Quantumness of correlations, quantumness of ensembles and quantum data hiding

We adopt the following general definition for disturbance:

Definition 2.1. Given a measure $D[\cdot ,\cdot ]$ of distance between quantum states, and a set $\mathcal{L}$ of measurement strategies, we define the quantumness of an ensemble $\mathcal{E}\;:=\;\{({{p}_{i}},{{\rho }^{(i))}}\}_{i=1}^{n}$ under $\mathcal{L}$ as measured by D, or simply the $(D,\mathcal{L})$-quantumness of $\mathcal{E}$, as

$$\begin{eqnarray}&&{{Q}_{D,\mathcal{L}}}[\mathcal{E}]\;:=\;\mathop{{\rm inf} }\limits_{\Lambda \in \mathcal{L}}\mathop{\sum }\limits_{i=1}^{n}{{p}_{i}}D[{{\rho }^{(i)}},\Lambda [{{\rho }^{(i)}}]],\end{eqnarray} \tag{ 1 }$$

$\Lambda [\rho ]$

$D\left[ {{\rho }^{(i)}},\Lambda [{{\rho }^{(i)}}] \right]$

Two meaningful distance measures to consider are the trace distance, ${{D}_{1}}[\sigma ,\tau ]=1/2\parallel \sigma -\tau {{\parallel }_{1}}$, with $\parallel X{{\parallel }_{1}}={\rm Tr}\sqrt{{{X}^{\dagger }}X}$, and the relative entropy⁵ , $S[\sigma \parallel \tau ]={\rm Tr}(\sigma ({{{\rm log} }_{2}}\sigma -{{{\rm log} }_{2}}\tau ))$, because they have well-understood operational meanings in terms of state distinguishability (Nielsen and Chuang 2000). For the sake of simplicity, we will use the notation ${{Q}_{{{D}_{1}},\mathcal{L}}}$ and ${{Q}_{S,\mathcal{L}}}$, respectively. Another sensible choice would be the Bures distance $\sqrt{1-F(\sigma ,\tau )}$ (or its square), with the fidelity $F(\sigma ,\tau )={\rm Tr}\sqrt{\sqrt{\sigma }\tau \sqrt{\sigma }}$ (Uhlmann 1976, Jozsa 1994), and we do use it in section 4, but we will mostly focus on the trace distance and the relative entropy.

As for $\mathcal{L},$ in order for our definition of ensemble quantumness to make sense, we must restrict it to sets of operations that admit a meaningful 'post-measurement state' from whence the distance measure is rendered meaningful. In principle, if the states in the ensemble are density matrices acting on a Hilbert space $\mathcal{X},$ any subset of the set of channels $C(\mathcal{X})\;:=\;\{\Lambda :L(\mathcal{X})\to L(\mathcal{X}),\Lambda \}\;{\rm completely}\;{\rm positive}\;{\rm and}\;{\rm trace}\;{\rm preserving}$, where $L(\mathcal{X})$ is the set of operators from $\mathcal{X}$ to $\mathcal{X},$ would be a potential mathematically sound choice. Nonetheless, we aim here at capturing the idea of 'informative' measurement, either constrained or unconstrained, that necessarily leads to some disturbance for most states (see, e.g., (D'Ariano 2003, Kretschmann et al 2008, Luo 2008b) and references therein). Furthermore, we have in mind the following notions of classicality, for ensembles and for correlations, respectively (Fuchs and Sasaki 2003, Horodecki et al 2005, 2006, Groisman et al 2007, Piani et al 2008):

Definition 2.2. A set of states $\{{{\rho }^{(i)}}\}$ is classical if all the states in the set commute, i.e. $[{{\rho }^{(i)}},{{\rho }^{(j)}}]=0$ for all $i,j$, so that the states can be diagonalized simultaneously.

Definition 2.3. An n-partite state ${{\rho }_{{{A}_{1}}{{A}_{2}}\ldots {{A}_{n}}}}$ is classical on A_k if ${{\rho }_{{{A}_{1}}{{A}_{2}}\ldots {{A}_{n}}}}={{\sum }_{i}}{{p}_{i}}|i\rangle \langle i{{|}_{{{A}_{k}}}}\otimes {{\langle i{{|}_{{{A}_{k}}}}{{\rho }_{{{A}_{1}}{{A}_{2}}\ldots {{A}_{n}}}}|i\rangle }_{{{A}_{k}}}}$ for some orthonormal basis $\{|i\rangle \}$ of A_k. In particular, a bipartite state ${{\rho }_{AB}}$ is called classical-quantum if it is classical on A and classical-classical (or fully classical) if it is classical on both A and B.

${{\sum }_{i}}{{p}_{i}}\rho _{A}^{(i)}\otimes \rho _{B}^{(i)}$

For these reasons, we mostly focus on complete projective measurements $\Pi \;:=\;{{\{{{P}^{(k)}}\}}_{k}}$ acting as $\Pi [\sigma ]={{\sum }_{k}}{{P}^{(k)}}\sigma {{P}^{(k)}}$, and on complete local projective measurements ${{\Pi }_{A}}\;:=\;{{\{P_{A}^{(j)}\}}_{j}}$ (on subsystem A; similarly for subsystem B) or complete bilocal ones ${{\Pi }_{A}}\otimes {{\Pi }_{B}}$ (and generalizations thereof for multipartite systems) when we want to focus on the quantumness of correlations. Of course, one could consider other generalizations, for example, to incomplete measurements, or to channels whose Kraus operators have some specified rank (Luo and Fu 2013, Gharibian 2012, Brodutch 2013), but for the sake of concreteness, we will limit ourselves to the explicit cases above.

For the sake of concreteness, we explicitly list the quantifiers of ensemble quantumness corresponding to the set of operations $\{\Pi \}$, $\{{{\Pi }_{A}}\}$ and $\{{{\Pi }_{A}}\otimes {{\Pi }_{B}}\}$, and to the two distance measures mentioned above:

• ensemble quantumness for single systems:
  $$\begin{eqnarray}&&{{Q}_{{{D}_{1}},\{\Pi \}}}[\mathcal{E}]\;:=\;\mathop{{\rm min} }\limits_{\Pi }\mathop{\sum }\limits_{i=1}^{n}{{p}_{i}}\frac{1}{2}\left\|{{\rho }^{(i)}}-\Pi [{{\rho }^{(i)}}]\right \|{{}_{1}},\end{eqnarray} \tag{ 2 }$$
  $$\begin{eqnarray}&&{{Q}_{S,\{\Pi \}}}[\mathcal{E}]\;:=\;\mathop{{\rm min} }\limits_{\Pi }\mathop{\sum }\limits_{i=1}^{n}{{p}_{i}}S\left[ {{\rho }^{(i)}} \right.\left. \Pi [{{\rho }^{(i)}}] \right]=\mathop{{\rm min} }\limits_{\Pi }\mathop{\sum }\limits_{i=1}^{n}{{p}_{i}}\left[ S(\Pi [{{\rho }^{(i)}}])-S({{\rho }^{(i)}}) \right];\end{eqnarray} \tag{ 3 }$$
• ensemble quantumness of correlations:
  • one-sided:
    $$\begin{eqnarray}&&{{Q}_{{{D}_{1}},\{{{\Pi }_{A}}\}}}[\mathcal{E}]\;:=\;\mathop{{\rm min} }\limits_{{{\Pi }_{A}}}\mathop{\sum }\limits_{i=1}^{n}{{p}_{i}}\frac{1}{2}\left\|{{\rho }^{(i)}}-{{\Pi }_{A}}[{{\rho }^{(i)}}]\right \|{{}_{1}},\end{eqnarray} \tag{ 4 }$$
    $$\begin{eqnarray}&&{{Q}_{S,\{{{\Pi }_{A}}\}}}[\mathcal{E}]\;:=\;\mathop{{\rm min} }\limits_{{{\Pi }_{A}}}\mathop{\sum }\limits_{i=1}^{n}{{p}_{i}}S\left[ {{\rho }^{(i)}} \right.\left. {{\Pi }_{A}}[{{\rho }^{(i)}}] \right]\end{eqnarray} \tag{ 5 }$$
    $$\begin{eqnarray}&&\qquad \qquad \ \ \ {\mkern 1mu} =\;\mathop{{\rm min} }\limits_{{{\Pi }_{A}}}\mathop{\sum }\limits_{i=1}^{n}{{p}_{i}}\left[ S(\Pi [{{\rho }^{(i)}}])-S({{\Pi }_{A}}[{{\rho }^{(i)}}]) \right];\end{eqnarray} \tag{ 6 }$$
  • two-sided:
    $$\begin{eqnarray}&&{{Q}_{{{D}_{1}},\{{{\Pi }_{A}}\otimes {{\Pi }_{B}}\}}}[\mathcal{E}]\;:=\;\mathop{{\rm min} }\limits_{{{\Pi }_{A}}\otimes {{\Pi }_{B}}}\mathop{\sum }\limits_{i=1}^{n}{{p}_{i}}\frac{1}{2}\left\|{{\rho }^{(i)}}-({{\Pi }_{A}}\otimes {{\Pi }_{B}})[{{\rho }^{(i)}}]\right \|{{}_{1}}.\end{eqnarray} \tag{ 7 }$$
    $$\begin{eqnarray}&&{{Q}_{S,\{{{\Pi }_{A}}\otimes {{\Pi }_{B}}\}}}[\mathcal{E}]\;:=\;\mathop{{\rm min} }\limits_{{{\Pi }_{A}}\otimes {{\Pi }_{B}}}\mathop{\sum }\limits_{i=1}^{n}{{p}_{i}}S\left[ {{\rho }^{(i)}} \right.\left. ({{\Pi }_{A}}\otimes {{\Pi }_{B}})[{{\rho }^{(i)}}] \right]\end{eqnarray} \tag{ 8 }$$
    $$\begin{eqnarray}&&\qquad \qquad \qquad \ \;=\mathop{{\rm min} }\limits_{{{\Pi }_{A}}\otimes {{\Pi }_{B}}}\mathop{\sum }\limits_{i=1}^{n}{{p}_{i}}\left[ S(\Pi [{{\rho }^{(i)}}])-S({{\Pi }_{A}}\otimes {{\Pi }_{B}}[{{\rho }^{(i)}}]) \right].\end{eqnarray} \tag{ 9 }$$

Note that we have used the fact that for projective measurements the infimum in (1) is in fact a minimum. In addition, for the measures based on relative entropy, we made use of the relation $S(\rho \parallel \Pi [\rho ])=S(\Pi [\rho ])-S(\rho )$, valid for any (not necessarily complete) projective measurement Π, with $S(\sigma )\;:=\;-{\rm Tr}(\sigma {{{\rm log} }_{2}}\sigma )$ the von Neumann entropy (Nielsen and Chuang 2000).

