Quantum entanglement, the most nonclassical manifestation of quantum mechanics, has found use in a variety of physical tasks in quantum information processing, quantum cryptography, thermodynamics, and quantum computing Horodecki2009a . A natural and fundamental problem is to develop a theoretical framework to quantify and describe it. In spite of remarkable recent progress in the resource theory of entanglement (for reviews see, e.g., Plenio2007 ; Horodecki2009a ), many fundamental challenges have remained open.
One of the most important aspects of the resource theory of entanglement consists of the interconversions of states, with respect to a class of free operations. In particular, the problem of entanglement dilution asks: given a target bipartite state and a canonical unit of entanglement represented by the Bell state (or ebit) , what is the minimum rate at which we can produce copies of from copies of under a chosen set of free operations?
The entanglement cost Bennett1996c was introduced to quantify the minimal rate of converting to with an arbitrarily high fidelity in the limit as becomes large. When local operations and classical communication (LOCC) are allowed for free, the authors of Hayden2001 proved that the entanglement cost is equal to the regularized entanglement of formation Bennett1996c . When the free operations consist of quantum operations that completely preserve positivity of the partial transpose (the PPT-preserving operations of R99 ; R01 ), it is known that the entanglement cost is not equal to the regularized entanglement of formation Audenaert2003 ; F03 ; H06book .
The exact entanglement cost Audenaert2003 is an alternative and natural way to quantify the cost of entanglement dilution, being defined as the smallest asymptotic rate at which is required in order to reproduce exactly. The exact entanglement cost under PPT-preserving operations (PPT entanglement cost) was introduced and solved for a large class of quantum states in Audenaert2003 , but it has hitherto remained unknown for general quantum states.
The above resource-theoretic problems can alternatively be phrased as simulation problems: How many copies of are needed to simulate copies of a given bipartite state ? As discussed above, the simulation can be either approximate, such that a verifier has little chance of distinguishing the simulation from the ideal case, while it can also be exact, such that a verifier has no chance at all for distinguishing the simulation from the ideal case.
With this perspective, it is natural to consider the simulation of a quantum channel, when allowing some set of operations for free and metering the entanglement cost of the simulation. The authors of BBCW13 defined the entanglement cost of a channel to be the smallest rate at which is needed, along with the free assistance of LOCC, in order to simulate the channel , in such a way that a verifier would have little chance of distinguishing the simulation from the ideal case of . In BBCW13 , it was shown that the regularized entanglement of formation of the channel is equal to its entanglement cost, thus extending the result of Hayden2001 in a natural way.
In a recent work Wilde2018 , it was observed that the channel simulation task defined in BBCW13 is actually a particular kind of simulation, called a parallel channel simulation. The paper Wilde2018 then defined an alternative notion of channel simulation, called sequential channel simulation, in which the goal is to simulate uses of the channel in such a way that the most general verification strategy would have little chance of distinguishing the simulation from the ideal uses of the channel. Although a general formula for the entanglement cost in this scenario was not found, it was determined for several key channel models, including erasure, dephasing, three-dimensional Holevo–Werner, and single-mode pure-loss and pure-amplifier bosonic Gaussian channels.
i.2 Summary of results
In this paper, we solve significant questions in the resource theory of entanglement, one of which has remained open since the inception of entanglement theory over two decades ago. Namely, we prove that the exact PPT-entanglement cost for both quantum states and channels have efficiently computable, single-letter formulas, reflecting the fundamental entanglement structures of bipartite quantum states and channels. Notably, this is the first time that an entanglement measure has been shown to be both efficiently computable and to possess a direct operational meaning. Furthermore, we prove that the exact parallel and sequential entanglement costs of quantum channels are given by the same efficiently computable, single-letter formula.
Our paper is structured as follows. We first introduce the -entanglement measure of a bipartite state, and we prove that it satisfies several desirable properties, including monotonicity under completely-PPT-preserving channels, additivity, normalization, faithfulness, non-convexity, and non-monogamy. For finite-dimensional states, it is also efficiently computable by means of a semi-definite program.
Next, we prove that the -entanglement is equal to the exact entanglement cost of a quantum state. This direct operational interpretation and the fact that both convexity and monogamy are violated for the -entanglement measure calls into question whether these properties are truly necessary for entanglement.
We evaluate the -entanglement (and the exact entanglement cost) for several bipartite states of interest, including isotropic states, Werner states, maximally correlated states, some states supported on the antisymmetric subspace, and all bosonic Gaussian states.
We then extend the -entanglement measure from bipartite states to point-to-point quantum channels. We prove that it also satisfies several desirable properties, including non-increase under amortization, monotonicity under PPT superchannels, additivity, normalization, faithfulness, and non-convexity. For finite-dimensional channels, it is also efficiently computable by means of a semi-definite program.
