Upper bounds on entangling rates of bipartite Hamiltonians
We discuss upper bounds on the rate at which unitary evolution governed by a non-local Hamiltonian can generate entanglement in a bipartite system. Given a bipartite Hamiltonian coupling two finite dimensional particles and , the entangling rate is shown to be upper bounded by , where is the smallest dimension of the interacting particles, is the operator norm of , and is a constant close to . Under certain restrictions on the initial state we prove analogous upper bound for the ancilla-assisted entangling rate with a constant that does not depend upon dimensions of local ancillas. The restriction is that the initial state has at most two distinct Schmidt coefficients (each coefficient may have arbitrarily large multiplicity). Our proof is based on analysis of a mixing rate — a functional measuring how fast entropy can be produced if one mixes a time-independent state with a state evolving unitarily.
Consider two remote parties Alice and Bob controlling finite-dimensional quantum systems and . Suppose and interact with each other according to a time-independent Hamiltonian . Unless is a sum of local Hamiltonians, a unitary evolution is a non-local operation capable of creating entanglement between and . The goal of the present paper is to get an upper bound on the rate at which the entanglement between Alice and Bob can increase or decrease as a function of time.
We consider ancilla-assisted entangling, see Figure 1. It means that Alice’s laboratory consists of two subsystems: and , such that the Hamiltonian acts only on the subsystem . Similarly, Bob’s laboratory is partitioned into and , where acts only on the subsystem . The subsystems and are the local ancillas held by Alice and Bob. In the ancilla-assisted entangling Alice and Bob start from a pure state of the composite system which may be already entangled.
Time evolution of the composite system is described by a unitary operator . Thus the joint state of Alice and Bob always remains pure. Accordingly, the entanglement between Alice and Bob at any time can be quantified by entanglement entropy
The quantity we are interested in is entangling rate
Understanding properties of the entangling rate is crucial for optimal generation of entanglement , computing capacities of bidirectional quantum communication channels , and for describing dynamics of entanglement in quantum spin lattice models .
Our goal is to get an upper bound on that would not explicitly depend upon dimensions of the local ancillas and . Upper bounds of this kind can be easily generalized via the Trötter decomposition to arbitrary multipartite Hamiltonians decomposable into a sum of few-party interactions.
1.1 Previous work
The maximal entangling rate has been studied by many authors for the case when and are qubits. The optimal state maximizing has been found by Dür, Vidal et al  for a general Hamiltonian assuming . These authors have also observed that for some Hamiltonians local ancillas are capable of increasing the maximal entangling rate . A powerful technique of getting upper bounds on for arbitrarily large and was proposed by Childs, Leung et al . It was used to identify a subclass of two-qubit Hamiltonians, including the Ising interaction, for which the maximum does not depend upon and (and thus can be achieved without local ancillas). An upper bound on the entangling rate of the Ising interaction has been also obtained by Cirac, Dür et al.  by showing that the time evolution with the Ising Hamiltonian for time can be implemented by LOCC protocol consuming e-bits of pre-shared entanglement.
Some progress has been also achieved for larger dimensions of and . The technique of  has been generalized by Wang and Sanders  to prove that for any product Hamiltonian if have eigenvalues . This result applies to arbitrary dimensions of and . An upper bound for the ancilla-assisted case has been proved for arbitrary bipartite product Hamiltonians by Childs, Leung, and Vidal . Finally, it was shown by Bennett, Harrow et al.  that , where and does not depend upon and . This result relies on a decomposition of an arbitrary bipartite Hamiltonian into a sum of product Hamiltonians. Thus for fixed and the entangling rate has a constant upper bound independent on how large are dimensions and . The authors of  also proved that the supremum of over all dimensions of local ancillas coincides with the asymptotic capacity of to generate entanglement by any protocol in which unitary evolution with is interspersed with LOCC. It is unknown whether the supremum over and can be actually achieved for finite dimensional ancillas.
It is not known whether , i.e., whether the maximal entangling and disentangling rates of a given Hamiltonian coincide. The results by Linden, Smolin, and Winter  applicable to finite unitary operators suggest that this equality might be wrong.
1.2 Summary of results
We start from observing that the unitary evolution with any Hamiltonian can not increase (decrease) the entanglement by more than , where , see . This property can be called small total entangling, as it says that the total increase (decrease) of entanglement throughout the unitary evolution remains bounded as the dimensions of local ancillas and go to infinity. Is there an analogue of the small total entangling property for infinitely small time intervals? The following conjecture was initially proposed by Kitaev . We call it small incremental entangling.
