Collapse of the quantum correlation hierarchy links entropic uncertainty to entanglement creation

# Collapse of the quantum correlation hierarchy links entropic uncertainty to entanglement creation

Patrick J. Coles Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA Centre for Quantum Technologies, National University of Singapore, Singapore
###### Abstract

Quantum correlations have fundamental and technological interest, and hence many measures have been introduced to quantify them. Some hierarchical orderings of these measures have been established, e.g. discord is bigger than entanglement, and we present a class of bipartite states, called premeasurement states, for which several of these hierarchies collapse to a single value. Because premeasurement states are the kind of states produced when a system interacts with a measurement device, the hierarchy collapse implies that the uncertainty of an observable is quantitatively connected to the quantum correlations (entanglement, discord, etc.) produced when that observable is measured. This fascinating connection between uncertainty and quantum correlations leads to a reinterpretation of entropic formulations of the uncertainty principle, so-called entropic uncertainty relations, including ones that allow for quantum memory. These relations can be thought of as lower-bounds on the entanglement created when incompatible observables are measured. Hence, we find that entanglement creation exhibits complementarity, a concept that should encourage exploration into “entanglement complementarity relations”.

###### pacs:
03.67.Mn, 03.65.Ta, 03.67.Hk

## I Introduction

As researchers attempt to develop the ultimate theory of information, encompassing both classical and quantum information, it is becoming increasingly apparent that quantum correlations - correlations that go beyond classical correlations - are of great fundamental and technological interest. Questions like, what gives the quantum advantage in computing tasks DatShaCav08 (), have motivated the definition and study of many quantitative measures of quantum correlations, ranging from entanglement HHHH09 () to discord OllZur01 () and other related measures ModiEtAl2011review (). Some of these measures are operationally motivated, e.g. the number of Einstein-Podolsky-Rosen (EPR) pairs that can be distilled from the state, others are geometrically motivated like the distance to the nearest separable state or the nearest classical state, while others are motivated due to their ease of calculation. The zoo of quantum correlation measures is vast, and yet the story is simple for bipartite pure states, where the entropy of the reduced state pretty much captures it all. While it would be nice if the correlations of mixed states shared the simplicity of those of pure states, in general, we must settle for a hierarchical ordering of the various correlation measures, e.g., discord is bigger than entanglement PianiAdessoPRA.85.040301 (); HorEtAl05 (), which in turn is bigger than coherent information DevWin05 ().

In the present article, we consider a class of bipartite states for which this zoo dramatically simplifies to a single number; various quantum correlation measures which are in general related by a hierarchy of inequalities become equal for these states, so we say that these states “collapse the quantum correlation hierarchy”. Hence these states are like pure states in that their correlations are “simple”, even though the set includes not only pure states but also some mixed states. Interestingly, the set of states that collapse the quantum correlation hierarchy corresponds precisely to the set of states that can be produced when a system interacts with a measurement device. These states have been called premeasurement states, since the unitary interaction (called premeasurement) that potentially correlates the system to the measurement device is the first step in the measurement process ZurekReview (). The fact that premeasurement states collapse the quantum correlation hierarchy has significant consequences, and much of this article is devoted to exploring these consequences.

The most interesting consequence is a connection to uncertainty and the uncertainty principle. While the study of quantum correlations has seen a revolution of sorts recently, so has the study of the uncertainty principle. In quantitative expressions of the uncertainty principle, so-called uncertainty relations, researchers have replaced the standard deviation, the uncertainty measure employed in the original formulations Heisenberg (); Robertson (), with entropy measures, leading to a variety of different entropic uncertainty relations (EURs) EURreview1 (), which are more readily applied to information-processing tasks. Allowing the observer to possess “quantum memory” (a quantum system that may be entangled to the system of interest) has led to EURs RenesBoileau (); BertaEtAl (); TomRen2010 (); ColesEtAl (); ColesColbeckYuZwolak2012PRL () with direct application in entanglement witnessing LXXLG (); PHCFR () and cryptography TLGR ().

Our results allow us to establish a precise and general connection between the uncertainty of an observable and the quantum correlations, such as entanglement, created when that observable is measured (more precisely, premeasured). As a consequence, a wide variety of EURs, including those allowing for quantum memory, are subject to reinterpretation. The conventional interpretation is that EURs are lower bound on our inability to predict the outcomes of incompatible measurements, but our results imply that EURs can be thought of as lower bounds on the entanglement created in incompatible measurements.