We first remark that the use of one specific distance measure, rather than another one in general, strongly depends on the context and, potentially, on the convenience of calculation. For example, in the following, we will often focus on the quantity ${{Q}_{{{D}_{1}},\{{{\Pi }_{A}}\}}}$ because of its natural connection with the task of discriminating distributed states via LOCC. On the other hand, it is natural to expect ${{Q}_{S,\mathcal{L}}}$ to be more relevant from an information-theoretical point of view (see section 2.3). Also, the quantumness measures ${{Q}_{{{D}_{1}},\mathcal{L}}}$ are always bounded above by unity, and could be considered to be not so helpful in providing insight on the role of quantumness with increasing dimensions of the systems under scrutiny. We do not believe this to be a strong contraindication to the adoption of measures in the class ${{Q}_{{{D}_{1}},\mathcal{L}}}$; furthermore, one can always reinstate a scaling with dimensions via the composition with the logarithm function. For example, in section 3 we will find less trivial upper bounds for ${{Q}_{{{D}_{1}},\{{{\Pi }_{A}}\}}}$, and show that, for fixed local dimension d, ${{Q}_{{{D}_{1}},{{\Pi }_{A}}}}$ is maximized by maximally entangled states—even in the case of the trivial ensemble made of only one state—assuming in such a case the value $1-1/d$. This suggests to define a logarithmic version of disturbance as

$$\begin{eqnarray*}&&L{{Q}_{{{D}_{1}},{{\Pi }_{A}}}}:=-{\rm log} (1-{{Q}_{{{D}_{1}},{{\Pi }_{A}}}}).\end{eqnarray*}$$

Such a quantity varies between 0 for ensembles of states classical on A, to ${\rm log} d$ for (ensembles of) maximally entangled states (in dimension d).

In general, the properties of the measure ${{Q}_{D,\mathcal{L}}}[\mathcal{E}]$ will strongly depend on the properties of the distance D and of the class of operations $\mathcal{L}$. Some basic properties like invariance under unitaries or local unitaries are easily assessed for relevant choices of D and $\mathcal{L}$. Interestingly, if we suppose that D is jointly convex, as it happens for any norm-based distance—like the trace distance—and for the relative entropy, then ${{Q}_{D,\mathcal{L}}}[\mathcal{E}]$ is a monotone under coarse graining, independently of the choice of $\mathcal{L}$. More precisely, if $\mathcal{E}^{\prime} =\{(p_{i}^{\prime },\rho {{^{\prime} }^{(i)}})\}$ is such that $p_{i}^{\prime }\rho {{^{\prime} }^{(i)}}={{\sum }_{k\in {{I}_{i}}}}{{p}_{k}}{{\rho }^{(k)}}$ for some starting ensemble $\mathcal{E}=\{({{p}_{j}},{{\rho }^{(j)}})\}_{j=1}^{n}$ and some partition $\left\{ {{I}_{i}}:{{\cup }_{i}}{{I}_{i}}=\{1,2,\ldots ,n\},{{I}_{i}}\cap {{I}_{j}}=\varnothing \;\forall i\ne j \right\}$, then

$$\begin{eqnarray*}&&{{Q}_{D,\mathcal{L}}}[\mathcal{E}^{\prime} ]\leqslant {{Q}_{D,\mathcal{L}}}[\mathcal{E}].\end{eqnarray*}$$

A final remark is that through Pinskerʼs inequality (Hiai et al 1981, Schumacher and Westmoreland 2002), $\parallel \rho -\sigma \parallel _{1}^{2}\leqslant ({\rm ln} 4)S[\rho \parallel \sigma ]$, one easily derives the relation

$$\begin{eqnarray}&&{{Q}_{{{D}_{1}},\mathcal{L}}}[\mathcal{E}]\leqslant \sqrt{\frac{{\rm ln} 2}{2}}\sqrt{{{Q}_{S,\mathcal{L}}}[\mathcal{E}]}\;\quad \forall \mathcal{L},\mathcal{E}.\end{eqnarray} \tag{ 10 }$$

In this section, we relate the family of quantities introduced in section 2 with quantifiers of quantumness already present in the existing literature.

2.3.1. Quantumness of correlations of a single state

In the case in which the ensemble $\mathcal{E}$ is trivial and contains only one state, i.e., $\mathcal{E}=\{(1,\rho )\}$, the ensemble measures become single-state measures, and we write ${{Q}_{D,\mathcal{L}}}[\rho ]$ for ${{Q}_{D,\mathcal{L}}}(\{(1,\rho )\})$. In particular, the quantities introduced above trivially vanish in the case in which we consider single systems and 'global' complete von Neumann measurements, i.e.,

$$\begin{eqnarray*}&&{{Q}_{{{D}_{1}},\{\Pi \}}}[\rho ]={{Q}_{S,\{\Pi \}}}[\rho ]=0,\end{eqnarray*}$$

because one can always consider projective measurements in the eigenbasis of ρ. On the other hand, for a distributed state $\rho ={{\rho }_{AB}}$, all the following are non-trivial measures of the quantumness of correlations of ${{\rho }_{AB}}$:

$$\begin{eqnarray}&&{{Q}_{{{D}_{1}},\{{{\Pi }_{A}}\}}}[{{\rho }_{AB}}]=\mathop{{\rm min} }\limits_{{{\Pi }_{A}}}\frac{1}{2}\parallel {{\rho }_{AB}}-{{\Pi }_{A}}[{{\rho }_{AB}}]{{\parallel }_{1}},\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{{Q}_{S,\{{{\Pi }_{A}}\}}}[{{\rho }_{AB}}]=\mathop{{\rm min} }\limits_{{{\Pi }_{A}}}S\left[ {{\rho }_{AB}} \right.\left. {{\Pi }_{A}}[{{\rho }_{AB}}] \right]=\mathop{{\rm min} }\limits_{{{\Pi }_{A}}}S({{\Pi }_{A}}[{{\rho }_{AB}}])-S({{\rho }_{AB}}),\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{{Q}_{{{D}_{1}},\{{{\Pi }_{A}}\otimes {{\Pi }_{B}}\}}}[{{\rho }_{AB}}]=\mathop{{\rm min} }\limits_{{{\Pi }_{A}}\otimes {{\Pi }_{B}}}\frac{1}{2}\parallel {{\rho }_{AB}}-({{\Pi }_{A}}\otimes {{\Pi }_{B}})[{{\rho }_{AB}}]{{\parallel }_{1}},\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&{{Q}_{S,\{{{\Pi }_{A}}\otimes {{\Pi }_{B}}\}}}[{{\rho }_{AB}}]=\mathop{{\rm min} }\limits_{{{\Pi }_{A}}\otimes {{\Pi }_{B}}}S\left[ {{\rho }_{AB}} \right.\left. ({{\Pi }_{A}}\otimes {{\Pi }_{B}})[{{\rho }_{AB}}] \right]\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&\qquad \qquad \qquad \quad {\mkern 1mu} =\mathop{{\rm min} }\limits_{{{\Pi }_{A}}\otimes {{\Pi }_{B}}}S(({{\Pi }_{A}}\otimes {{\Pi }_{B}})[{{\rho }_{AB}}])-S({{\rho }_{AB}}).\end{eqnarray} \tag{ 15 }$$

These four quantifiers correspond to the measurement-induced disturbance (Luo 2008b) due to one-sided or two-sided measurement, measured either entropically or by means of the trace-distance. The latter case, corresponding to a 'trace-norm discord', has been defined and studied in (Debarba et al 2012, Rana and Parashar 2013, Paula et al 2013, Nakano et al 2013); the entropic measures were instead introduced first as quantum deficits (Horodecki et al 2005). Such measures also correspond, either in general (in the entropic case) (Modi et al 2010) or at least in relevant special cases (for the trace-norm) (Paula et al 2013, Nakano et al 2013), to the distance of the given state from the set of classical–quantum or classical–classical states. Another interpretation worth mentioning of such measures, again valid either in general or in special cases, is that in terms of entanglement generated in a quantum measurement (Streltsov et al 2011, Piani et al 2011, Gharibian et al 2011, Piani and Adesso 2012, Adesso et al 2014). We refer to (Nakano et al 2013) for a recent and more extensive summary of the relevant properties and relations between these measures.