Small Incremental Entangling (SIE): There exists a constant such that
for all dimensions , and for all states of a composite system .
It is not known whether SIE is true or false. Our first results is a proof of SIE for the special case when has at most two distinct Schmidt coefficients with respect to a partition (each Schmidt coefficient may have arbitrarily large multiplicity). Our proof yields the constant although the actual value of might be much smaller. The constraint on serves technical purposes and was introduced in order to make the problem tractable. It should be mentioned that the upper bound Eq. (3) can be confirmed by straightforward calculation if (no local ancillas), see Section 2. In this case the optimal pair can be found explicitly. It turns out that the optimal state has only two distinct Schmidt coefficients and thus falls into the category that we consider, see Section 2 for details. We don’t know whether the optimal state has only two Schmidt coefficients in the ancilla-assisted case.
It is known that in some cases local ancillas and pre-shared entanglement can lead to counter-intuitive effects, such as entanglement embezzling  or locking of classical correlations . In particular, it was demonstrated by DiVincenzo, Horodecki et al.  that sending a single qubit from Alice to Bob can increase their classical mutual information by an arbitrarily large amount in the presence of local ancillas. Thus SIE may be violated if locking effects also occur in the infinitesimal unitary transformations if one measures correlations by entanglement entropy. For future references let us give an explicit expression for the entangling rate that can be easily obtained by computing the derivative in Eq. (2).
Our proof of SIE goes by getting an upper bound on a mixing rate. In order to define a mixing rate, consider a probabilistic ensemble of (mixed) states defined on a finite-dimensional Hilbert space (lacking any tensor product structure). Let be the average state corresponding to . For any Hamiltonian define a time dependent state . Define a mixing rate as
Here is the von Neumann entropy of . Basic properties of the von Neumann entropy imply that , where is the average entropy for the ensemble and is the Shannon entropy. This property can be called small total mixing as it says that the total increase (decrease) of the entropy throughout the unitary evolution goes to zero as one of the probabilities or goes to zero. Is there an analogue of the small total mixing property for infinitely small time intervals? A naive generalization would be as follows.
Small Incremental Mixing (SIM): There exists a constant such that
for any probabilistic ensemble .
Here it is meant that the constant is independent from the dimension of the Hilbert space. It is not known whether SIM is true or false. Although SIM might seem completely unrelated to ancilla-assisted entangling and SIE, it turns out that SIM is a stronger version of SIE. More strictly, we prove that SIM with a constant implies SIE with a constant , see Section 3.
Our second result is a proof of SIM for the special case when has at most two distinct eigenvalues (each eigenvalue may have arbitrarily large multiplicity), see Section 4. We shall refer to eigenvalues obeying this constraint as a binary spectrum. Our proof yields , although the actual value of might be smaller.
The connection between SIE to SIM described in Section 3 has a peculiar property that the number of distinct eigenvalues of in SIM is exactly the same as the number of distinct Schmidt coefficients of the initial state in SIE. Thus a proof of SIM for the case when has a binary spectrum implies SIE for the case when the initial state has only two distinct Schmidt coefficients. In the general case we prove that the mixing rate of any ensemble has an upper bound , so it does not explicitly depend on the dimension of the Hilbert space, see Section 4. For future references let us give an explicit expression for the mixing rate:
The paper is organized as follows. In Section 2 we find the the maximal entangling rate and the optimal pair for the case when Alice and Bob do not use local ancillas. Section 3 proves that SIM implies SIE. Section 4 contains the main results of the paper. It proves SIM for the case when has a binary spectrum. Section 5 reports results of numerical maximization aimed at verifying SIM. The data obtained in numerical simulations are consistent with SIM.