It is helpful to illustrate this connection with a simple example. Consider a qubit in state , then the unitary associated with a -measurement is a controlled-not (CNOT) acting on a register qubit that is initially in state . In this case, the overall state evolves trivially: , producing no entanglement. But if instead we did an -measurement, with a CNOT controlled by the basis, then the state evolves as , which is maximally entangled. Note that the uncertainty of the () observable was zero (maximal), which is connected to the final entanglement being zero (maximal). This example shows the connection of uncertainty to entanglement creation, and it also shows the complementarity of entanglement creation: the measurement must create entanglement because the measurement does not.

We remark that the entanglement created in measurements has been an area of interest previously ZurekReview (); VedralPRL2003 (), and there is renewed interest in this as it provides a general framework for quantifying discord PianiEtAl11 (); StrKamBru11 (); PianiAdessoPRA.85.040301 (). It should, therefore, be of interest that our reinterpretation of EURs implies that the entanglement (and discord) created in measurements exhibits complementarity. This idea, which seems to be a general principle, suggests that there are classes of inequalities that capture the complementarity of quantum mechanics, which have yet to be explored and involve entanglement (or discord) creation. There is generally a trade-off; for a given quantum state, if one avoids creating quantum correlations in one measurement, then a complementary measurement will necessarily create such correlations.

In summary, we emphasize three main concepts in this article: (1) the quantum correlation hierarchy dramatically simplifies for premeasurement states, (2) an observable’s uncertainty quantifies the entanglement created upon measuring that observable, and (3) entanglement creation exhibits complementarity. Mathematically speaking, concept (1) implies concept (2) which in turn implies concept (3), as we will discuss.

The rest of the manuscript is organized as follows. In Section II we define various classes of bipartite quantum states, including premeasurement states. In Section III we consider several different quantum correlation hierarchies, and we show that premeasurement states collapse these hierarchies. In particular, we consider hierarchies of measures based on a generic relative entropy, measures related to the von Neumann entropy, and measures related to smooth entropies. In Section IV, we use these results to connect an observable’s uncertainty to the quantum correlations created when that observable is measured. Then we argue that this gives a reinterpretation for EURs in Section V, focusing particularly on the complementarity of entanglement creation. Section VI gives a few more implications of our results and discusses the future outlook for “entanglement complementarity relations”. Section VII gives some concluding remarks.

## Ii Classes of bipartite states

### ii.1 Classical, separable, and entangled states

Since we will be considering various correlation measures, it is helpful to define particular classes of bipartite quantum states. First, consider the set of all separable states, hereafter denoted Sep, which have the general form of a convex combination of tensor products:

 ρAB=∑jpjρA,j⊗ρB,j, (1)

where is some probability distribution and and are density operators on systems and . Entangled states are defined as those states that are not separable; we denote this set as Ent, the complement of Sep.

A special kind of separable state is a classical state, often called a classical-classical or CC state, with the general form:

 ρAB=∑j,kpj,k|j⟩⟨j|⊗|k⟩⟨k|, (2)

which is like the embedding of a classical joint probability distribution in a Hilbert space, where and are orthonormal bases on and , respectively. More generally, a state can be classical with respect to one of the subsystems, e.g., of the form:

 ρAB=∑jpj|j⟩⟨j|⊗ρB,j, (3)

in which case it is called classical-quantum or CQ, and naturally is called quantum-classical or QC if it is classical with respect to system . The following relations between these sets should be clear from the above definitions:

 CQ⊂Sep,QC⊂Sep,CQ∩QC=CC, (4)

and a Venn diagram in Fig. 1 depicts these relations.

### ii.2 Pure states

Pure states can be either separable or entangled, though if a pure state is separable, it is necessarily a classical state (more specifically, a product state), in other words,

 (Pure∩Sep)⊂CC, (5)

as depicted in Fig. 1. The correlations of pure states are very well understood, e.g., see HHHH09 (); NieChu00 (), and one of our contributions is to characterize a set of states whose correlations are somewhat analogous to those of pure states, a set that encompasses, but goes beyond, pure states. We discuss this set below.

### ii.3 Premeasurement states

Consider the following set of bipartite states:

 MQ:={ρAB:ρAC∈CQ for pure ρABC}. (6)

Here, is any purification of , so we are considering the set of states such that there exists a purification whose marginal is of the general form of (3), i.e., classical with respect to system . (If for some purification of , then the same will be true for all other purifications.) If and change roles in (6), i.e., if , then we denote this set as QM, and if both and are CQ, then we say that . In other words,

 MQ∩QM=MM. (7)

It turns out, as we will see below, that MM corresponds precisely to the “maximally correlated states”, introduced by Rains RainsPhysRevA.60.179 ().