2.3.2. Quantumness of single-system ensembles as quantumness of correlations

Given a generic ensemble $\mathcal{E}=\{({{p}_{i}},\rho _{S}^{(i)})\}_{i=1}^{n}$ for a system S, one can associate with it a bipartite state ${{\rho }_{SX}}(\mathcal{E})={{\sum }_{i}}{{p}_{i}}\rho _{S}^{(i)}\otimes |i\rangle \langle i{{|}_{X}}$. Using the fact that for relative entropy (Nielsen and Chuang 2000, Cover and Thomas 2012, Piani 2009)

$$\begin{eqnarray}&&S\left[ \mathop{\sum }\limits_{i}{{p}_{i}}\rho _{S}^{(i)}\otimes |i\rangle \langle i{{|}_{X}} \right.\left. \mathop{\sum }\limits_{i}{{p}_{i}}\sigma _{S}^{(i)}\otimes |i\rangle \langle i{{|}_{X}} \right]=\mathop{\sum }\limits_{i}{{p}_{i}}S\left[ \rho _{S}^{(i)} \right.\left. \sigma _{S}^{(i)} \right],\end{eqnarray} \tag{ 16 }$$

and that the trace norm of a block diagonal matrix is equal to the sum of the trace norms of the blocks, i.e., $\left\|{{\oplus }_{i}}{{M}_{i}}\right \|{{}_{1}}={{\sum }_{i}}\left\|{{M}_{i}}\right \|{{}_{1}}$ (Bhatia 1997), so that

$$\begin{eqnarray}&&\left\|\mathop{\sum }\limits_{i}{{p}_{i}}\rho _{S}^{(i)}\otimes |i\rangle \langle i{{|}_{X}}-\mathop{\sum }\limits_{i}{{p}_{i}}\sigma _{S}^{(i)}\otimes |i\rangle \langle i{{|}_{X}}\right \|{{}_{1}}=\mathop{\sum }\limits_{i}{{p}_{i}}\parallel \rho _{S}^{(i)}-\sigma _{S}^{(i)}{{\parallel }_{1}},\end{eqnarray} \tag{ 17 }$$

we have the identities

$$\begin{eqnarray}&&{{Q}_{{{D}_{1}},\{{{\Pi }_{S}}\}}}[{{\rho }_{SX}}(\mathcal{E})]={{Q}_{{{D}_{1}},\{\Pi \}}}(\mathcal{E})={{Q}_{{{D}_{1}},\{{{\Pi }_{S}}\otimes {{\Pi }_{X}}\}}}[{{\rho }_{SX}}(\mathcal{E})],\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&{{Q}_{S,\{{{\Pi }_{S}}\}}}[{{\rho }_{SX}}(\mathcal{E})]={{Q}_{S,\{\Pi \}}}(\mathcal{E})={{Q}_{S,\{{{\Pi }_{S}}\otimes {{\Pi }_{X}}\}}}[{{\rho }_{SX}}(\mathcal{E})],\end{eqnarray} \tag{ 19 }$$

where the last equality in each of the above equations is due to the fact that ${{\Pi }_{X}}$ can be chosen to project in the basis ${{\{|i\rangle }_{X}}\}$. On the other hand, the first equality in each equation holds independently of the class of operations $\mathcal{L}$ considered, that is, for example,

$$\begin{eqnarray*}&&{{Q}_{{{D}_{1}},{{\mathcal{L}}_{S}}}}[{{\rho }_{SX}}(\mathcal{E})]={{Q}_{{{D}_{1}},\mathcal{L}}}(\mathcal{E}),\quad \forall \mathcal{L}.\end{eqnarray*}$$

The approach consisting of using tools originally introduced to quantify the quantumness of correlations to instead quantify the quantumness of ensembles was already put forward in, e.g., (Luo et al 2010, 2011, Yao et al 2013), in particular making use of the relative entropy. Here, we emphasize that this is a general fact that actually applies to any distance measure that respects a 'direct sum' rule such as (16) and (17), and fits in a general paradigm as the one we laid out in section 2. For example, in the case where S is a composite system itself, that is, S = AB, one can have similar relations for the (bipartite) quantumness of correlations of an ensemble of states $\{({{p}_{i}},\rho _{AB}^{(i)})\}$ of $AB$ and the (tripartite) quantumness of correlations of a tripartite state ${{\rho }_{ABX}}(\mathcal{E})={{\sum }_{i}}{{p}_{i}}\rho _{AB}^{(i)}\otimes |i\rangle \langle i{{|}_{X}}$, like

$$\begin{eqnarray*}&&{{Q}_{{{D}_{1}},\{{{\Pi }_{A}}\otimes {{\Pi }_{B}}\}}}[{{\rho }_{ABX}}(\mathcal{E})]={{Q}_{{{D}_{1}},\{{{\Pi }_{A}}\otimes {{\Pi }_{B}}\otimes {{\Pi }_{X}}\}}}[{{\rho }_{ABX}}(\mathcal{E})]={{Q}_{{{D}_{1}},\{{{\Pi }_{A}}\otimes {{\Pi }_{B}}\}}}\left[ \left\{ ({{p}_{i}},\rho _{AB}^{(i)}) \right\} \right].\end{eqnarray*}$$

2.3.3. Quantumness of ensembles for a single mixed state and convex-roof constructions

In section 2.3.1, we have seen how our general ensemble quantifiers can reduce to quantifiers for a single state in the case of a trivial ensemble. There are other, less trivial ways to use our ensemble measures when we deal with a single state.

One possibility is that of considering ensemble realizations of that state. For example, we can consider an arbitrary pure-state ensemble $\mathcal{E}(\rho )=\left\{ ({{p}_{i}},|{{\psi }^{(i)}}\rangle \langle {{\psi }^{(i)}}|) \right\}$ such that $\rho ={{\sum }_{i}}{{p}_{i}}|{{\psi }^{(i)}}\rangle \langle {{\psi }^{(i)}}|$. The idea, then, is that of defining a single-state quantifier of quantum correlations based on a minimization over such a decomposition, i.e.,

$$\begin{eqnarray*}&&Q_{D,\mathcal{L}}^{{\rm ens}}(\rho )\;:=\;\mathop{{\rm min} }\limits_{\mathcal{E}(\rho )}{{Q}_{D,\mathcal{L}}}(\mathcal{E}(\rho )).\end{eqnarray*}$$

As discussed in section 2.2, if D is jointly convex, then ${{Q}_{D,\mathcal{L}}}(\mathcal{E})$ is monotonic under coarse-graining of $\mathcal{E},$ and one has the relation

$$\begin{eqnarray*}&&Q_{D,\mathcal{L}}^{{\rm ens}}(\rho )\geqslant {{Q}_{D,\mathcal{L}}}(\rho ).\end{eqnarray*}$$

We notice that a state that is classical on, let us say, A will admit pure-state ensemble decompositions, where all the pure states in the ensemble are classical in the same basis. So, for example, ${{Q}_{{{D}_{1}},\{{{\Pi }_{A}}\}}}({{\rho }_{AB}})=0$ implies $Q_{{{D}_{1}},\{{{\Pi }_{A}}\}}^{{\rm ens}}({{\rho }_{AB}})=0$.

On the other hand, given a mixed state, one can again use pure-state ensemble realizations of that state, but consider the so-called convex-roof construction

$$\begin{eqnarray}&&Q_{D,\mathcal{L}}^{{\rm cr}}(\rho )\;:=\;\mathop{{\rm min} }\limits_{\mathcal{E}(\rho )}\mathop{\sum }\limits_{i}{{p}_{i}}{{Q}_{D,\mathcal{L}}}(|{{\psi }^{(i)}}\rangle \langle {{\psi }^{(i)}}|).\end{eqnarray} \tag{ 20 }$$

This construction is the standard one used to extend many entanglement measures from bi- and multi-partite pure states to mixed states (Horodecki et al 2009), and indeed $Q_{D,\mathcal{L}}^{{\rm cr}}(\rho )$ is an entanglement measure if ${{Q}_{D,\mathcal{L}}}(|\psi \rangle \langle \psi |)$ is an entanglement monotone on pure states $|\psi \rangle$ (Vidal 2000, Horodecki 2001). Note that in the following, we often use the shorthand notation $\psi =|\psi \rangle \langle \psi |$, hence writing ${{Q}_{D,\mathcal{L}}}(\psi )$.

Notice that the difference between $Q_{D,\mathcal{L}}^{{\rm cr}}(\rho )$ and $Q_{D,\mathcal{L}}^{{\rm ens}}(\rho )$, both defined for a single state ρ, is that the infimum entering in definition (1) is or is not, respectively, adapted to each element of the pure-state ensemble of ρ. This automatically implies

$$\begin{eqnarray}&&Q_{D,\mathcal{L}}^{{\rm ens}}(\rho )\geqslant Q_{D,\mathcal{L}}^{{\rm cr}}(\rho ),\end{eqnarray} \tag{ 21 }$$

independently of the convexity properties of D.

2.3.4. Quantumness and entanglement

When we say that we are interested in a quantumness of correlations that is more general than entanglement, we have in mind a hierarchy that, above all, is qualitative, but may also be cast in quantitative terms, depending on the choice of quantifiers for entanglement and general quantumness. What we expect is that the general quantumness of correlations will be larger than entanglement also quantitatively. A generic approach that leads to a consistent quantitative hierarchy, i.e., a hierarchy in which the quantumness of correlations is always greater than entanglement, is the one based on the mapping of said quantumness into entanglement itself through a measurement interaction (Streltsov et al 2011, Piani et al 2011, Gharibian et al 2011, Piani and Adesso 2012, Coles 2012, Nakano et al 2013, Adesso et al 2014). Another such hierarchy is the one that naturally arises by considering distance measures from various sets that form a hierarchy themselves (Modi et al 2010).