2 Maximal entangling rate in the absence of ancillas
The maximal entangling rate can be easily found in the absence of local ancillas, i.e., when . In this case we can always write the initial state of Alice and Bob using the Schmidt decomposition with . Denote the reduced density matrix of Alice. Using the general formula Eq. (4) for the entangling rate one gets
Since the entangling rate is a linear function of we can assume that . Using the fact that for any Hermitian operator
one can carry out the maximization over ,
For any vectors one has the following identity
Substituting one gets
We can assume that all (otherwise replace by ). Then the maximum of can be found by solving extremal point equations . After simple algebra one gets
(Here and stand for base two and natural logarithm.) The equality for the minus sign in the r.h.s. of Eq. (13) is possible only for one value of , since it implies . Let us agree that . Then for all , that is the state must have a binary spectrum with multiplicities . Introduce a variable such that
Let be the state with Schmidt coefficients . Using Eq. (12) the maximal entangling rate with the initial state can be written as
The optimal value and the optimal entangling rate have to be found by maximizing Eq. (15) over . For example, if , numerical maximization yields and . It coincides with the maximal entangling rate of product two-qubit Hamiltonians found in [1, 4]. It follows that under normalization condition product two-qubit Hamiltonians are capable of generating entanglement with the largest rate.
One can easily infer from Eq. (15) that and for sufficiently large . It proves that in the absence of local ancillas. Moreover, in the limit of large one can explicitly write down the optimal state and the optimal Hamiltonian . Namely,
Thus the optimal state is a superposition of a product state and a maximally entangled state with locally orthogonal supports. The optimal Hamiltonian is a generator for a rotation in the corresponding two-dimensional subspace.
For a finite the optimal value and the corresponding entangling rate can be found numerically, see Figure 2. For large one has , while entanglement entropy of scales as .
3 SIM implies SIE
In this section we assume that SIM is true and show that this assumption implies SIE. Consider ancilla-assisted entangling with dimensions and some initial state . Let us assume that . Then SIE is equivalent to an upper bound
Since the r.h.s. of this inequality does not depend on and , it is enough to prove Eq. (18) in the case since we can always extend to . Now we have a tripartite system initially prepared in a pure state and evolving under a Hamiltonian acting on and . Using formula Eq. (4) for the entangling rate one gets
where and are the reduced states of , and a state is defined as
In order to define a probabilistic ensemble needed for a reduction to SIM we shall need the following lemma.
For any mixed state there exists a mixed state such that
Proof: Since the partial trace is a linear operation, it suffices to prove the lemma for the case when is a pure state. The statement of the lemma is then equivalent to inequality
Let be the Schmidt rank of . Obviously, . Consider a state , where and are the local bases of and that diagonalize and . Then
Taking into account that , one concludes that . Multiplying this inequality by on the left and on the right we get which implies Eq. (22).
Define an ensemble of states such that , , , . Here is the state that appears in the decomposition Eq. (21). This ensemble has the average state . Let be a Hamiltonian that appears in Eq. (19). Assuming that SIM is true one gets
(Note that for any one has .) On the other hand, formula Eq. (7) for the mixing rate leads to
The reduction Eq. (20) has a peculiar property that the number of distinct eigenvalues of and are the same. Since we have used identification , the number of distinct eigenvalues of coincides with the number of distinct Schmidt coefficients of . Thus if we can prove SIM with having only two distinct eigenvalues, we will prove SIE for the case when has only two distinct Schmidt coefficients.
4 Upper bounds on the mixing rate
Let us start from proving a weaker (compared to SIM) upper bound on the mixing rate.
For any ensemble and Hamiltonian one has
Proof: Suppose , where the states and a Hamiltonian are defined on a -dimensional Hilbert space. Consider ancilla-assisted entangling protocol shown on Figure 1 with , , , . Choose a Hamiltonian coupling and as
Choose the initial state as
where is a purification of . Then the state evolves in time as
This lemma implies that the whole difficulty of proving SIM (if it is true) concerns the limiting cases or . Note also that , where ensemble is obtained from by interchanging with . Thus we can assume that .
In the following we shall represent an ensemble using the data , where
One can easily check that a triple represents some ensemble iff
Indeed, the only non-trivial statement is that . It can be obtained from inequality by multiplying it by on the left and on the right. Using the representation Eq. (26) one can rewrite the mixing rate as
It is worth mentioning that for given the upper bound Eq. (6) is true for sufficiently small . Indeed, applying the triangle inequality for the trace norm to Eq. (29) one gets . Since for any operators it follows that . If is smaller than the smallest eigenvalue of , one has and thus .
In the rest of the section we prove SIM under the assumption that has binary spectrum,
Here can be arbitrary integer between and . Define projectors and such that
After some algebra one gets the following identity
In order to upper bound the trace norm of the commutator we shall use the following.