It is clear that if is pure, then any purifying system will be in a tensor product with (uncorrelated with) , hence both marginals and will be classical. So all pure states are in MM,

 Pure⊂MM. (8)

Figure 1 schematically depicts Eqs. (7) and (8). Also captured by this figure is an extension of (5) to MQ and QM states:

 (MQ∩Sep)⊂CC,(QM∩% Sep)⊂CC. (9)

While (9) is not at all obvious, it is a consequence of our results proven in Sec. III.

Our curious notation MQ is motivated by the fact that one of the subsystems, namely system in (6), is behaving like a measurement device in a way that we elaborate on below. Thus, one can read MQ as “measurement device - quantum”, analogous to how one reads CQ as “classical - quantum”.

To make the connection to measurement, it is helpful to switch to a more intuitive notation for the various subsystems. We consider the interaction of system with a device that measures observable of , where the are orthogonal projectors that sum to the identity on . [We emphasize that the are not necessarily rank-one; is a general projection valued measure (PVM).] This can be modeled by considering a set of orthonormal states on , and if is hit by projector , then goes to the state , as follows:

 |0⟩MX|ψ⟩S→∑j|j⟩MX(Xj|ψ⟩)S=VX|ψ⟩S, (10)

which is essentially a controlled-shift operation, and the notation is simplified by defining the isometry:

 VX=∑j|j⟩MX⊗(Xj)S. (11)

In (10) we assumed that both and were initially described by pure states. More generally either state could be mixed, although we could always lump the measurement device’s environment into system and hence purify the state of and call it the state. We make this simplification throughout, although see VedralPRL2003 () for a treatment allowing the measurement device to be in a mixed state. On the other hand, we find it convenient and natural to the think of the system’s initial state as being a (possibly mixed) density operator ; then the final state after the interaction with is:

 ~ρMXS=VXρSV†X. (12)

The circuit diagram for this process is depicted in Fig. 2, using the controlled-not (CNOT) symbol even though the process is slightly more general. Also shown is the quantum system that purifies , called . Because it is the first step in performing a measurement, this process has been called “premeasurement”, and the resulting states that are produced have been called “premeasurement states” ZurekReview ().

Now suppose we consider the set of all premeasurement states, i.e., the set of all bipartite states that can be thought of as resulting from a process like that depicted in Fig. 2. It turns out that this set is precisely equivalent to MQ, as shown in Appendix A. To write MQ as the set of all premeasurement states, we revert to the notation and for the two subsystems, since we are being general and abstract again. Denote the set of all orthonormal bases on as , denote the set of all PVMs on system as , and denote the set of all premeasurement isometries as

 V={VX:(∃W∈WA)(∃X∈XB)(VX=∑j|Wj⟩⊗Xj)}. (13)

Denoting the set of (normalized) density operators on as , then we have (see Appendix A for the proof)

 MQ={ρAB:(∃σB∈DB)(∃VX∈V)(ρAB=VXσBV†X)}. (14)

In other words, the general form for states in MQ is:

 ρAB=∑j,k|Wj⟩⟨Wk|⊗XjσBXk (15)

for some , , and . It is clear from (15) that if all the projectors are rank-one and hence can be thought of as an orthonormal basis, then the state is a “maximally correlated state”, in the sense that the basis on is perfectly correlated with the basis on . So maximally correlated states are a special kind of MQ state, corresponding to MM. But more generally, we can think of MQ states as being one-way maximally correlated in the sense that an orthonormal basis on is perfectly correlated to some projective observable (not necessarily a basis) on ; again and are the two observables playing this role in (15).

We remark that MQ is a strict subset of a set of states considered in Ref. CorDeOFan11 (), defined as follows

 mQ:={ρAB:ρAC∈Sep for pure ρABC}, (16)

i.e., the set of states where is separable for any purification . Since , it is clear that . We believe it is important to make this connection with Ref. CorDeOFan11 (), because they showed an analog of (9), namely that , a consequence of the fact that states in mQ partially collapse the quantum correlation hierarchy. However, we note in Sec. III.4 that mQ states do not necessarily collapse the “full” quantum correlation hierarchy, and that the restriction of mQ to MQ is precisely what is needed in order to obtain the “full” collapse.