In the latter spirit, if one considers the relative entropy of entanglement (Vedral et al 1997, Vedral and Plenio 1998)

$$\begin{eqnarray*}&&{{E}_{R}}\left[ {{\rho }_{AB}} \right]:=\mathop{{\rm min} }\limits_{{{\sigma }_{AB}}\;{\rm separable}}S[{{\rho }_{AB}}\parallel {{\sigma }_{AB}}],\end{eqnarray*}$$

it is easy to see that ${{Q}_{S,\{{{\Pi }_{A}}\}}}({{\rho }_{AB}})\geqslant {{E}_{R}}[{{\rho }_{AB}}]$, since ${{\Pi }_{A}}[{{\rho }_{AB}}]$ is necessarily separable for ${{\Pi }_{A}}$ a complete von Neumann measurement. On the other hand, ${{Q}_{S,\{{{\Pi }_{A}}\}}}$ and the entanglement of formation (Bennett et al 1996)

$$\begin{eqnarray*}&&{{E}_{F}}\left[ {{\rho }_{AB}} \right]:=\mathop{{\rm min} }\limits_{\mathcal{E}({{\rho }_{AB}})}\mathop{\sum }\limits_{i}{{p}_{i}}S({\rm T}{{{\rm r}}_{A}}(|\psi _{AB}^{(i)}\rangle \langle \psi _{AB}^{(i)}|))\end{eqnarray*}$$

do not respect a hierarchy (Luo 2008a), i.e., there exist states ${{\rho }_{AB}}$ and ${{\sigma }_{AB}}$ such that

$$\begin{eqnarray*}&&{{Q}_{S}},\{{{\Pi }_{A}}\}({{\rho }_{AB}})\lt {{E}_{F}}({{\rho }_{AB}})\quad {\rm and}\quad {{Q}_{S,\{{{\Pi }_{A}}\}}}({{\sigma }_{AB}})\gt {{E}_{F}}({{\sigma }_{AB}}).\end{eqnarray*}$$

We observe here that instead

$$\begin{eqnarray*}&&Q_{S,\{{{\Pi }_{A}}\}}^{{\rm ens}}({{\rho }_{AB}})\geqslant {{E}_{F}}[{{\rho }_{AB}}].\end{eqnarray*}$$

Indeed, for a pure state, ${{Q}_{S,\{{{\Pi }_{A}}\}}}(|{{\psi }_{AB}}\rangle \langle {{\psi }_{AB}}|)=S\left( {\rm T}{{{\rm r}}_{A}}(|{{\psi }_{AB}}\rangle \langle {{\psi }_{AB}}|) \right)={{E}_{F}}(|{{\psi }_{AB}}\rangle \langle {{\psi }_{AB}}|)$ (Luo 2008b), so that $Q_{S,\{{{\Pi }_{A}}\}}^{{\rm cr}}({{\rho }_{AB}})={{E}_{F}}[{{\rho }_{AB}}]$, and we can invoke the general relation (21).

Here, we prove that a large class of measures ${{Q}_{D,{{\mathcal{L}}_{A}}}}$, including ${{Q}_{S,\{{{\Pi }_{A}}\}}}$ and ${{Q}_{D,\{{{\Pi }_{A}}\}}}$, when restricted to single bipartite pure states (see section 2.3.1), are entanglement monotones. By this, we mean that said quantifiers do not increase on average under stochastic LOCC (SLOCC) (Vidal 2000).

Theorem 2.4. For any distance measure D that

• (i)  
  is invariant under unitaries,
• (ii)  
  is monotonic under general quantum operations (this condition actually comprises (i)),
• (iii)  
  respects the 'flags' condition⁷ $D({{\sum }_{i}}{{p}_{i}}{{\rho }_{i}}\otimes |i\rangle \langle i|,{{\sum }_{i}}{{p}_{i}}{{\sigma }_{i}}\otimes |i\rangle \langle i|)={{\sum }_{i}}{{p}_{i}}D({{\rho }_{i}},{{\sigma }_{i}})$, for $\{{{p}_{i}}\}$ a probability distribution, $\{{{\rho }_{i}}\}$, $\{{{\sigma }_{i}}\}$ states, and $\{|i\rangle \}$ orthogonal flags,

and for any class ${{\mathcal{L}}_{A}}$ of local operations ${{\Lambda }_{A}}$ that is closed under conjugation by unitaries, i.e., if ${{\Lambda }_{A}}$ is in ${{\mathcal{L}}_{A}}$ then also $U_{A}^{\dagger }{{\Lambda }_{A}}[{{U}_{A}}\cdot U_{A}^{\dagger }]{{U}_{A}}$ is in ${{\mathcal{L}}_{A}}$ for all U_A, one has that ${{Q}_{D,{{\mathcal{L}}_{A}}}}({{\psi }_{AB}})$ is an entanglement monotone on average, i.e.

$$\begin{eqnarray*}&&{{Q}_{D,{{\mathcal{L}}_{A}}}}({{\psi }_{AB}})\geqslant \mathop{\sum }\limits_{i}{{p}_{i}}{{Q}_{D,{{\mathcal{L}}_{A}}}}(\phi _{AB}^{i}),\end{eqnarray*}$$

where $\{{{p}_{i}},\phi _{AB}^{i}\}$ is a pure-state ensemble obtained from ${{\psi }_{AB}}$ by local operations and classical communication.

Proof. We first note that ${{Q}_{D,{{\mathcal{L}}_{A}}}}({{\psi }_{AB}})$ depends only on the Schmidt coefficients $\{{{p}_{i}}\}$ of $|{{\psi }_{AB}}\rangle ={{U}_{A}}\otimes {{U}_{B}}|{{\phi }_{AB}}\rangle$ with $|{{\phi }_{AB}}\rangle =\sum \sqrt{{{p}_{i}}}|ii{{\rangle }_{AB}}$. This follows from the unitary invariance of D and from $U_{A}^{\dagger }{{\Lambda }_{A}}[{{U}_{A}}\cdot U_{A}^{\dagger }]{{U}_{A}}=\Lambda _{A}^{\prime }[\cdot ]\in {{\mathcal{L}}_{A}}$:

$$\begin{eqnarray*}\begin{array}{rcl} {{Q}_{D,{{\mathcal{L}}_{A}}}}({{\psi }_{AB}}) & = & \mathop{{\rm min} }\limits_{{{\Lambda }_{A}}\in {{\mathcal{L}}_{A}}}D(U_{A}^{\dagger }\otimes U_{B}^{\dagger }{{\psi }_{AB}}{{U}_{A}}\otimes {{U}_{B}},U_{A}^{\dagger }\otimes U_{B}^{\dagger }{{\Lambda }_{A}}[{{\psi }_{AB}}]{{U}_{A}}\otimes {{U}_{B}}) \\ {} & = & \mathop{{\rm min} }\limits_{\Lambda _{A}^{\prime }\in {{\mathcal{L}}_{A}}}D({{\phi }_{AB}},\Lambda _{A}^{\prime }[{{\phi }_{AB}}]))=f(\{{{p}_{i}}\}). \\ \end{array}\end{eqnarray*}$$

Given the symmetry of the state $|{{\phi }_{AB}}\rangle$ if is clear that

$$\begin{eqnarray}&&{{Q}_{D,{{\mathcal{L}}_{A}}}}({{\psi }_{AB}})=\mathop{{\rm min} }\limits_{{{\Lambda }_{A}}\in {{\mathcal{L}}_{A}}}D({{\psi }_{AB}},{{\Lambda }_{A}}[{{\psi }_{AB}}])=\mathop{{\rm min} }\limits_{{{\Lambda }_{B}}\in {{\mathcal{L}}_{B}}}D({{\psi }_{AB}},{{\Lambda }_{B}}[{{\psi }_{AB}}])={{Q}_{D,{{\mathcal{L}}_{B}}}}({{\psi }_{AB}}),\end{eqnarray} \tag{ 22 }$$

i.e., we can consider indifferently a minimization over projections on Aliceʼs or Bobʼs side. In order to prove monotonicity on average, it is sufficient to prove monotonicity under unilocal transformations, i.e., stochastic operations defined through

$$\begin{eqnarray}&&|\phi _{AB}^{i}\rangle =C_{A}^{i}|\psi {{\rangle }_{AB}}/\sqrt{{{p}_{i}}}\qquad {{p}_{i}}={\rm Tr}(C_{A}^{i}|\psi \rangle \langle \psi |C_{A}^{i\dagger }),\end{eqnarray} \tag{ 23 }$$

where the C_Aⁱʼs are the Kraus operators of a generic quantum operation on Aliceʼs side. If monotonicity holds under such operations, and the same holds for operations on Bobʼs side, then monotonicity on average under general LOCC follows, because an LOCC protocol is just a sequence of adaptive unilocal operations (Vidal 2000). To this end,

$$\begin{eqnarray*}\begin{array}{rcl} {{Q}_{D,{{\mathcal{L}}_{B}}}}({{\psi }_{AB}}) & = & \mathop{{\rm min} }\limits_{{{\Lambda }_{B}}\in {{\mathcal{L}}_{B}}}D({{\psi }_{AB}},{{\Lambda }_{B}}[{{\psi }_{AB}}]) \\ {} & \geqslant & \mathop{{\rm min} }\limits_{{{\Lambda }_{B}}\in {{\mathcal{L}}_{B}}}D(\mathop{\sum }\limits_{i}C_{A}^{i}{{\psi }_{AB}}C_{A}^{i\dagger }\otimes |i\rangle \langle i{{|}_{A^{\prime} }},\mathop{\sum }\limits_{i}C_{A}^{i}{{\Lambda }_{B}}[{{\psi }_{AB}}]C_{A}^{i\dagger }\otimes |i\rangle \langle i{{|}_{A^{\prime} }}) \\ {} & = & \mathop{{\rm min} }\limits_{{{\Lambda }_{B}}\in {{\mathcal{L}}_{B}}}D(\mathop{\sum }\limits_{i}{{p}_{i}}{{\phi }_{i}}\otimes |i\rangle \langle i{{|}_{A^{\prime} }},\mathop{\sum }\limits_{i}{{p}_{i}}{{\Lambda }_{B}}[{{\phi }_{i}}]\otimes |i\rangle \langle i{{|}_{A^{\prime} }}) \\ \end{array}\end{eqnarray*}$$

$$\begin{eqnarray*}\begin{array}{clc} {} & = & \mathop{{\rm min} }\limits_{{{\Lambda }_{B}}\in {{\mathcal{L}}_{B}}}\mathop{\sum }\limits_{i}{{p}_{i}}D({{\phi }_{i}},{{\Lambda }_{B}}[{{\phi }_{i}}])) \\ {} & \geqslant & \mathop{\sum }\limits_{i}{{p}_{i}}\mathop{{\rm min} }\limits_{{{\Lambda }_{B}}\in {{\mathcal{L}}_{B}}}D({{\phi }_{i}},{{\Lambda }_{B}}[{{\phi }_{i}}]) \\ {} & = & \mathop{\sum }\limits_{i}{{p}_{i}}{{Q}_{D,{{\mathcal{L}}_{B}}}}({{\phi }_{i}}). \\ \end{array}\end{eqnarray*}$$

The first inequality is due to monotonicity of the distance D under quantum operations, in this case the application of local Kraus operators, with the 'which-operator' information stored in a classical local flag. Notice that the quantum operation on Alice and the projection on Bob commute. Simply moving the measuring operation to Aliceʼs side thanks to (22), similar steps can be taken in the case of a unilocal operation on Bobʼs side.

