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First, we construct a resource theory analogous to bipartite local operations (LO), under which CMI does not increase. The main ingredients to any resource theory are:

1. A set of free operations that may consume or convert, but not create the quantity defined as a resource. 2. A set of free states that have zero resource value.

3. One or more monotones, which are zero for free states, non-increasing under free operations, and interpreted as quantifying the resource value of a given state.

See [2] for background on the defining aspects of a resource theory. Resource theories abound in thermo- dynamics, where one converts and expends reserves of energy to perform heat or work. Another famous resource theory in quantum information is that of entanglement, which is non-increasing under local opera- tions and classical communication. For a bipartite state ρAB_{, local operations would consist of all channels}

acting entirely on either A or B.

We generalize local operations to individual operations on a pair of algebras:

Definition 4.1 (Individual Operations, based on definitions from [23]). Let S, T ⊆ S ∨ T and ρ ∈ S1(S ∨ T )

be a density. We define an S-operation (S-op for short) as a sequence of the following:

2. We may transform ρ → Φ(ρ) for any channel Φ : S1(S ∨ T ) → S1(S ∨ T ) that is T -preserving up to

isometry and satisfies [Φ, ES] = 0, without explicitly changing S or T .

3. We may reduce S → ˜S, ρ → E_S∨T˜ (ρ) without changing T for ˜S satisfying ˜S ⊆ S, S ∩ T = ˜S ∩ T , and

E_S∨T˜ ES = E_S˜.

We define T -operations analogously. Directly following, we find:

Theorem 4.1 (based on theorem 2.12 in [23]). Under an S-operation for which S → ˜S and ρ → ˜ρ,

I(S : T )ρ≥ I( ˜S : T )ρ˜. (4.1)

As a consequence, I is non-increasing under any sequence of S-ops and T -ops.

proof of Theorem 4.1. We prove this by showing the monotonicity for each class of operations:

1. We use the additivity of entropy for product states. H(T ) and H(S ∩ T ) each gain contributions of log |C|, which cancel. H(S) and H(S ∨ T ) each gain contributions of H(σ), which cancel between them.

2. Under the assumption that ET(Φ(ρ)) = U ET(ρ)U† for some unitary U , H(T )ρ = H(T )Φ(ρ) is un-

changed. Moreover, by [Φ, ES] = 0,

ES∩T(Φ(ρ)) = ETESΦ(ρ) = ETΦES(ρ) = U ET ∩S(ρ)U† , (4.2)

and H(S ∩ T )ρ = H(S ∩ T )Φ(ρ). For the other two terms H(S)ρ−H(S ∨T )ρ= DS(ρ) is non-increasing

under Φ that commutes with ES.

3. H(S ∩ T ) = H( ˜S ∩ T ) and H(T ) are unchanged. Again by the assumption E_S∨T˜ ES = E_S˜,

H(S)ρ− H(S ∨ T )ρ= DS(ρ) ≥ D(E_S∨T˜ (ρ)kE_S∨T˜ (ES(ρ))) ≥ D(E_S∨T˜ (ρ)kE_S˜(ρ)) .

As should be expected, individual operations restrict to local operations if we impose the constraint that S, T must be factors. If they overlap, then their intersection becomes an auxiliary system. We thereby recover monotonicity of CMI under local operations. In general, S, T need not be subsystems. A canonical

such example is when we choose S = X , the algebra of observables in the Pauli X basis, and T = Z, that in the Pauli Z basis. In this case we still find that X ∩ Z = C, so there is no overlap between subsystems. Nonetheless, X ∨ Z = H2, a qubit Hilbert space.

