Introduction

Quantum coherence, originating from the quantum pure state superposition principle, is one of the most fundamental properties of quantum mechanics. It is increasingly recognized as a vital resource in a range of scenarios, including quantum reference frames1,2,3, transport in biological systems4,5,6 and quantum thermodynamics7,8,9. How to measure coherence is an essential problem in both quantum theory and quantum information and has recently attracted much attention10,11,12,13,14. The quantification of coherence in a single quantum system depends on both the quantum state and a fixed basis for the density matrix of the system10,11. The fixed basis is usually chosen as the eigenbasis of the Hamiltonian or another observable. In either case, the quantified coherence is not an intrinsic property of the single-party quantum state itself. The dynamics of quantum coherence under certain noisy channels has also attracted a lot of research attention15,16 and is connected to the dynamics of quantum correlations17.

When Alice and Bob share a correlated quantum system, a measurement by Alice can ‘steer’ the quantum state of Bob. Quantum steering, especially Einstein-Podolsky-Rosen (EPR) steering, has long been noted as a distinct nonlocal quantum effect18 and has attracted recent research interest both theoretically and experimentally19,20,21,22. The quantum steering ellipsoid (QSE)23,24,25,26,27, defined as the whole set of Bloch vectors to which Bob’s qubit can be steered by a positive-operator valued measurement (POVM) on Alice’s qubit, provides a faithful geometric presentation for two-qubit states. Using the QSE formalism we have studied a class of two-qubit states whose quantum discord can be increased by local operations28. Interestingly, arbitrarily small mutual information is sufficient for the QSE of a pure two-qubit state to be the whole Bloch ball. Since mutual information is an upper bound of quantum correlation measures such as entanglement and discord, the power that one qubit has to steer another cannot be fully characterized by the quantum correlation between the two qubits. A measure that quantifies the power of generating quantum coherence by steering is therefore necessary.

In this paper we consider a bipartite quantum state ρ with non-degenerate reduced state ρB and study the coherence of Bob’s steered state, which is obtained by Alice’s POVM. Here the eigenbasis of ρB is employed as the fixed basis in which to calculate the coherence of the steered state. The significance of this choice of basis is that Bob’s initial state is incoherent. When Alice performs a local measurement, she can steer Bob’s state to one that is coherent in the eigenbasis of ρB, i.e. Alice generates Bob’s coherence. By we denote the maximum coherence that Alice can generate through local measurement and classical communication. In contrast to existing quantifiers of coherence, is an intrinsic property of the bipartite quantum state ρ, because the reference basis of coherence, chosen as the eigenbasis of ρB, is inherent to the bipartite state. Furthermore, we find that gives a different ordering of states compared to quantum entanglement or discord; this indicates that describes remote quantum properties distinct from these measures of quantum correlation. Properties of are also studied. The maximal steered coherence is found to vanish only for classical states and can be created and increased by local quantum channels. Given that coherence plays a central role in a diverse range of quantum information processing tasks, we can also consider how steered coherence might be used as a resource. We close our discussion by presenting one such scenario.

We note that, shortly after this paper first appeared, Mondal et al. presented a study on the steerability of local quantum coherence29. We consider our works to be complementary: though examining a similar topic, our approaches are very different (Mondal et al. consider steering from the existence of a local hidden state model rather than from the perspective of the QSE formalism).

Results

Definition

We consider a bipartite quantum state ρ, where the reduced state ρB is non-degenerate with eigenstates . When Alice obtains the POVM element M as a measurement outcome, Bob’s state is steered to with probability , where denotes the single qubit identity operator. Baumgratz et al.10 gives the the quantum coherence C of in the basis as the summation of the absolute values of off-diagonal elements:

Here we maximize the coherence over all possible POVM operators M and define the maximal steered coherence as

When ρB is degenerate, is not uniquely defined; however, we can take the infimum over all possible eigenbases for Bob and define the maximal steered coherence as

It is worth noting that is an intrinsic property of the bipartite quantum state ρ. When fixing the basis in which to calculate the coherence, we need not choose an observable that is independent of the state; the basis we choose here is inherent to the state ρ.