The -entanglement of channels has a direct operational meaning as the entanglement cost of both parallel and sequential channel simulation. Thus, the theory of channel simulation significantly simplifies for the setting in which completely-PPT-preserving channels are allowed for free. In addition to all of the properties that it satisfies, this operational interpretation solidifies the -entanglement of a channel as a foundational measure of the entanglement of a quantum channel.
As the last contribution of this paper, we evaluate the -entanglement (and exact entanglement cost) of several important channel models, including erasure, depolarizing, dephasing, and amplitude damping channels. We also leverage recent results in the literature LMGA17 , regarding the teleportation simulation of bosonic Gaussian channels, in order to evaluate the -entanglement and exact entanglement cost for several fundamental bosonic Gaussian channels. We remark that these latter results provide a direct operational interpretation of the Holevo–Werner quantity HW01 for these channels.
Finally, we conclude with a summary and some open questions.
Ii -entanglement measure for bipartite states
We now introduce an entanglement measure for a bipartite state, here called the -entanglement measure:
Definition 1 (-entanglement measure)
Let be a bipartite state acting on a separable Hilbert space. The -entanglement measure is defined as follows:
In the case that the state acts on a finite-dimensional Hilbert space, then is calculable by a semi-definite program, and thus it is efficiently computable with respect to the dimension of the Hilbert space.
ii.1 Monotonicity under completely-PPT-preserving channels
Throughout this paper, we consider completely-PPT-preserving operations R99 ; R01 , defined as a bipartite operation (completely positive map) such that the map is also completely positive, where and denote the partial transpose map acting on the input system and the output system , respectively. If is also trace preserving, such that it is a quantum channel, and is also completely positive, then we say that is a completely-PPT-preserving channel.
The most important property of the -entanglement measure is that it does not increase under the action of a completely-PPT-preserving channel. Note that an LOCC channel Bennett1996c ; CLMOW14 , as considered in entanglement theory, is a special kind of completely-PPT-preserving channel, as observed in R99 ; R01 .
Theorem 1 (Monotonicity)
Let be a quantum state acting on a separable Hilbert space, and let be a set of completely positive, trace non-increasing maps that are each completely PPT-preserving, such that the sum map is quantum channel. Then the following entanglement monotonicity inequality holds
where . In particular, for a completely-PPT-preserving quantum channel , the following inequality holds
Proof. Let be such that
Since is completely-PPT-preserving, we have that
which reduces to the following for all such that :
Furthermore, since and is completely positive, we conclude the following for all such that :
Thus, the operator is feasible for . Then we find that
The first equality follows from the assumption that the sum map is trace preserving. The first inequality follows from concavity of the logarithm. The second inequality follows from the definition of and the fact that satisfies (6) and (7). Since the inequality holds for an arbitrary satisfying (4), we conclude the inequality in (2).
ii.2 Dual representation and additivity
The optimization problem dual to in Definition 1 is as follows:
which can be found by the Lagrange multiplier method (see, e.g., (Watrous2011b, , Section 1.2.2)). By weak duality (Watrous2011b, , Section 1.2.2), we have for any bipartite state acting on a separable Hilbert space that
For all finite-dimensional states , strong duality holds, so that
This follows as a consequence of Slater’s theorem.
By employing the strong duality equality in (14) for the finite-dimensional case, along with the approach from FAR11 , we conclude that the following equality holds for all bipartite states acting on a separable Hilbert space:
Both the primal and dual SDPs for are important, as the combination of them allows for proving the following additivity of
with respect to tensor-product states.
Proposition 2 (Additivity)
For any two bipartite states and acting on separable Hilbert spaces, the following additivity identity holds
Proof. From Definition 1, we can write as
Let be an arbitrary operator satisfying , and let be an arbitrary operator satisfying . Then it follows that
Since the inequality holds for all and satisfying the constraints above, we conclude that
To see the super-additivity of , i.e., the opposite inequality, let and be arbitrary operators satisfying the conditions in (12) for and , respectively. Now we choose
One can verify from (12) that
which implies that is a feasible solution to (12) for . Thus, we have that
Since the inequality has been shown for arbitrary and satisfying the conditions in (12) for and , respectively, we conclude that
ii.3 Relation to logarithmic negativity
Let be a bipartite state acting on a separable Hilbert space. Then
If satisfies the binegativity condition , then
Proof. Consider from the dual formulation of in (12) that
Using the fact that the transpose map is its own adjoint, we have that
Then by a substitution, we can write this as
Consider a decomposition of into its positive and negative part
Let be the projection onto the positive part, and let be the projection onto the negative part. Consider that
Then we can pick and in (34), to find that
Furthermore, we have for this choice that
ii.4 Normalization, faithfulness, no convexity, no monogamy
In this section, we prove that is normalized on maximally entangled states, and for finite-dimensional states, that it achieves its largest value on maximally entangled states. We also show that is faithful, in the sense that it is non-negative and equal to zero if and only if the state is a PPT state. Finally, we provide simple examples that demonstrate that is neither convex nor monogamous.