It will be important in this work to establish bounds on the quantumness of states and ensembles. For this purpose, it is useful to note that maximal quantumness corresponds to maximal entanglement.

Corollary 2.5. For a given fixed dimension of A, d_A, maximally entangled states are maximally quantum-correlated with respect to any measure ${{Q}_{D,{{\mathcal{L}}_{A}}}}({{\rho }_{AB}})$ that respects the conditions of theorem 2.4.

Proof. In (Streltsov et al 2012) it was already proven that a measure of correlations Q that does not increase under operations on at least one side must be maximal on pure states. It is easy to verify that monotonicity under operations of D implies monotonicity under operations on B for any ${{Q}_{D,{{\mathcal{L}}_{A}}}}$. Furthermore, any pure state of A and B can be obtained via LOCC—in particular, with one-way communication from Bob to Alice—from a maximally entangled state of Schmidt rank d_A.

The following lemma will be the key in the study of the maximum disturbances induced by a complete projective measurement on system A, for the case of a bipartite pure state.

Lemma 3.1. In the case of a bipartite pure state $|\psi {{\rangle }_{AB}}$, the disturbance caused by a one-sided complete projective measurement ${{\Pi }_{A}}$ on A, as measured by the trace distance, is given by the positive c such that

$$\begin{eqnarray}&&\mathop{\sum }\limits_{i}\frac{{{p}_{i}}}{c+{{p}_{i}}}=1,\end{eqnarray} \tag{ 25 }$$

for ${{p}_{i}}=\langle i|{{\rho }_{A}}|i\rangle$, with ${{\rho }_{A}}={\rm T}{{{\rm r}}_{B}}|\psi \rangle \langle \psi |$, the probability of obtaining outcome i by measuring in the local orthonormal basis $\{|i\rangle \}$.

Proof. Let $|\psi \rangle ={{\sum }_{i}}|i{{\rangle }_{A}}|{{w}_{i}}{{\rangle }_{B}}$, with the vectors $|{{w}_{i}}\rangle$ not necessarily orthogonal and satisfying $\langle {{w}_{i}}|{{w}_{i}}\rangle ={{p}_{i}}$. Then ${{\Pi }_{A}}[|\psi \rangle \langle \psi |]={{\sum }_{i}}|i\rangle \langle i|\otimes |{{w}_{i}}\rangle \langle {{w}_{i}}|$. We observe that for any vector $|v\rangle$ and any positive-semidefinite A, the positive-definite part of $|v\rangle \langle v|-A$ can have a rank of at most 1, and consequently, if $|v\rangle \langle v|-A$ is traceless, then $\parallel |v\rangle \langle v|-A{{\parallel }_{1}}=2\parallel |v\rangle \langle v|-A{{\parallel }_{\infty }}$. In our case then,

$$\begin{eqnarray*}&&\frac{1}{2}\parallel |\psi \rangle \langle \psi |-{{\Pi }_{A}}[|\psi \rangle \langle \psi |]{{\parallel }_{1}}=\parallel |\psi \rangle \langle \psi |-{{\Pi }_{A}}[|\psi \rangle \langle \psi |]{{\parallel }_{\infty }}.\end{eqnarray*}$$

To further simplify things, let us consider a generic normalized vector $|\phi \rangle ={{\sum }_{i}}|i\rangle |{{z}_{i}}\rangle$, where, similarly as before, the vectors $|{{z}_{i}}\rangle$ are not necessarily orthogonal and satisfy $\langle {{z}_{i}}|{{z}_{i}}\rangle ={{q}_{i}}$, with q_i the probability of the outcome i when measuring A in the basis $\{|i\rangle \}$. One has

$$\begin{eqnarray*}\begin{array}{rcl} \parallel |\psi \rangle \langle \psi |-{{\Pi }_{A}}[|\psi \rangle \langle \psi |]{{\parallel }_{\infty }} & = & \mathop{{\rm max} }\limits_{|\phi \rangle }\langle \phi |(|\psi \rangle \langle \psi |-{{\Pi }_{A}}[|\psi \rangle \langle \psi |])|\phi \rangle \\ {} & = & \mathop{{\rm max} }\limits_{|\phi \rangle }|\mathop{\sum }\limits_{i}\langle {{z}_{i}}|{{w}_{i}}\rangle {{|}^{2}}-\mathop{\sum }\limits_{i}|\langle {{z}_{i}}|{{w}_{i}}\rangle {{|}^{2}} \\ {} & = & \mathop{{\rm max} }\limits_{|\phi \rangle }\mathop{\sum }\limits_{i\gt j}\left( \langle {{z}_{i}}|{{w}_{i}}\rangle \langle {{w}_{j}}|{{z}_{j}}\rangle +\langle {{z}_{j}}|{{w}_{j}}\rangle \langle {{w}_{i}}|{{z}_{i}}\rangle \right) \\ \end{array}\end{eqnarray*}$$

$$\begin{eqnarray*}\begin{array}{clc} {} & = & 2\mathop{{\rm max} }\limits_{|\phi \rangle }\mathop{\sum }\limits_{i\gt j}\Re (\langle {{z}_{i}}|{{w}_{i}}\rangle \langle {{w}_{j}}|{{z}_{j}}\rangle ) \\ {} & = & 2\mathop{{\rm max} }\limits_{|q\rangle }\mathop{\sum }\limits_{i\gt j}\sqrt{{{q}_{i}}{{p}_{i}}{{q}_{j}}{{p}_{j}}} \\ {} & = & \mathop{{\rm max} }\limits_{|q\rangle }{{\left( \mathop{\sum }\limits_{i}\sqrt{{{q}_{i}}{{p}_{i}}} \right)}^{2}}-\mathop{\sum }\limits_{i}{{q}_{i}}{{p}_{i}} \\ {} & = & \mathop{{\rm max} }\limits_{|q\rangle }\langle q|(|p\rangle \langle p|-\mathop{\sum }\limits_{i}{{p}_{i}}|i\rangle \langle i|)|q\rangle , \\ \end{array}\end{eqnarray*}$$

where we have introduced the notation $|p\rangle ={{\sum }_{i}}\sqrt{{{p}_{i}}}|i\rangle$ (similarly for $|q\rangle$), and used that $\Re (\langle {{z}_{i}}|{{w}_{i}}\rangle \langle {{w}_{j}}|{{z}_{j}}\rangle )\leqslant \sqrt{{{q}_{i}}{{p}_{i}}{{q}_{j}}{{p}_{j}}}$, with equality achieved for $|{{z}_{i}}\rangle =(\sqrt{{{q}_{i}}}/\sqrt{{{p}_{i}}})|{{w}_{i}}\rangle$. Thus, it is sufficient to find the largest (positive) eigenvalue of the operator $|p\rangle \langle p|-{{\sum }_{i}}{{p}_{i}}|i\rangle \langle i|$, which we know will be achieved by $|q\rangle$ of the form $|q\rangle ={{\sum }_{i}}\sqrt{{{q}_{i}}}|i\rangle$. With this ansatz, we set

$$\begin{eqnarray*}&&\left( |p\rangle \langle p|-\mathop{\sum }\limits_{i}{{p}_{i}}|i\rangle \langle i| \right)|q\rangle =c|q\rangle ,\end{eqnarray*}$$

from which we find the scalar relations

$$\begin{eqnarray*}&&\langle p|q\rangle \sqrt{{{p}_{i}}}-{{p}_{i}}\sqrt{{{q}_{i}}}=c\sqrt{{{q}_{i}}},\end{eqnarray*}$$

i.e.,

$$\begin{eqnarray*}&&\sqrt{{{q}_{i}}}=\langle p|q\rangle \frac{\sqrt{{{p}_{i}}}}{c+{{p}_{i}}}.\end{eqnarray*}$$

Imposing consistency for ${{\sum }_{i}}\sqrt{{{p}_{i}}{{q}_{i}}}=\langle p|q\rangle$, i.e., imposing

$$\begin{eqnarray*}&&\mathop{\sum }\limits_{i}\sqrt{{{p}_{i}}{{q}_{i}}}=\mathop{\sum }\limits_{i}\sqrt{{{p}_{i}}}\left( \langle p|q\rangle \frac{\sqrt{{{p}_{i}}}}{c+{{p}_{i}}} \right)=\langle p|q\rangle \mathop{\sum }\limits_{i}\frac{{{p}_{i}}}{c+{{p}_{i}}}\equiv \langle p|q\rangle ,\end{eqnarray*}$$

and noticing that (25) is monotonically decreasing (and hence invertible) for $c\geqslant 0$, we arrive at the condition of the statement.

We now make another preliminary observation.