Lemma 3 (Hölder inequality)
Let be a projector and be a positive semi-definite operator. Then
Let be a projector of rank and be a Hermitian operator such that
. Then there exists a Hermitian operator such that
We shall postpone the proof of the two lemmas above until the end of the section. Let us apply Lemma 3 to the commutator in Eqs. (30) where is replaced by from Lemma 4. Then we can upper bound the mixing rate in Eq. (29) as
In order to analyze the expression above introduce new variables such that
Expressing and in terms of and one can rewrite Eq. (32) as
Let us find constraints on the variables . Noting that and taking the trace with one gets . Besides, condition (iii) of Lemma 4 implies that and thus . By obvious reasons one also has . Summarizing,
Now the problem of getting upper bound on the mixing rate reduces to maximizing a function in Eq. (34) under constraints Eq. (35). We prove (see Lemma 5 at the end of the section) that as long as . Using condition (i) of Lemma 4 we get . As was mentioned in the beginning of the section, we can assume that and thus . Summarizing, we get
where we used the fact that a function is monotone increasing on the interval . Thus we have proved SIM under the assumption that has a binary spectrum.
Proof of Lemma 3: Hölder inequality asserts that
for any operators and , see . In order to choose proper and let us use an identity and triangle inequality for the trace norm:
Substituting and into Eq. (36) and noting that , are projectors we get the inequality stated in the lemma.
Proof of Lemma 4: Any Hermitian operator satisfying can be written as a convex combination of projectors:
Suppose we can find the operator promised in the lemma for every projector in the sum, that is we can find such that (i) , (ii) , and (iii) . Then we can choose the desired operator as . Thus it suffices to prove the lemma for the case when is a projector. Let be the dimension of the Hilbert space. Consider a direct sum decomposition
where is the range of , so that . Then we can write
Here and are Hermitian operators on and and is some operator . The requirement that is a projector implies
Consider a decomposition , where and , while and have only eigenvalues . It follows from Eq. (38) that
Thus for any from the range of and for any from the range of . Accordingly, the projector has the following block structure
Let be the central block in . Clearly is a projector and satisfies conditions (i) and (ii) of the lemma. It remains to check that . Let be the dimension of the block . By definition, . Since is a projector, the operators obey the same constraint as Eq. (38), that is
Since and we conclude that and are non-singular matrices and thus the dimensions of the blocks and both equal to . Also from Eq. (40) we infer that , that is the spectrum of coincides with the spectrum of including multiplicities. Accordingly, . Therefore satisfies all three conditions of the lemma.
Let be a real number. Consider a function
Suppose and . Then .
Let us consider three cases:
case 1: , .
Then . Using an upper bound we arrive to
The last inequality follows from monotonicity of a function on the interval .
case 2: , .
First note that . Take into account that
case 3: , .
Introduce new variable such that , that is . Then
One can check that for all . Using inequality
and noticing that is monotone increasing for we conclude that
for any . Thus
Combining all three cases we get
5 Numerical maximization of the mixing rate
This section describes numerical simulations aimed at verifying SIM. Let us start from expression Eq. (28) for the mixing rate and the constraints Eq. (27). It is convenient to represent the Hamiltonian as . Note that iff . Then where
For a fixed average state SIM is equivalent to an upper bound
where the maximization is subject to
We found the maximum of numerically for the average states
with the dimension and for several values of between and , see Figure 3 (as was mentioned in Section 4, it is enough to consider between the smallest eigenvalue of and ).
The motivation for this particular choice of the average state comes from the fact that the states can “embezzle” any other mixed state for sufficiently large , see . More strictly, it was proved in  that for any state there exists an isometry such that in the sense that fidelity between the two states goes to as goes to infinity. In particular this is true for the optimal state corresponding to the optimal ensemble maximizing the mixing rate in Eq. (29) for a fixed . On the other hand, the state is at least as good as as far as the mixing rate is concerned. Indeed, one can easily verify that the mixing rate for the ensemble is the same as the mixing rate for the ensemble , where and . Thus a verification of SIM for the family of states is a good test for general validity of SIM.
The data obtained in the numerical maximization of are presented in Figure 3. They are consistent with the conjecture Eq. (42) with a constant . The numerical algorithm that we used is based on reformulation of the maximization problem as a semidefinite program. The details of the algorithm are described in Appendix A.