Our notation mQ is motivated by the following observation. Unlike MQ there are states in mQ of the form:

 ρAB=∑j,k|ϕj⟩⟨ϕk|⊗XjσBXk=~VXσB~V†X (17)

where the are non-orthogonal pure states, , , and is an isometry. States of the form of (17) can be viewed as resulting from a sort of premeasurement, but where the conditional states on the measurement device , associated with the different projectors on the system being measured, are not necessarily orthogonal. Hence these states are obtained from doing a “weak” or “soft” premeasurement (i.e., not fully extracting the information), and the lower-case m in mQ emphasizes this.

## Iii Collapse of quantum correlation hierarchy

### iii.1 Four types of quantum correlation measures

To what degree is the correlation between two systems different from that of a classical joint probability distribution - this is the basic question one aims to answer with quantum correlation measures. This difference can be quantified in a wide variety of ways, but let us consider four common paradigms. (This introduction is for completeness only, please see HHHH09 (); ModiEtAl2011review () for review articles.)

One can quantify how far the quantum state is from the set of classical states, CC, either in terms of a distance or in terms of the information content of the states. These are sometimes called two-way quantumness or two-way discord measures, since they measure non-classicality with respect to both subsystems.

A second paradigm is to quantify how far the state is from either CQ or QC, these are one-way quantumness (or discord) measures, since they measure the non-classicality with respect to just one subsystem.

A third paradigm is to quantify the distance to Sep, these are called entanglement measures. In practice, the label “entanglement measure” is restricted to those measures that are non-increasing under local operations and classical communication (LOCC) HHHH09 (), though there is some connection between this criterion and quantifying the distance to Sep VedrPlen1998 ().

Finally there are measures of the form of the negative of a conditional entropy, which quantify the distance to a state of the form where is the maximally mixed state (see below), and in the case of von Neumann entropy, the measure is called coherent information.

### iii.2 Basic structure of results

Within each of these four paradigms, there are different quantitative measures, and below we discuss measures based on relative entropies, measures related to the von Neumann entropy, and measures related to smooth entropies. But in each case there is a basic structure that negative conditional entropy (coherent information) lower bounds entanglement which lower bounds one-way discord which lower bounds two-way discord. We call this the quantum correlation hierarchy, e.g., Ref. PianiAdessoPRA.85.040301 () discussed this idea. Many of the inequalities in these hierarchies are well-known, although some require proof.

In what follows, we present our main technical results, that premeasurement states collapse the quantum correlation hierarchy. For these states, which we also call MQ states, defined by (6) or (14), the inequalities relating coherent information, entanglement, one-way discord, and two-way discord turn into equalities.

Geometrically speaking, the collapse is some reflection of the fact that the closest separable state to a premeasurement state is a CC state. One can verify this claim (Appendix B) using the Bures distance BenZyc06 (), a true metric, though in what follows we observe this phenomenon using various relative entropies as (pseudo) measures of distance.

### iii.3 Collapse of measures based on relative entropy

Here we use the relative entropy to express various correlation measures as a distance to a certain class of states ModiEtAl2010 (). In particular, we consider a generalized relative entropy , a function that maps two positive-semidefinite operators and to the real numbers, that satisfies the following two properties (also considered in ColesColbeckYuZwolak2012PRL ()):

1. Non-increasing under quantum channels : .

2. Being unaffected by null subspaces: , where denotes direct sum.

These properties are satisfied by several important examples ColesColbeckYuZwolak2012PRL (), and so there is power in formulating a general result that relies only on the properties. Examples include the von Neumann relative entropy VedralReview02 (); NieChu00 ():

 D(P||Q):=Tr(PlogP)−Tr(PlogQ), (18)

the Renyi relative entropies Renyi (); Petz84 () within the range :

 Dα(P||Q):=1α−1logTr(PαQ1−α), (19)

and the relative entropies associated with the min- and max-entropies RennerThesis05 (); KonRenSch09 (), respectively,

 Dmax(P||Q) :=logmin{λ:P⩽λQ}, (20) Dfid(P||Q) :=−2logTr[(√PQ√P)1/2]. (21)

We label (20) as (even though it is associated with the min-entropy) because in general Datta09 (), and we label (21) as because it is closely related to the fidelity.