Lemma 3.2. Let $\{{{p}_{i}}\}$ indicate a probability distribution. Then

$$\begin{eqnarray*}&&{{f}_{c}}(\{{{p}_{i}}\})\;:=\;\mathop{\sum }\limits_{i}\frac{{{p}_{i}}}{c+{{p}_{i}}}\end{eqnarray*}$$

is Schur-concave in $\{{{p}_{i}}\}$ for fixed $c\geqslant 0$. It is also monotonically decreasing for fixed $\{{{p}_{i}}\}$ and for increasing positive c. Define also the function

$$\begin{eqnarray*}&&c(\{{{p}_{i}}\})\;:=\;{\rm the}\;{\rm unique}\;c\geqslant 0\;{\rm such}\;{\rm that}\;{{f}_{c}}(\{{{p}_{i}}\})=\mathop{\sum }\limits_{i}\frac{{{p}_{i}}}{c+{{p}_{i}}}=1\end{eqnarray*}$$

on probability vectors. The function $c(\{{{p}_{i}}\})$ is Schur-concave in $\{{{p}_{i}}\}$.

Proof. The Schur-concavity of f_c is a simple consequence of the concavity of $x/(c+x)$ in $x\geqslant 0$ for $c\geqslant 0$, and of the symmetry of f_c in the p_iʼs. Monotonicity in c is evident.

Consider now a probability distribution $p_{i}^{\prime }={{\sum }_{j}}{{B}_{ij}}{{p}_{j}}$ obtained from the probability distribution ${{p}_{i}}_{i}$ by multiplication by a bistochastic matrix B_ij. Because of the Schur concavity of ${{f}_{c}}(\{{{p}_{i}}\})$ in $\{{{p}_{i}}\}$ for fixed c, we have

$$\begin{eqnarray*}\begin{array}{rcl} 1 & = & {{f}_{c(\{{{p}_{i}}\})}}(\{{{p}_{i}}\}) \\ {} & = & \mathop{\sum }\limits_{j}\frac{{{p}_{j}}}{c(\{{{p}_{i}}\})+{{p}_{j}}} \\ {} & \leqslant & \mathop{\sum }\limits_{i}\frac{p_{i}^{\prime }}{c(\{{{p}_{i}}\})+p_{i}^{\prime }} \\ {} & = & {{f}_{c(\{{{p}_{i}}\})}}(\{p_{i}^{\prime }\}). \\ \end{array}\end{eqnarray*}$$

Because of the monotonicity of ${{f}_{c}}(\{{{p}_{i}}\})$ in c for fixed $\{{{p}_{i}}\}$, we conclude that $c(\{p_{i}^{\prime }\})\geqslant c(\{{{p}_{i}}\})$. This proves that $c(\{{{p}_{i}}\})$ is a Schur-concave function of $\{{{p}_{i}}\}$.

Thus we arrive at:

Theorem 3.3. The one-sided trace-norm disturbance

$$\begin{eqnarray*}&&{{Q}_{{{D}_{1}},\{{{\Pi }_{A}}\}}}(|\psi \rangle \langle \psi |)=\mathop{{\rm min} }\limits_{{{\Pi }_{A}}}\frac{1}{2}\parallel |\psi \rangle \langle \psi |-{{\Pi }_{A}}[|\psi \rangle \langle \psi |]{{\parallel }_{1}}\end{eqnarray*}$$

is a bona fide entanglement monotone for the bipartite pure state $|\psi \rangle ={{\sum }_{i}}\sqrt{{{p}_{i}}}|i\rangle |i\rangle$, here expressed in its Schmidt decomposition. The minimum disturbance is obtained by measuring in the local Schmidt basis and is equal to the positive c such that

$$\begin{eqnarray}&&\mathop{\sum }\limits_{i}\frac{{{p}_{i}}}{c+{{p}_{i}}}=1.\end{eqnarray} \tag{ 26 }$$

The same holds for the two-sided trace-norm disturbance

$$\begin{eqnarray*}&&{{Q}_{{{D}_{1}},\{{{\Pi }_{A}}\otimes {{\Pi }_{B}}\}}}(|\psi \rangle \langle \psi |)=\mathop{{\rm min} }\limits_{{{\Pi }_{A}}\otimes {{\Pi }_{B}}}\frac{1}{2}\parallel |\psi \rangle \langle \psi |-({{\Pi }_{A}}\otimes {{\Pi }_{B}})[|\psi \rangle \langle \psi |]{{\parallel }_{1}},\end{eqnarray*}$$

so that we have ${{Q}_{{{D}_{1}},\{{{\Pi }_{A}}\otimes {{\Pi }_{B}}\}}}(|\psi \rangle \langle \psi |)={{Q}_{{{D}_{1}},\{{{\Pi }_{A}}\}}}(|\psi \rangle \langle \psi |)$ for all $|\psi {{\rangle }_{AB}}$.

Proof. From lemma 3.1, we have that $\frac{1}{2}\parallel |\psi \rangle \langle \psi |-{{\Pi }_{A}}[|\psi \rangle \langle \psi |]{{\parallel }_{1}}$ is equal to $c(\{{{q}_{i}}\})$ where q_i is the probability of outcome i in the local projective measurement. Let the latter take place in the local Schmidt basis of $|\psi \rangle ={{\sum }_{i}}\sqrt{{{p}_{i}}}|i\rangle |i\rangle$, so that q_i = p_i. Let $\{|{{u}_{j}}\rangle \}$ be any other orthonormal basis. The probability of the outcome j in such an alternative basis is

$$\begin{eqnarray*}\begin{array}{rcl} p_{j}^{\prime } & = & \langle {{u}_{j}}|{{\rho }_{A}}|{{u}_{j}}\rangle \\ {} & = & \langle {{u}_{j}}|(\mathop{\sum }\limits_{i}{{p}_{i}}|i\rangle \langle i|)|{{u}_{j}}\rangle \\ {} & = & \mathop{\sum }\limits_{i}{{p}_{i}}|\langle {{u}_{j}}|i\rangle {{|}^{2}}. \\ \end{array}\end{eqnarray*}$$

Since the coefficients ${{B}_{ij}}=|\langle {{u}_{j}}|i\rangle {{|}^{2}}$ form the entries of a bistochastic matrix, the Schur concavity of $c(\{{{q}_{i}}\})$ (lemma 3.2) lets us conclude that measurement in the Schmidt basis is optimal for the sake of disturbance.

Note that, similarly, the Schur concavity of $c(\{{{q}_{i}}\})$ in $\{{{q}_{i}}\}$ ensures that ${{Q}_{{{D}_{1}},\{{{\Pi }_{A}}\}}}(|\psi \rangle \langle \psi |)=c(\{{{p}_{i}}\})$, for $\{\sqrt{{{p}_{i}}}\}$ the Schmidt coefficients of $|\psi \rangle$, is a bona fide entanglement measure on pure states (Nielsen 1999, Vidal 2000) for deterministic LOCC transformations. Recall that theorem 2.4 already shows that a wide class of disturbance measures, including ${{Q}_{{{D}_{1}},\{{{\Pi }_{A}}\}}}$, are entanglement monotones on pure states, i.e., they are non-increasing on average under (non-deterministic) LOCC. Hence, it provides a proof for a stronger form of monotonicity.

Finally, to see that ${{Q}_{{{D}_{1}},\{{{\Pi }_{A}}\otimes {{\Pi }_{B}}\}}}(|\psi \rangle \langle \psi |)={{Q}_{{{D}_{1}},\{{{\Pi }_{A}}\}}}(|\psi \rangle \langle \psi |)$, realize that it again holds

$$\begin{eqnarray*}&&\frac{1}{2}\parallel |\psi \rangle \langle \psi |-({{\Pi }_{A}}\otimes {{\Pi }_{B}})[|\psi \rangle \langle \psi |]{{\parallel }_{1}}=\mathop{{\rm max} }\limits_{|\phi \rangle }\langle \phi |\left( |\psi \rangle \langle \psi |-({{\Pi }_{A}}\otimes {{\Pi }_{B}})[|\psi \rangle \langle \psi |] \right)|\phi \rangle .\end{eqnarray*}$$

Consider ${{\Pi }_{A}}$ projecting in the local Schmidt basis and $|\phi \rangle ={{\sum }_{i}}\sqrt{{{q}_{i}}}|i\rangle |i\rangle$ optimal choices for the sake of achieving ${{{\rm max} }_{|\phi \rangle }}\langle \phi |\left( |\psi \rangle \langle \psi |-{{\Pi }_{A}}[|\psi \rangle \langle \psi |] \right)|\phi \rangle \;=$ ${{Q}_{{{D}_{1}},\{{{\Pi }_{A}}\}}}(|\psi \rangle \langle \psi |)=|{{\sum }_{i}}\sqrt{{{p}_{i}}{{q}_{i}}}{{|}^{2}}-{{\sum }_{i}}{{p}_{i}}{{q}_{i}}$ (see the proof lemma 3.1), in the case of the one-sided measurement. Then, coming back to two-sided projective measurements,

$$\begin{eqnarray*}&&\begin{array}{l} \mathop{{\rm max} }\limits_{|\phi \rangle }\langle \phi |\left( |\psi \rangle \langle \psi |-({{\Pi }_{A}}\otimes {{\Pi }_{B}})[|\psi \rangle \langle \psi |] \right)|\phi \rangle \\ \quad \geqslant \;\langle \phi |\left( |\psi \rangle \langle \psi |-({{\Pi }_{A}}\otimes {{\Pi }_{B}})[|\psi \rangle \langle \psi |] \right)|\phi \rangle \\ \quad \geqslant |\mathop{\sum }\limits_{i}\sqrt{{{p}_{i}}{{q}_{i}}}{{|}^{2}}-\mathop{\sum }\limits_{i}{{p}_{i}}{{q}_{i}}{\rm Tr}({{\Pi }_{B}}[|i\rangle \langle i|]{{\Pi }_{B}}[|i\rangle \langle i|]). \\ \end{array}\end{eqnarray*}$$

Since ${\rm Tr}({{\Pi }_{B}}[|i\rangle \langle i|]{{\Pi }_{B}}[|i\rangle \langle i|])\leqslant 1$ for all choices of ${{\Pi }_{B}}$, and since ${{\Pi }_{A}}$ was optimal for ${{Q}_{{{D}_{1}},\{{{\Pi }_{A}}\}}}(|\psi \rangle \langle \psi |)$, we have proven ${{Q}_{{{D}_{1}},\{{{\Pi }_{A}}\otimes {{\Pi }_{B}}\}}}(|\psi \rangle \langle \psi |)\geqslant {{Q}_{{{D}_{1}},\{{{\Pi }_{A}}\}}}(|\psi \rangle \langle \psi |)$. The last inequality can be saturated for ${{\Pi }_{B}}$, a projection in the local Schmidt basis of B.