Properties

We prove that the following important properties hold for maximal steered coherence.

(E1) vanishes if and only if ρ is a classical state (zero discord for Bob), i.e. can be written as

The proof of this is given in Methods.

(E2) reaches a maximum for all pure entangled states with full Schmidt rank, i.e. states that can be written in as with . Here dB is the dimension of Bob’s state. For a single quantum system of dimension dB, the maximally coherent state is 10; Bob is steered to this when Alice obtains the measurement outcome , where is the state after normalisation.

(E3) is invariant under local unitary operations. When the unitary operator acts on a bipartite state ρ, the eigenbasis of ρB is rotated by UB, so that the off-diagonal elements of become

From Eq. (2) it is clear that .

(E4) can be increased by Bob performing a local quantum channel prior to Alice’s steering. Property (E4) holds owing to the fact that a local channel ΛB, under certain conditions30, can transform a classical state with vanishing into a discordant state with strictly positive . Note, however, that a channel ΛA performed by Alice prior to steering cannot increase . (This follows because ΛA can be performed by applying a unitary operation to A and an ancilla A′ and then discarding A′; the unitary operation does not affect the set of Bob’s steered states, while discarding A′ may limit Alice’s ability to steer Bob’s state. Thus ΛA performed by Alice does not alter Bob’s reduced state ρB but shrinks the set of his steered states; such a channel cannot increase .)

Let us also note an important consequence of property (E2): is distinct from the entanglement E31 and discord-type quantum correlations 32. In fact, gives a different ordering of states from E or . We demonstrate this by considering states and , where and are both pure entangled states with full Schmidt rank of dimension d and . According to (E2), reaches the maximum; whereas for ρ2, Bob’s steered state is always mixed and hence not maximally coherent state in any given basis. We therefore have . Meanwhile, and E(ρ1) can be made arbitrarily small by taking δ to be small enough, whilst and E(ρ2) approach 1 for small δ. Hence δ exists such that and .

General expression for two-qubit states

We now derive the general form of for two-qubit states. The state of a single qubit can be written as , where , σi with i = 1, 2, 3 are Pauli matrices, and is Bob’s Bloch vector. The norm of the vector b is denoted by b. The quantum coherence of ρB in a given basis , where , is

Let B and N be the points associated with the vectors b and n respectively and let O be the origin. Since is the area of ΔOBN and the line segment is unit length, C(ρB, n) is simply the perpendicular distance between the point B and the line . Similarly, we can write a two-qubit state in the Pauli basis as , where the coefficients form a block matrix . Here a and b are Alice and Bob’s Bloch vectors respectively and T is a 3 × 3 matrix. Note that when ρB is non-degenerate we have b ≠ 0. We ignore the trivial case that a = 1, when ρA is pure and hence ρ is a product state.

When the POVM operator is obtained on Alice’s qubit, Bob’s state becomes

Here and can be any point on or inside the Bloch sphere. The set of bM forms the QSE . When ρB is non-degenerate, we have b≠0. According to Eq. (6), the coherence of bM in the basis is , with nB = b/b; this represents the perpendicular distance from the point BM to the line (Fig. 1a). Hence the maximal steered coherence , as defined in Eq. (2), is the maximal perpendicular distance between a point on the surface of and . Explicitly, we have and

Figure 1
figure 1

An illustration of the geometric interpretation of maximal steered coherence for two-qubit states ρ using the QSE .

For simplicity we take a = 0. The point B representing Bob’s Bloch vector is indicated by a green blob and the line is also shown in green; states lying along this line are incoherent in the basis ρB. is given by the maximal perpendicular distance between a point on and ; this is shown by the red arrow. (a) Theorem 2 shows that for any canonical state, is bounded by the longest semiaxis of the QSE. (b) A state of the form (15), which achieves maximal for a given b. The QSE is a chord perpendicular to . (c) When ρ is an X state, lies along an axis of the QSE and is the length of the longest of the other two semiaxes. (d) When ρ is a Werner state, is a ball centred on the origin. In this case, even though ρB is degenerate, is well-defined as the radius of the ball.