Consider an entanglement measure VedrPlen1998 () based on :

 EA|BK(ρAB):=minσAB∈SepDK(ρAB||σAB). (22)

Property 1 implies, for any LOCC ,

 EA|BK(ρAB)⩾EA|BK(Λ(ρAB)), (23)

which is the well-known monotonicity property HHHH09 (). Let us also define one-way and two-way measures of quantumness (a.k.a. discord) ModiEtAl2011review ():

 Δ−−→A|BK(ρAB):=minσAB∈CQDK(ρAB||σAB), (24) Δ←−→A|BK(ρAB):=minσAB∈CCDK(ρAB||σAB). (25)

Finally, let us define a conditional entropy ColesColbeckYuZwolak2012PRL (),

 HK(A|B):=maxσB[−DK(ρAB||\openone⊗σB)], (26)

where the maximization is over all (normalized) density operators on .

To prove our result, we note two additional properties, which were discussed in ColesColbeckYuZwolak2012PRL (). If satisfies 1 and 2, and if , then

 DK(P||Q)⩾DK(P||~Q), (27)

and if is a projector onto a space that includes the support of , then

 DK(P||Q)⩾DK(P||ΠPQΠP). (28)

We now show that the correlation measures defined above form a hierarchy. This hierarchy, though interesting in itself, will be useful below in proving that the various correlation measures become equal in the special case of premeasurement states.

###### Lemma 1.

Let satisfy 1 and 2, then for any ,

 −HK(A|B)⩽EA|BK⩽Δ−−→A|BK,Δ−−→B|AK⩽Δ↔A|BK. (29)
###### Proof.

The left-most inequality is proven by supposing achieves the minimization in , then

 EA|BK(ρAB) =DK(ρAB||σAB) ⩾DK(ρAB||\openone⊗σB)⩾−HK(A|B)

where we invoked (27) and the fact that, if is separable, then with . The other inequalities follow from . ∎

Now we can state one of our main technical results, that the hierarchy in (29) collapses onto a single value for MQ states, which we also call premeasurement states (see Sec. II.3) since system plays the role of a measurement device and is the system being measured.

###### Theorem 2.

Let satisfy 1 and 2, then for any premeasurement state ,

 −HK(MX|S)=EMX|SK=Δ−−−−→MX|SK=Δ−−−−→S|MXK=Δ←−−→MX|SK. (30)
###### Proof.

Let be the state that achieves the optimization in , then

 −HK(MX|S) =DK(~ρMXS||\openone⊗σS) ⩾DK(~ρMXS||VXV†X(\openone⊗σS)VXV†X) =DK(~ρMXS||∑j|j⟩⟨j|⊗XjσSXj) ⩾Δ←−−→MX|SK.

We used (28) in the second line, and the last inequality notes that . (Expand the blocks in their eigenbasis to verify this.) But (29) gives an inequality in the reverse direction, so the entire hierarchy in (29) must collapse onto the same value. ∎

Theorem 2 applies to all the relative entropies listed in (18) through (21). For example, in the case of von Neumann relative entropy, the quantities in (30), from left to right, are the coherent information NieChu00 (), the relative entropy of entanglement VedrPlen1998 (), the one-way information deficit HorEtAl05 (), and the relative entropy of quantumness HorEtAl05 (). In the next subsection, we further extend our results for these von Neumann measures, including other measures into the hierarchy collapse.

### iii.4 Collapse of von Neumann measures

#### iii.4.1 Long list of measures involved in the collapse

Here we elaborate on the hierarchy collapse for von Neumann measures, giving a long list of the measures involved. While operational or conceptual meanings of many of the measures can be found in HHHH09 (); ModiEtAl2011review (), this article is more concerned with the fact that they form a hierarchy and that this hierarchy collapses for MQ states. To illustrate the dramatic effect of the collapse, we attempt to demonstrate it for as many measures as possible here, even though it comes at the expense of having to define many quantities.

In the previous subsection, we considered the coherent information NieChu00 (), relative entropy of entanglement VedrPlen1998 (), one-way information deficit HorEtAl05 (), and relative entropy of quantumness HorEtAl05 (), respectively defined by:

 I−−→A|Bc(ρAB) :=−H(A|B)=D(ρAB||\openone⊗ρB), EA|BR(ρAB) :=minσAB∈SepD(ρAB||σAB), Δ−−→A|B(ρAB) :=minσAB∈CQD(ρAB||σAB), Δ←−→A|B(ρAB) :=minσAB∈CCD(ρAB||σAB). (31)

We note that appears in the expression for the quantum capacity of a quantum channel Lloyd97 (), is related to the entanglement distillable through one-way hashing DevWin05 (), and has been interpreted as the entanglement gained in quantum state merging HorOppWin05 ().