We will call ${{Q}_{{{D}_{1}},\{{{\Pi }_{A}}\}}}$ the entanglement of disturbance when considering it on pure states, and denote it as ${{E}^{{\rm disturbance}}}({{\psi }_{AB}})$. Since it is an entanglement monotone on average, it can be naturally extended to an entanglement measure on mixed states by a convex-roof construction⁸ :

$$\begin{eqnarray}&&{{E}^{{\rm disturbance}}}({{\rho }_{AB}})\;:=\;Q_{{{D}_{1}},\{{{\Pi }_{A}}\}}^{{\rm cr}}({{\rho }_{AB}})=\mathop{{\rm min} }\limits_{\mathcal{E}(\rho )}\mathop{\sum }\limits_{i}{{p}_{i}}{{E}^{{\rm disturbance}}}(|{{\psi }^{(i)}}\rangle \langle {{\psi }^{(i)}}|).\end{eqnarray} \tag{ 27 }$$

We remark that Bravyi (Bravyi 2003) also studied and quantified entanglement of pure states as a property that implies non-vanishing disturbance under local measurements, but chose entropy—equivalently, relative entropy—as a disturbance quantifier.

It is instructive to consider some simple cases to get a flavor of the new measure of entanglement, in particular to see how the formula (26) plays out. Obviously, in the case of a factorized state $|\alpha \rangle |\beta \rangle$, we have only one non-zero p_i, which is equal to 1. So, condition (26) becomes $1/(c+1)=1$, which is satisfied by c = 0, as expected. In the case of a maximally entangled state of two qudits, one has ${{p}_{i}}=1/d$ for $i=1,...,d$. So, (26) becomes $d\frac{1/d}{c+1/d}=1$, which is solved by $c=1-1/d$; this is the maximal value of minimal disturbance due to local projective measurements on pure states of two qudits. For a pure state of two qubits, the probability distribution reads $\{{{p}_{i}}\}=\{p,1-p\}$, and (26) becomes

$$\begin{eqnarray*}&&\frac{p}{c+p}+\frac{1-p}{c+1-p}=1,\end{eqnarray*}$$

which is satisfied by $c=\sqrt{p(1-p)}$. The latter is the same as the negativity of entanglement (Vidal and Werner 2001) [and, up to a constant factor, concurrence (Wootters 1998)], as to be expected from (Nakano et al 2013, Ciccarello et al 2014).

We conclude this section by providing some explicit upper and lower bounds for $E={{E}^{{\rm disturbance}}}$. We recall that the main result of theorem 3.3 can be restated as the fact that the entanglement of disturbance of $|{{\psi }_{AB}}\rangle$ is the positive number E such that

$$\begin{eqnarray*}&&\mathop{\sum }\limits_{i}\frac{{{p}_{i}}}{E+{{p}_{i}}}=1,\end{eqnarray*}$$

where the p_iʼs are the Schmidt coefficients of $|{{\psi }_{AB}}\rangle$. This is an analytic expression for E, as E can simply be considered the inverse of the function $y=f(x)={{\sum }_{i}}\frac{{{p}_{i}}}{x+{{p}_{i}}}$—with the Schmidt coefficients considered as parameters—evaluated at y = 1. Nonetheless, we provide here some bounds in terms of more standard functions.

In order to find an upper bound to E, we can consider the following steps:

$$\begin{eqnarray*}\begin{array}{rcl} 1 & = & \mathop{\sum }\limits_{i}\frac{{{p}_{i}}}{E+{{p}_{i}}} \\ {} & = & \frac{{{p}_{1}}}{E+{{p}_{1}}}+(R-1)\mathop{\sum }\limits_{i=2}^{R}\frac{1}{R-1}\left( \frac{{{p}_{i}}}{E+{{p}_{i}}} \right) \\ {} & \leqslant & \frac{{{p}_{1}}}{E+{{p}_{1}}}+(R-1)\frac{\sum _{i=2}^{R}\frac{1}{R-1}{{p}_{i}}}{E+\sum _{i=2}^{R}\frac{1}{R-1}{{p}_{i}}} \\ {} & = & \frac{{{p}_{1}}}{E+{{p}_{1}}}+\frac{1-{{p}_{1}}}{E+\frac{1-{{p}_{1}}}{R-1}}, \\ \end{array}\end{eqnarray*}$$

where the inequality is due to the concavity of $x/(1+x)$, p₁ is the largest probability, and R is the rank (the number of non-vanishing p_iʼs). From this, we find the bound

$$\begin{eqnarray}\begin{array}{rcl} E & \leqslant & \frac{-2+2{{p}_{1}}+R-{{p}_{1}}R+\sqrt{4-4{{p}_{1}}-4R+4p_{1}^{2}R+{{R}^{2}}+2{{p}_{1}}{{R}^{2}}-3p_{1}^{2}{{R}^{2}}}}{2(-1+R)} \\ {} & \leqslant & \frac{1}{2}\left( 1-{{p}_{1}}+\sqrt{-3p_{1}^{2}+1+2{{p}_{1}}} \right), \\ \end{array}\end{eqnarray} \tag{ 28 }$$

where the second bound is obtained from the first in the limit $R\to \infty$ (see figure 1). On the other hand, to find a lower bound, we can consider

$$\begin{eqnarray}\begin{array}{rcl} 1 & = & \mathop{\sum }\limits_{i}\frac{{{p}_{i}}}{E+{{p}_{i}}} \\ {} & = & \frac{{{p}_{1}}}{E+{{p}_{1}}}+\mathop{\sum }\limits_{i=2}^{R}\frac{{{p}_{i}}}{E+{{p}_{i}}} \\ {} & \geqslant & \frac{{{p}_{1}}}{E+{{p}_{1}}}+\mathop{\sum }\limits_{i=2}^{R}\frac{{{p}_{i}}}{E+{{p}_{2}}} \\ {} & = & \frac{{{p}_{1}}}{E+{{p}_{1}}}+\frac{1-{{p}_{1}}}{E+{{p}_{2}}}. \\ \end{array}\end{eqnarray} \tag{ 29 }$$

Here p₂ is the second largest probability, hence the inequality. From this, we can find

$$\begin{eqnarray*}&&E\geqslant \frac{1}{2}\left( 1-{{p}_{1}}-{{p}_{2}}+\sqrt{-3p_{1}^{2}+{{(-1+{{p}_{2}})}^{2}}+2{{p}_{1}}(1+{{p}_{2}})} \right)\geqslant 1-{{p}_{1}},\end{eqnarray*}$$

where the rightmost bound is obtained by setting ${{p}_{2}}\equiv {{p}_{1}}$, i.e., loosening the inequality in (29).

Both the upper and the lower bound can be checked to be good, in that they converge to the actual value of ${{E}^{{\rm disturbance}}}$ in both the limit of an unentangled state and a maximally entangled one.

We would like to remark on the difference between calculating (bounds for) the maximal disturbance on one distributed state, and on an ensemble of distributed states. This is because the measurement in the first case can be tailored to the particular state. More concretely, while

$$\begin{eqnarray*}&&{{Q}_{{{D}_{1}},\{{{\Pi }_{A}}\otimes {{\Pi }_{B}}\}}}(\psi )\leqslant \mathop{{\rm max} }\limits_{|{{\phi }_{AB}}\rangle }\mathop{{\rm min} }\limits_{\{{{\Pi }_{A}}\otimes {{\Pi }_{B}}\}}\frac{1}{2}\parallel |\phi \rangle \langle \phi {{|}_{AB}}-({{\Pi }_{A}}\otimes {{\Pi }_{B}})[|\phi \rangle \langle \phi {{|}_{AB}}]{{\parallel }_{1}},\end{eqnarray*}$$

we have (see equation (24))

$$\begin{eqnarray*}&&{{Q}_{{{D}_{1}},\{{{\Pi }_{A}}\otimes {{\Pi }_{B}}\}}}(\mathcal{E})\leqslant \mathop{{\rm min} }\limits_{\{{{\Pi }_{A}}\otimes {{\Pi }_{B}}\}}\mathop{{\rm max} }\limits_{|{{\phi }_{AB}}\rangle }\frac{1}{2}\parallel |\phi \rangle \langle \phi {{|}_{AB}}-({{\Pi }_{A}}\otimes {{\Pi }_{B}})[|\phi \rangle \langle \phi {{|}_{AB}}]{{\parallel }_{1}}.\end{eqnarray*}$$

In particular, theorem 3.3 implies

$$\begin{eqnarray}&&{{Q}_{{{D}_{1}},\{{{\Pi }_{A}}\otimes {{\Pi }_{B}}\}}}(\psi )\leqslant 1-1/{{d}_{A}},\end{eqnarray} \tag{ 30 }$$

to be compared with the ensemble bound (35) of corollary 3.6 below, here reported for the convenience of the reader:

$$\begin{eqnarray*}&&{{Q}_{{{D}_{1}},\{{{\Pi }_{A}}\otimes {{\Pi }_{B}}\}}}(\mathcal{E})\leqslant 1-1/({{d}_{A}}{{d}_{B}}).\end{eqnarray*}$$

Nonetheless, as anticipated, in our quest for bounds for the quantumness of ensembles, we will be able to take advantage of the results and techniques developed in section 3.1.