Full size image

The maximization needs to be performed only over all projective measurements with m = 1 because steered states on the surface of correspond to measurements m on the surface of the Bloch sphere.

When ρB is degenerate, b = 0 and nB is arbitrary; the infimum can then be taken over all nB to give the maximal steered coherence of a two-qubit state as

Properties for two-qubit states

We now study two-qubit states in more detail; this allows us to identify some important features of the maximal steered coherence, as well as giving a clear geometric interpretation of using the steering ellipsoid formalism.

As demonstrated by property (E4), a trace-preserving channel ΛB performed by Bob may increase ; we now study an explicit example. Say that Alice and Bob share the classical two-qubit state

with and . When Bob applies the single-qubit amplitude damping channel, the state transforms as , where with and . We then find that the maximal steered coherence of the transformed state is

vanishes when γ = 0, becomes positive for 0 < γ < 1 and then vanishes again at γ = 1.

Maximal steered coherence can be increased by Bob’s local amplitude damping channel even when Alice and Bob share a non-classical state. Consider the two-qubit state

where 0 < p < 1 and . The QSE for such a state is an ellipsoid centered at with semiaxes of length , aligned with the coordinate axes ( is in fact a prolate spheroid as c1 > c2 = c3). Bob’s Bloch vector is , which lies on the x axis. The maximal steered coherence is therefore

For , the state ρp has zero entanglement but nonzero . Note also that is related to both the fraction of and the entanglement associated with .

Figure 2 shows the evolution of under the channel , i.e. , where . By altering p and θ we alter the ratio of the axes c3/c1. The results indicate that the potential for increasing under Bob’s local amplitude damping is related to the ratio c3/c1: the smaller the ratio, the stronger the local increase of . In other words the effect is strongest when the QSE is highly prolate (‘baguette-shaped’).

Figure 2
figure 2

The evolution of maximal steered coherence under Bob’s local amplitude damping channel: , with ρp given by Eq. (12).

The parameters for the four curves are p = 0.9, θ = 0.2π for the cyan dashed line; p = 0.9, θ = 0.1π for the red dotted line; p = 0.7, θ = 0.1π for the green solid line; and p = 0.5, θ = 0.1π for the blue dash-dotted line. The corresponding semiaxes ratios, which give a measure of the prolateness of , are c3/c1 = 0.980, 0.859, 0.629 and 0.496 respectively. The effect of locally increasing is stronger for more prolate .

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In fact, it is possible to formulate a necessary and sufficient condition for the increase of maximal steered coherence for two-qubit states.

Theorem 1. Bob’s local qubit channel ΛB can increase maximal steered coherence for some input two-qubit state if and only if ΛB is neither unital nor semi-classical.

The proof is given in Methods. We therefore see that the behavior of maximal steered coherence under local operations is similar to that of quantum discord . The set of local channels that can increase for some two-qubit state is the same as the set of local channels that can increase . Moreover, can be increased when the QSE is very prolate; we showed in Reference28 that the quantum discord of Bell-diagonal states with such baguette-shaped can be increased by the local amplitude damping channel. We therefore conjecture the local increase in quantum correlations originates from the increase in steered coherence.

We now investigate the set of so-called canonical states, which have particular significance in the steering ellipsoid formalism24,25,33. Here, a canonical state ρcan corresponds to one for which Alice’s marginal is maximally mixed (a = 0). This implies that the QSE is centered at B (Fig. 1a). Let c1, c2 and c3 be the lengths of the semiaxes of ordered such that c1 ≥ c2 ≥ c3.

Theorem 2. For any canonical state ρ can the maximal steered coherence is bounded by the longest semiaxis. This in turn is bounded by the length of Bob’s Bloch vector as

The bound is saturated if and only if is a chord perpendicular to b meeting the surface of the Bloch sphere at   and . This represents a canonical state of the form

where .