We will also consider discord measures OllZur01 (); WuEtAl2009 () based on a difference of quantum mutual informations , defined as follows

 δ−−→A|B(ρAB):=minY{I(ρAB)−I[(Y⊗\openone)(ρAB)]}, δ←−→A|B(ρAB):=minY,Y′{I(ρAB)−I[(Y⊗Y′)(ρAB)]}. (32)

Here, we suppose that and are positive operator valued measures (POVMs) on and , respectively, and the quantum channels and associated with these POVMs are defined such that

 (Y⊗\openone)(ρAB) :=∑j|j⟩⟨j|⊗TrA(YjρAB), (Y⊗Y′)(ρAB) :=∑j,kTr[(Yj⊗Y′k)ρAB]|j⟩⟨j|⊗|k⟩⟨k|,

with being the standard (orthonormal) basis.

Now we define regularized versions of these measures:

 I−−→A|Bc,∞(ρAB) :=limN→∞(1/N)I−−−−−−−−→A⊗N|B⊗Nc(ρ⊗NAB), EA|BR,∞(ρAB) :=limN→∞(1/N)EA⊗N|B⊗NR(ρ⊗NAB), Δ−−→A|B∞(ρAB) :=limN→∞(1/N)Δ−−−−−−−−→A⊗N|B⊗N(ρ⊗NAB), Δ←−→A|B∞(ρAB) :=limN→∞(1/N)Δ←−−−−−−→A⊗N|B⊗N(ρ⊗NAB), δ−−→A|B∞(ρAB) :=limN→∞(1/N)δ−−−−−−−−→A⊗N|B⊗N(ρ⊗NAB), δ←−→A|B∞(ρAB) :=limN→∞(1/N)δ←−−−−−−→A⊗N|B⊗N(ρ⊗NAB). (33)

From the additivity of the von Neumann relative entropy, we have

 I−−→A|Bc,∞=I−−→A|Bc,

and it was shown in Devetak07 () and discussed in HorEtAl05 () that

 δ−−→A|B∞=Δ−−→A|B∞. (34)

In asymptotia, uniquely characterises the amount of entanglement in a state when all non-entangling transformations are allowed BranPlen2008 (), while has been linked to entanglement irreversibility (when dilution and distillation are respectively done by LOCC and hashing) in a tripartite scenario CorDeOFan11 ().

In what follows, we also consider the distillable entanglement and the distillable secret key HHHH09 (), both of which are asymptotic rates for conversion of many copies of into some resource, where the resource is EPR pairs and bits of secret correlation, respectively, for and .

Now we consider some hierarchies satisfied by the above measures. As mentioned, the basic structure for these hierarchies is that coherent information lower bounds entanglement which lower bounds one-way discord which lower bounds two-way discord, and indeed (35)–(37) below each have this form. Equation (35) involves discord based on relative entropy whereas (36) involves discord based on a difference of mutual informations, and Eq. (37) involve regularised versions of these measures. Thus, individually, each equation in the following Lemma, proved in Appendix C, can be regarded as a quantum correlation hierarchy.

###### Lemma 3.

For any bipartite state ,

 I−−→A|Bc⩽EA|BD⩽KA|BD⩽EA|BR⩽Δ−−→A|B,Δ−−→B|A⩽Δ←−→A|B, (35) I−−→A|Bc⩽EA|BD⩽KA|BD⩽δ−−→A|B,δ−−→B|A⩽δ←−→A|B⩽Δ←−→A|B, (36) I−−→A|Bc⩽EA|BR,∞⩽δ−−→A|B∞,δ−−→B|A∞⩽δ←−→A|B∞⩽Δ←−→A|B∞⩽Δ←−→A|B. (37)

We now see that each of these hierarchies (35)–(37) collapses in the special case where the state is MQ. In fact, the hierarchies themselves are useful in proving the collapse. In Theorem 2, we showed that, if , then

 I−−−−→MX|Sc=Δ←−−→MX|S,

so combining this with Lemma 3 immediately implies the following result.

###### Theorem 4.

For any state in MQ, i.e., any premeasurement state ,

 I−−−−→MX|Sc =EMX|SD=KMX|SD=EMX|SR,∞=EMX|SR =δ−−−−→MX|S∞=δ−−−−→S|MX∞=δ−−−−→MX|S=δ−−−−→S|MX =Δ−−−−→MX|S=Δ−−−−→S|MX=δ←−−→MX|S∞=Δ←−−→MX|S∞ =δ←−−→MX|S=Δ←−−→MX|S. (38)

While the list in Theorem 4 is very long, we note that not all measures participate in the collapse for MQ. For example, need not be equal to the other correlation measures appearing above. One can see this as follows. For the state which is purified by to the state , we have:

 H(S|MX)−H(MX|S)=H(E|MX)=∑jpjH(ρE,j),

where . Hence if the are non-pure, then will not collapse onto the other measures. Also, see CorDeOFan11 () for a discussion of entanglement of formation and entanglement cost.

