)
+expect(op::AbstractOperator, state::AbstractOperator) =
+    (check_multiplicable(state, state); check_multiplicable(op,state); tr(op*state))
 
 """
     variance(index, op, state)
 
 If an `index` is given, it assumes that `op` is defined in the subsystem specified by this number
 """
-function variance(indices, op::AbstractOperator{B,B}, state::AbstractOperator{BC,BC}) where {B,BC<:CompositeBasis}
-    N = length(state.basis_l.shape)
-    indices_ = complement(N, indices)
-    variance(op, ptrace(state, indices_))
-end
+variance(indices, op::AbstractOperator, state::AbstractOperator) =
+    variance(op, ptrace(state, complement(nsubsystems(state), indices)))
+
+variance(index::Integer, op::AbstractOperator, state::AbstractOperator) = variance([index], op, state)
 
-variance(index::Integer, op::AbstractOperator{B,B}, state::AbstractOperator{BC,BC}) where {B,BC<:CompositeBasis} = variance([index], op, state)
 variance(op::AbstractOperator, states::Vector) = [variance(op, state) for state=states]
+
 variance(indices, op::AbstractOperator, states::Vector) = [variance(indices, op, state) for state=states]
 
-function variance(op::AbstractOperator{B,B}, state::AbstractOperator{B,B}) where B
-    expect(op*op, state) - expect(op, state)^2
+function variance(op::AbstractOperator, state::AbstractOperator)
+    check_multiplicable(op,op)
+    check_multiplicable(state,state)
+    check_multiplicable(op,state)
+    @compatiblebases expect(op*op, state) - expect(op, state)^2
 end
diff --git a/src/julia_base.jl b/src/julia_base.jl
index 2d8e085..2eb11fa 100644
--- a/src/julia_base.jl
+++ b/src/julia_base.jl
@@ -14,7 +14,6 @@ addnumbererror() = throw(ArgumentError("Can't add or subtract a number and an op
 *(a::StateVector, b::Number) = b*a
 copy(a::T) where {T<:StateVector} = T(a.basis, copy(a.data)) # FIXME issue #12
 length(a::StateVector) = length(a.basis)::Int # FIXME issue #12
-basis(a::StateVector) = a.basis # FIXME issue #12
 adjoint(a::StateVector) = dagger(a)
 
 
@@ -33,8 +32,6 @@ Base.broadcastable(x::StateVector) = x
 ##
 
 length(a::AbstractOperator) = length(a.basis_l)::Int*length(a.basis_r)::Int  # FIXME issue #12
-basis(a::AbstractOperator) = (check_samebases(a); a.basis_l) # FIXME issue #12
-basis(a::AbstractSuperOperator) = (check_samebases(a); a.basis_l[1]) # FIXME issue #12
 
 # Ensure scalar broadcasting
 Base.broadcastable(x::AbstractOperator) = Ref(x)
diff --git a/src/linalg.jl b/src/linalg.jl
index b48f458..0fd8a26 100644
--- a/src/linalg.jl
+++ b/src/linalg.jl
@@ -4,6 +4,50 @@
 
 const BASES_CHECK = Ref(true)
 
+"""
+    @compatiblebases
+
+Macro to skip checks for compatible bases. Useful for `*`, `expect` and similar
+functions.
+"""
+macro compatiblebases(ex)
+    return quote
+        BASES_CHECK.x = false
+        local val = $(esc(ex))
+        BASES_CHECK.x = true
+        val
+    end
+end
+
+function check_addible(b1, b2)
+    if BASES_CHECK[] && !addible(b1, b2)
+        throw(IncompatibleBases())
+    end
+end
+
+function check_multiplicable(b1, b2)
+    if BASES_CHECK[] && !multiplicable(b1, b2)
+        throw(IncompatibleBases())
+    end
+end
+
+addible(a::AbstractQObjType, b::AbstractQObjType) = false
+addible(a::AbstractBra, b::AbstractBra) = (basis(a) == basis(b))
+addible(a::AbstractKet, b::AbstractKet) = (basis(a) == basis(b))
+addible(a::AbstractOperator, b::AbstractOperator) =
+    (basis_l(a) == basis_l(b)) && (basis_r(a) == basis_r(b))
+
+multiplicable(a::AbstractQObjType, b::AbstractQObjType) = false
+multiplicable(a::AbstractBra, b::AbstractKet) = (basis(a) == basis(b))
+multiplicable(a::AbstractOperator, b::AbstractKet) = (basis_r(a) == basis(b))
+multiplicable(a::AbstractBra, b::AbstractOperator) = (basis(a) == basis_l(b))
+multiplicable(a::AbstractOperator, b::AbstractOperator) = (basis_r(a) == basis_l(b))
+
+basis(a::StateVector) = throw(ArgumentError("basis() is not defined for this type of state vector: $(typeof(a))."))
+basis_l(a::AbstractOperator) = throw(ArgumentError("basis_l() is not defined for this type of operator: $(typeof(a))."))
+basis_r(a::AbstractOperator) = throw(ArgumentError("basis_r() is not defined for this type of operator: $(typeof(a))."))
+basis(a::AbstractOperator) = (basis_l(a) == basis_r(a); basis_l(a))
+
 ##
 # tensor, reduce, ptrace
 ##