The proof is given in Methods and an example QSE for an optimal state of the form (15) is shown in Fig. 1b). Note that this bound is remarkably simple and geometrically intuitive: it depends only on the longest semiaxis of and not on the orientation or position of the QSE. Theorem 2 is in the same vein as bounds presented in Reference33 that relate several other measures of quantum correlation to geometric features of QSEs.

We also note that optimal states of the form (15) have the highest quantum discord among discordant states with a given b that are obtained from classical states by a local trace-preserving channel. As shown in Reference34, when we take a two-qubit B-side classical (zero discord) state and apply a channel ΛB to Bob’s qubit, in order to create maximal B-side quantum discord in the output state, the optimal input state is of the form and the channel ΛB should have Kraus operators , , where and are determined by b.

Examples

Let us now examine some interesting classes of two-qubit states for which maximal steered coherence is easy both to find analytically and to interpret geometrically using QSEs.

X states

When nB lies along an axis of the QSE it is straightforward to see that is simply the length of the longest of the other two semiaxes (Fig. 1c). All ρ which are X states, i.e. have non-zero entries only in the characteristic X shape in the computational basis35, will have such QSEs26.

Werner states

As a special case of the above, when is a ball of radius r centered on O′ and nB is collinear with , we have . Furthermore, when is an origin-centered ball, we have regardless of the value of b. This allows us to evaluate for Werner states36, which do not in fact satisfy the non-degenerate condition b≠0. For a Werner state with , is an origin-centered ball of radius p and hence (Fig. 1d).

Discordant states locally created from a classical state

We know from property (E1) that vanishes for classical states; for a classical two-qubit state, all steered states must have the same orientation and the QSE is therefore a radial line segment. For a state obtained locally from a classical state, ρdlc, the QSE is a nonradial line segment25. b can be any point on this segment except for the two ends of , which we call b1 and b2, where b1 ≥ b2. By definition varies for different b; in general, we find that

where θ is the angle between b1 and b2 and θ1 is determined by . From Eq. (16), we see that is strictly larger than zero. In fact, can reach unity when b1 = 1 and .

Maximally obese states

The general form of a maximally obese state is given by33

where . This is a canonical state (a = 0) with centered at (0, 0, b) and semiaxes of length , aligned with the coordinate axes. We therefore have . It should be noted that maximally obese states maximize several measures of quantum correlation (CHSH nonlocality, singlet fraction, concurrence and negativity) over the set of all canonical states with a given marginal for Bob33. Interestingly, however, they do not achieve the maximum possible .

Discussion

We have studied the maximal steered coherence for a bipartite state ρ. When Alice obtains a POVM outcome M, Bob’s state is steered to ; is defined as the coherence of the steered state in Bob’s original basis, maximized over all possible M. The general form of is derived for two-qubit states. By calculating the maximal steered coherence for some important classes of two-qubit state, we find that gives a different ordering of states from quantum entanglement or discord-like correlations. This means that is a distinct and new measure for characterizing the remote quantum properties of bipartite states.

The maximal steered coherence vanishes only when ρ is a classical state and can be increased by local trace-preserving channels. For a two-qubit state ρ we derive a necessary and sufficient condition for a local qubit channel to be capable of increasing . This is in fact identical to the condition for increasing quantum discord, suggesting that local increase of quantum discord might be used in a protocol for increasing steered coherence.

Finally, we consider the relevance of from a more physical perspective by presenting a concrete example in which steered coherence can be exploited. Say that Alice and Bob share a two-qubit state of the form (15) with , and b = (0, 0, b) (which, as illustrated in Fig. 1b, will lie at the midpoint of the chord joining and . Suppose also that Alice’s and Bob’s systems are described by the local Hamiltonian . Let us restrict Alice’s and Bob’s local operations to those which are covariant with respect to time-translation symmetry37. For Alice, these operations are the ones for which ; and similarly for Bob. Physically, this restriction corresponds to local energy-conserving unitaries with the assistance of incoherent environmental ancillas38: Alice’s operation is covariant if and only if it can be written as where U is a unitary, HE is the Hamiltonian of the ancilla, and ; and similarly for Bob. The set of covariant operations is a strict subset of incoherent operations10 and a strict superset of thermal operations39.