#### iii.4.2 Is the collapse unique to Mq?

Here we give a simple argument that MQ is the only set of bipartite states for which , and hence the only set that collapses the full hierarchy as in Theorem 4. Let purify , then it is straightforward to show that:

 I−−→A|Bc=δ−−→A|B−δ−−→A|C, (39)

by noting that the optimization in is achieved by a rank-one POVM, and in fact the same rank-one POVM achieves the optimization in both and ColesEtAl (). From Ref. DattaArxiv2010 () and the definition of MQ, we have:

 δ−−→A|C=0⇔ρAC∈CQ⇔ρAB∈MQ.

Therefore, for any that is not in MQ, we have (for all purifications of ) and

 I−−→A|Bc<δ−−→A|B⩽Δ←−→A|B, (40)

showing that MQ is the only set of states for which , and hence the only set for which .

We wish to emphasize that other states besides MQ states may collapse “part” of the hierarchy. For example, consider a tensor product of maximally mixed states, say, of the form . Clearly all measures of entanglement and discord are zero for this state. But the coherent information is , and this state is not an MQ state.

Likewise, as mentioned in Section II.3, a superset of MQ, denoted mQ, partially collapses the hierarchy, as shown in CorDeOFan11 (). Specifically, Ref. CorDeOFan11 () showed that

 I−−→A|Bc=EA|BD=KA|BD=EA|BR,∞=δ−−→B|A∞=δ−−→B|A (41)

for . However, (40) indicates that, for those states in mQ that are not in MQ, there is a gap between the “collapsed measures” appearing in (41) and a particular one-way discord, .

### iii.5 Collapse of smooth measures

While there are various correlation hierarchies that we could investigate, we have been focusing on those that involve a conditional entropy as one of the measures. This is because we will ultimately be interested in using the hierarchy collapse to reinterpret entropic uncertainty relations (EURs), which are often formulated using conditional entropies. One such EUR has been formulated for smooth entropies TomRen2010 (), and so we will consider the correlation hierarchy related to smooth entropies in this subsection, again with the intention of giving a reinterpretation of this EUR.

Smooth entropies pose a dilemma in that they are highly powerful tools relevant to non-asymptotic information-processing tasks RennerThesis05URL (); TomamichelThesis2012 (), yet they are quite technical. We therefore give only the main results in this section, and relegate all proofs to (a lengthy) Appendix D.

 Hmin(A|B)ρ:=maxσB[−Dmax(ρAB||\openone⊗σB)], Hmax(A|B)ρ:=maxσB[−Dfid(ρAB||\openone⊗σB)],

where the maximization is over all normalized density operators , and and were defined in (20) and (21).

To define the smooth entropy of , we optimize the entropy over a ball of radius centered around in the space of subnormalized positive operators, denoted . We use the purified distance to define this ball TomColRen10 (), again with all the details in Appendix D. Then, the smooth min- and max-entropies are defined as TomamichelThesis2012 (); TomColRen10 ():

 Hϵmin(A|B)ρ:=maxσAB∈Bϵ(ρ)Hmin(A|B)σ, Hϵmax(A|B)ρ:=minσAB∈Bϵ(ρ)Hmax(A|B)σ.

Note that a maximisation (minimisation) is performed for the smooth min (max) entropy; in this form these are the relevant quantities for characterising the operational tasks involved in quantum key distribution RennerThesis05URL (); TomamichelThesis2012 ().

To obtain results that are mathematically analogous to Lemma 1 and Theorem 2, we will need to define smooth measures of entanglement and discord. We note that smooth measures of entanglement were considered, e.g., in BusDatPRL2011 (); BranDatt2011 (). Consider first the unsmooth measures of entanglement and discord (one-way and two-way) based on the max relative entropy, respectively given by:

 EA|Bmax(ρAB):=minσAB∈% SepDmax(ρAB||σAB), Δ−−→A|Bmax(ρAB):=minσAB∈CQDmax(ρAB||σAB), Δ←−→A|Bmax(ρAB):=minσAB∈CCDmax(ρAB||σAB).

and consider analogous quantities , , and defined similarly but with replaced by . We note that and are non-increasing under LOCC due to Property 1, and was characterised in Datta09 ().