Proposition 4 (Normalization)
Let be a maximally entangled state of Schmidt rank . Then
Furthermore, for any bipartite state , the following bound holds
where and denote the dimensions of systems and , respectively.
Proof. Consider that satisfies the binegativity condition because
where is the unitary swap operator, such that , with and the respective projectors onto the symmetric and antisymmetric subspaces. Thus, by Proposition 3, it follows that
To see (48), let us suppose without loss of generality that . Given the bipartite state , Bob can first locally prepare a state and teleport the system to Alice using a maximally entangled state shared with Alice, which implies that there exists a completely-PPT-preserving channel that converts to . Therefore, by the monotonicity of with respect to completely-PPT-preserving channels (Theorem 1), we find that
This concludes the proof.
Proposition 5 (Faithfulness)
For a state acting on a separable Hilbert space, we have that and if and only if .
Now suppose that . Then we can set in (1), so that the conditions and are satisfied. Then , so that . Combining with the fact that for all states, we conclude that if .
Finally, suppose that . Then, by Proposition 3, , so that . Decomposing into positive and negative parts as (such that and ), we have that . But we also have by assumption that . Subtracting these equations gives , which implies that . From this, we conclude that .
Proposition 6 (No convexity)
The -entanglement measure is not generally convex.
Proof. Due to Proposition 3
and the fact that the binegativity is always positive for any two-qubit stateIshizaka2004a , the non-convexity of boils down to finding a two-qubit example for which the logarithmic negativity is not convex. In particular, let us choose the two-qubit states , , and their average . By direct calculation, we obtain
Therefore, we have
which concludes the proof.
where the entanglement in is understood to be with respect to the bipartite cut between systems and . It is known that some entanglement measures satisfy the monogamy inequality above CKW00 ; KWin04 . However, the -entanglement measure is not monogamous, as we show by example in what follows.
Proposition 7 (No monogamy)
The -entanglement measure is not generally monogamous.
Proof. Consider a state of three qubits, where
Due the fact that can be written as
where , this state is locally equivalent to with respect to the bipartite cut . One then finds that . Furthermore, we have that , and , which implies that
This concludes the proof.
Iii Exact entanglement cost of quantum states
In this section, we prove that the -entanglement of a bipartite state is equal to its exact entanglement cost, when completely-PPT-preserving channels are allowed for free. After doing so, we evaluate the exact entanglement cost of several key examples of interest: isotropic states, Werner states, maximally correlated states, some states supported on the antisymmetric subspace, and bosonic Gaussian states. In particular, we conclude that the resource theory of entanglement (the exact PPT case) is irreversible by evaluating the max-Rains relative entropy of WD16pra and and showing that there is a gap between them.
iii.1 -entanglement measure is equal to the exact PPT-entanglement cost
Let represent a set of free operations, which can be either LOCC or PPT. The one-shot exact entanglement cost of a bipartite state , under the operations, is defined as
where represents the standard maximally entangled state of Schmidt rank . The exact entanglement cost of a bipartite state , under the operations, is defined as
The exact entanglement cost under LOCC operations was previously considered in N99 ; TH00 ; H06book ; YC18 , while the exact entanglement cost under PPT operations was considered in Audenaert2003 ; Matthews2008 .
In Audenaert2003 , the following bounds were given for :
the lower bound being the logarithmic negativity recalled in (29), and the upper bound defined as
Due to the presence of the dimension factor , the upper bound in (63) clearly only applies in the case that is finite-dimensional.
In what follows, we first recast as an optimization problem, by building on previous developments in Audenaert2003 ; Matthews2008 . After that, we bound in terms of , by observing that is a relaxation of the optimization problem for . We then finally prove that is equal to .
Let be a bipartite state acting on a separable Hilbert space. Then the one-shot exact PPT-entanglement cost is given by the following optimization:
Proof. The achievability part features a construction of a completely-PPT-preserving channel such that , and then the converse part demonstrates that the constructed channel is essentially the only form that is needed to consider for the one-shot exact PPT-entanglement cost task. The achievability part directly employs some insights of Audenaert2003 , while the converse part directly employs insights of Matthews2008 . In what follows, we give a proof for the sake of completeness.
Let be a positive integer and a density operator such that the following inequalities hold
Then we take the completely-PPT-preserving channel to be as follows:
The action of can be understood as a measure-prepare channel (and is thus a channel): first perform the measurement , and if the outcome occurs, prepare the state , and otherwise, prepare the state . To see that the channel is a completely-PPT-preserving channel, we now verify that the map is completely positive. Let be a positive semi-definite operator with isomorphic to and isomorphic to . Then consider that