Bob’s reduced state is incoherent in his energy eigenbasis and his local covariant operations alone cannot generate any coherence. However, by performing a σ3 measurement, which is a covariant operation and classically communicating the result to Bob, Alice steers him to either or , states that are manifestly coherent in the energy eigenbasis. gives a measure of the maximal coherence that Alice can induce on Bob’s system by steering. In this way, Alice remotely ‘activates’ a coherent state for Bob that he was unable to produce himself. Bob may now use this coherence as a resource for quantum information processing tasks, e.g. work extraction by a thermal machine, which is known to be enhanced in the presence of a coherent reference system40. Given the ever-increasing number of applications for coherence found throughout quantum information science, one can envisage a range of such scenarios in which steered coherence could be used as a resource.

Methods

Proof of property (E1)

The ‘if’ part is obvious: Bob’s reduced state is and the steered state is . These are both diagonal in the basis and hence .

For the ‘only if’ part, first consider a separable state . When , the steered states for different POVM operators M should commute with each other, which is equivalent to all commuting with each other. So vanishes only if it is in the form (4). For an entangled state ρe, we express ρe in the optimal pure state decomposition form as , so that the entanglement of formation is . Since ρe is entangled, at least one of the is entangled. Hence, for ρe, it is not possible for all of Bob’s steered states to share the same eigenbasis; this means that for any entangled ρe.

Proof of Theorem 1

A channel ΛB that is neither unital nor semi-classical can increase , because such channels can transform a classical state with vanishing into a discordant state with nonzero 30,41. We now focus on the ‘only if’ part and prove that a local unital channel or a local semi-classical channel cannot increase for any two-qubit input state.

A semi-classical channel Λsc 41, which maps any input state ρ to a state with zero coherence in a given basis , yields for any input state. As proved by King and Ruskai42, any unital channel is equivalent to , where 0 ≤ ei ≤ 1 and . The effect of this channel on a qubit state is to shrink the Bloch vector as , where and p2,3 are related to ei in a similar way. Let bM be a steered state for the input state ρ. Then the coherence of bM is . Under the action of Λu, the steered state and Bob’s reduced state become b′M and b′ respectively and the coherence of b′M in the eigenbasis of b′ is

If the inequality

holds then the maximal steered coherence for the output state , where Mopt is the optimal POVM operator to maximize (2) for the output state and is the corresponding input state for . Hence it is sufficient to prove that (19) holds for some bM and b. Note that . By using the fact that for 0 < B < D, 0 < A < C, 0 < A′ < C′ and , we arrive at , which is equivalent to (19).

Proof of Theorem 2

The steered state bM which achieves the maximum in Eq. (8) corresponds to a point BM on the surface of . We have , where is the perpendicular distance between BM and . To ensure that lies inside the Bloch sphere we require that .

To saturate the bound we take , but we must also demonstrate that c2 = c3 = 0, i.e. that cannot be an ellipsoid or an ellipse. We know that must meet the Bloch sphere at two points, corresponding to the pure states and . Firstly suppose that is a three-dimensional ellipsoid. Elementary geometry tells us that the surface of an ellipsoid at the end of any axis must be perpendicular to that axis. The points at the ends of the c1 axis on lie on the surface of the Bloch sphere. Since the surface of must lie perpendicular to the c1 axis at these points, must puncture the surface of the Bloch sphere. Such cannot represent a physical two-qubit state and so cannot be an ellipsoid. Now consider the case that is an ellipse. The nested tetrahedron condition tells us that any degenerate describing a physical state must fit inside a triangle inside the Bloch sphere25,26. Geometrically, no ellipse that touches the Bloch sphere at two points can satisfy this and so cannot be an ellipse. must therefore be a line, i.e. the chord going between and ; this corresponds to the state (15).

Additional Information

How to cite this article: Hu, X. et al. Quantum coherence of steered states. Sci. Rep. 6, 19365; doi: 10.1038/srep19365 (2016).