We now define smooth versions of these quantum correlation measures, as follows:

 ϵEA|Bmax(ρAB):=minσAB∈Bϵ(ρ)EA|Bmax(σAB), ϵΔ−−→A|Bmax(ρAB):=minσAB∈Bϵ(ρ)Δ−−→A|Bmax(σAB), ϵΔ←−→A|Bmax(ρAB):=minσAB∈Bϵ(ρ)Δ←−→A|Bmax(σAB), (42)

and

 ϵEA|Bfid(ρAB):=maxσAB∈Bϵ(ρ)EA|Bfid(σAB), ϵΔ−−→A|Bfid(ρAB):=maxσAB∈Bϵ(ρ)Δ−−→A|Bfid(σAB), ϵΔ←−→A|Bfid(ρAB):=maxσAB∈Bϵ(ρ)Δ←−→A|Bfid(σAB). (43)

A smooth max entanglement defined similarly to the one in (III.5) was previously given an operational meaning in terms of one-shot catalytic entanglement cost under non-entangling maps BranDatt2011 (). We note that performing a minimisation in (III.5) and a maximisation in (III.5) appears to be necessary to obtain the generalisation of our results to smooth measures.

We now state an analog of Lemma 1 for smooth measures, where Eqs. (44) and (45) below can be viewed as quantum correlation hierarchies involving the smooth min and max entropies, respectively.

###### Lemma 5.

For any bipartite state ,

 −Hϵmin(A|B)⩽ϵEA|Bmax⩽ϵΔ−−→A|Bmax,ϵΔ−−→B|Amax⩽ϵΔ←−→A|Bmax, (44) −Hϵmax(A|B)⩽ϵEA|Bfid⩽ϵΔ−−→A|Bfid,ϵΔ−−→B|Afid⩽ϵΔ←−→A|Bfid. (45)

Analogous to Theorem 2, we find that the hierarchies of smooth quantum correlation measures in (44) and (45) collapse in the special case of premeasurement states.

###### Theorem 6.

For any state in MQ, i.e., any premeasurement state ,

 −Hϵmin(MX|S) =ϵEMX|Smax=ϵΔ−−−−→MX|Smax =ϵΔ−−−−→S|MXmax=ϵΔ←−−→MX|Smax, (46) −Hϵmax(MX|S) =ϵEMX|Sfid=ϵΔ−−−−→MX|Sfid =ϵΔ−−−−→S|MXfid=ϵΔ←−−→MX|Sfid. (47)

We note that these smooth measures reduce to the corresponding non-smooth measures for . Hence, we had already proved Lemma 5 and Theorem 6 for the special case of in Section III.3, but the smooth versions of these results, valid for any , are a significant generalization. While superficially it seems simple to add an as a superscript or subscript, let the reader beware that the proof of this result for smooth measures is non-trivial.

## Iv Connection to uncertainty

We have investigated several quantum correlation hierarchies, and in each case we found that premeasurement states collapse the hierarchy. We would now like to take advantage of the dynamic view, shown schematically in Fig. 2, that these states are produced during the measurement process. In principle, premeasurement states can range from being maximally entangled to being only classically correlated to being completely uncorrelated. What features of the state prior to the controlled-shift operation in Fig. 2 determine the correlations of the premeasurement state? As we will see, it is the uncertainty of the observable being measured that ultimately determines the correlations produced during the premeasurement.

The key property that allows us to connect uncertainty to the quantum correlations of premeasurement states is the tripartite duality of conditional entropy functions. For example, for the von Neumann entropy, we have:

 H(A|B)=−H(A|C) (48)

for any pure state on . Let us apply this duality to the pure state shown in Fig. 2, giving:

 H(MX|E)~ρ=−H(MX|S)~ρ (49)

Now we note that the left side of (49) is the standard way of defining the uncertainty of an observable conditioned on quantum memory RenesBoileau (); BertaEtAl (); TomRen2010 (); ColesEtAl (); ColesColbeckYuZwolak2012PRL (). That is, , the uncertainty of observable when the observer is given access to system is defined as the quantum conditional entropy of given at the end of the process depicted in Fig. 2. In addition, Theorem 4 showed that the right side of (49) is equal to a long list of other quantum correlation measures, so we have:

 H(X|E) =EMX|SD=KMX|SD=EMX|SR,∞=EMX|SR =δ−−−−→MX|S∞=δ−−−−→S|MX∞=δ−−−−→MX|S=δ−−−−→S|MX =Δ−−−−→