Remark 4.1. Theorem 4.1 leads directly to Theorem 3.5, as we can rewrite I(ΦS : ΨT) as a I(S : T )Φ◦Ψ(ρ),

where Φ and Ψ are individual operations. We may also denote

I(S : T|S ∩ T ) ≡ I(S : T ) . (4.3)

This notation makes it clearer that I(S : T|S ∩ T ) does not allow the parties holding S and T to modify the intersection, and that S ∩ T should not be considered shared when studying correlations or operations. Remark 4.2. The individual operations defined in 4.1 essentially gives the intuition behind and proof for Theorem 3.5, and also suggest why it does not go much further than positivity of GCMI. In particular, we may rewrite

I(ΦS : ΨT)ρ= I(S : T )ΦΨ(ρ) ,

recalling that ΦΨ = ΨΦ. Showing this inequality is a simple matter of applying the commutation and absorption relations assumed by Theorem 3.5. The main reason why this works is that these channels are individual operations.

(a) (b)

Figure 4.1: (a) Alice and Bob are two spatially-separated parties. No information written by Alice would be observable to Bob until sufficient time has passed for light to propagate. (b) Alice and Bob are not spatially separated, but their intersection is locked by imposed conditions. Any information written by Alice remains invisible to Bob. Images use LibreOffice clipart.

Definition 4.1 might not be the most general form of operations under which I(S : T )ρ is monotonic.

We could instead attempt to define a notion of individual operations in a game-like setting. Let us consider a set of experiments available to Alice, {a1, ..., an} and respectively to Bob, {b1, ..., bm}. These might index

observables, or operators in a positive operator-valued measurement (POVM). We might then allow Alice and Bob to each perform any operations on the joint system in any order, as long as the other’s experiments

cannot tell whether those actions occurred. This guarantees that no communication occurs between Alice and Bob. Unfortunately, it may not entirely ensure monotonicity of GCMI - one could for example imagine that both players start with individual mixed states, but they replace those states by an entangled or correlated pair. What we will see shortly, however, is that we can add some free operations that change the algebras in more interesting ways. In this section, we also note:

Remark 4.3. Some I-non-increasing transformations are not really S and T -operations but change the state and algebra in compensatory ways.

1. I(S : T )ρ is unchanged under ρ → U ρU†, S → ˜S = U SU† and T → ˜T = U T U† for any isometry

U . If U -conjugation commutes with ES, ET and ES∨T, then S, T = ˜S, ˜T , and we may change only the

density. These are essentially coordinate changes.

2. I(S : T ) is non-increasing under change of the algebra T → ˜T , ρ → ρ for which T ⊆ ˜T , S and S ∨ T = S ∨ ˜T are unchanged, and both T , S and ˜T , S form commuting squares. This locks elements of S in S ∩ T .

3. Let S, T ⊆ S ∨ T be a commuting square. Let ER be a conditional expectation onto a subalgebra

R and ˜S be another algebra such that T ⊂ R, R ∩ S = R ∩ ˜S, and ER commutes with ES and E_S˜.

I(S : T ) is non-increasing under the transformation S → ˜S, ρ → ER(ρ) with T unchanged.

Since Alice and Bob are not in tensor position, the intuitive concept of spatial separation generally need not apply. For example, there is no way to have two parties share complementary bases of a single qubit without co-location. As illustrated in Figure 4.1, we replace spatial separation by secrecy. The individual channel Φ as in step 2 of Definition 4.1 have the interpretation of obeying the following constraints:

• [Φ, ES] = 0 implies all observables in S and S ∨ T perceive it as the same channel.

• ΦET = ETΦ = ET implies invisibility to all experiments in T .

The other steps in individual operations, as well as those from Remark 4.3 also forbid communication between an observer with access to S and one with access to T . At times this may forbid some procedures that would intuitively be available. For example, a pair of observers holding the Pauli observables X and Z would each be forbidden from measuring their own observable, which would be sometimes detectable by an observer holding the other. They could however agree that one will drop access to observables in their subalgebra as in step 3 of an individual operation, and the other measures. Local operations are mutually commuting, so they do not require time-ordering, while free operations in more general commuting square settings may

not commute. Even in the usual setting of local operations and classical communication, communication between the parties breaks the commutation of local operations and makes this class much more complicated [85]. In the rest of this chapter, we will see some applications of operations that may consume coordinated shared randomness or classical communication.