To begin, we remark how in the proof of lemma 3.1 we used a decomposition $|\psi \rangle ={{\sum }_{i}}|i\rangle |{{w}_{i}}\rangle$; this was to analyze projective measurements of the first system in the basis $\{|i\rangle \}$, with the $|{{w}_{i}}\rangle$ʼs neither orthonormal nor normalized. It is easy to convince oneself that all the calculations we did remain valid in the case of a single system, as long as we interpret the $|{{w}_{i}}\rangle$ʼs as complex numbers. Therefore, we find

Theorem 3.4. In the case of a single-system pure state ψ, the disturbance caused by a complete projective measurement Π in the orthonormal basis $\{|i\rangle \}$, as measured by the trace distance is given by the positive c such that

$$\begin{eqnarray}&&\mathop{\sum }\limits_{i}\frac{{{p}_{i}}}{c+{{p}_{i}}}=1.\end{eqnarray} \tag{ 31 }$$

for ${{p}_{i}}=|\langle i|\psi \rangle {{|}^{2}}$.

We are thus able to find a bound on the quantumness of ensembles, based on the maximum disturbance caused by a fixed projective measurement.

Corollary 3.5. The maximum disturbance of one state under a von Neumann measurement in dimension d, minimized over all measurements and maximized over all states, is given by:

$$\begin{eqnarray}&&\begin{array}{l} \mathop{{\rm min} }\limits_{\{|i\rangle \}_{i=1}^{d}}\ \mathop{{\rm max} }\limits_{\rho }\frac{1}{2}\left\|\rho -\mathop{\sum }\limits_{i=1}^{d}\left( |i\rangle \langle i|\rho |i\rangle \langle i| \right)\right \|{{}_{1}} \\ \quad =\ \mathop{{\rm max} }\limits_{\rho }\frac{1}{2}\left\|\rho -\mathop{\sum }\limits_{i=1}^{d}\left( |i\rangle \langle i|\rho |i\rangle \langle i| \right)\right \|{{}_{1}}=1-\frac{1}{d}, \\ \end{array}\end{eqnarray} \tag{ 32 }$$

where $\{|i\rangle \}_{i=1}^{d}$ denotes a general orthonormal basis spanning the space. Thus,

$$\begin{eqnarray}&&{{Q}_{{{D}_{1}},\{\Pi \}}}(\mathcal{E})\leqslant 1-\frac{1}{d}.\end{eqnarray} \tag{ 33 }$$

Proof. The maximum is attained by a pure state because of the convexity of the trace-norm. From theorem 3.4 and the Schur concavity of c as a function of the probabilities $\{{{p}_{i}}\}$ (lemma 3.2), it is clear that the maximum is attained for the flat probability distribution ${{p}_{i}}=1/d$, which can be obtained by measuring the state $(\sum _{i=1}^{d}|i\rangle )/\sqrt{d}$ in the basis $\{|i\rangle \}$.

Thus, as one may expect, the worst disturbance is obtained by considering a pure state and a projective measurement in a basis that is unbiased with respect to that state, so that $\parallel |\psi \rangle \langle \psi |-\Pi [|\Psi \rangle \langle \Psi |]{{\parallel }_{1}}=\parallel |\psi \rangle \langle \psi |-\mathbb{1} /d{{\parallel }_{1}}=2(1-1/d)$.

In the bipartite case, the upper bound can be achieved even without entanglement, by local projective measurements acting on appropriately 'skewed' local pure states. We thus have

Corollary 3.6. The maximum disturbance of one bipartite state under local complete von Neumann measurements in local dimensions d_A and d_B is given by:

$$\begin{eqnarray}&&\begin{array}{l} \mathop{{\rm min} }\limits_{{{\{|i\rangle }_{A}}\}_{i=1}^{{{d}_{A}}},{{\{|j\rangle }_{B}}\}_{i=1}^{{{d}_{B}}}}\mathop{{\rm max} }\limits_{{{\rho }_{AB}}}\frac{1}{2}\left\|\rho -\mathop{\sum }\limits_{i=1}^{{{d}_{A}}}\mathop{\sum }\limits_{j=1}^{{{d}_{B}}}(|i\rangle \langle i{{|}_{A}}\otimes |j\rangle \langle j{{|}_{B}}){{\rho }_{AB}}(|i\rangle \langle i{{|}_{A}}\otimes |j\rangle \langle j{{|}_{B}})\right \|{{}_{1}} \\ \quad =1-\frac{1}{{{d}_{A}}{{d}_{B}}}. \\ \end{array}\end{eqnarray} \tag{ 34 }$$

Thus,

$$\begin{eqnarray}&&{{Q}_{{{D}_{1}},\{{{\Pi }_{A}}\otimes {{\Pi }_{B}}\}}}(\mathcal{E})\leqslant 1-\frac{1}{{{d}_{A}}{{d}_{B}}}.\end{eqnarray} \tag{ 35 }$$

These results can be generalized to the multipartite case, and one can cast the bound on disturbance in the following general way.

Theorem 3.7. Consider a composite system ${{A}_{1}}{{A}_{2}}\ldots {{A}_{n}}$, with local dimensions ${{d}_{1}},{{d}_{2}},\ldots ,{{d}_{n}}$. The maximum disturbance under complete projective measurements on ${{A}_{{{k}_{1}}}}{{A}_{{{k}_{2}}}}\ldots {{A}_{{{k}_{m}}}}$, with $\{{{k}_{1}},{{k}_{2}},\ldots ,{{k}_{m}}\}\subseteq \{1,2,\ldots ,n\}$ is given by:

$$\begin{eqnarray}&&\mathop{{\rm min} }\limits_{{{\Pi }_{{{A}_{{{k}_{1}}}}{{A}_{{{k}_{2}}}}\ldots {{A}_{{{k}_{m}}}}}}}\ \mathop{{\rm max} }\limits_{{{\rho }_{{{A}_{1}}{{A}_{2}}\ldots {{A}_{n}}}}}\frac{1}{2}\left\|{{\rho }_{{{A}_{1}}{{A}_{2}}\ldots {{A}_{n}}}}-{{\Pi }_{{{A}_{{{k}_{1}}}}{{A}_{{{k}_{2}}}}\ldots {{A}_{{{k}_{m}}}}}}\left[ {{\rho }_{{{A}_{1}}{{A}_{2}}\ldots {{A}_{n}}}} \right]\right \|{{}_{1}}=1-\frac{1}{{{d}_{{{A}_{{{k}_{1}}}}}}{{d}_{{{A}_{{{k}_{2}}}}}}\cdots {{d}_{{{A}_{{{k}_{m}}}}}}},\end{eqnarray} \tag{ 36 }$$

independently of whether the minimization is over arbitrary complete projective measurements on ${{A}_{{{k}_{1}}}}{{A}_{{{k}_{2}}}}\ldots {{A}_{{{k}_{m}}}}$ or over local—with respect to an arbitrary grouping of ${{A}_{{{k}_{1}}}}{{A}_{{{k}_{2}}}}\ldots {{A}_{{{k}_{m}}}}$—ones. Thus,

$$\begin{eqnarray}&&{{Q}_{{{D}_{1}},\{{{\Pi }_{{{A}_{{{k}_{1}}}}{{A}_{{{k}_{2}}}}\ldots {{A}_{{{k}_{m}}}}}}\}}}(\mathcal{E})\leqslant 1-\frac{1}{{{d}_{{{A}_{{{k}_{1}}}}}}{{d}_{{{A}_{{{k}_{2}}}}}}\cdots {{d}_{{{A}_{{{k}_{m}}}}}}}.\end{eqnarray} \tag{ 37 }$$

We notice that it is possible to tighten all the above bounds on disturbance if one takes also into account the probabilities of the various states in the ensemble. In particular, using a consideration similar to the one used to bound the relative entropy of quantumness of classical-quantum states in (Gharibian et al 2011), one can derive the following.

Theorem 3.8. Consider a composite system ${{A}_{1}}{{A}_{2}}\ldots {{A}_{n}}$, with local dimensions ${{d}_{1}},{{d}_{2}},\ldots ,{{d}_{n}}$. Suppose $\mathcal{E}$ is an ensemble comprising, with probability q, a state $\rho ={{\rho }_{{{A}_{1}}{{A}_{2}}\ldots {{A}_{n}}}}$ which is classical on the individual systems ${{A}_{{{k}_{1}}}}{{A}_{{{k}_{2}}}}\ldots {{A}_{{{k}_{m}}}}$; then

$$\begin{eqnarray*}&&{{Q}_{{{D}_{1}},{{\Pi }_{{{A}_{{{k}_{1}}}}{{A}_{{{k}_{2}}}}\ldots {{A}_{{{k}_{m}}}}}}}}(\mathcal{E})\leqslant (1-q)(1-\frac{1}{{{d}_{{{A}_{{{k}_{1}}}}}}{{d}_{{{A}_{{{k}_{2}}}}}}\cdots {{d}_{{{A}_{{{k}_{m}}}}}}}),\end{eqnarray*}$$

independently of whether the minimization is over arbitrary complete projective measurements or over local ones. In particular, if there are n states in the ensemble and they all are classical on the individual systems ${{A}_{{{k}_{1}}}}{{A}_{{{k}_{2}}}}\ldots {{A}_{{{k}_{m}}}}$, then

$$\begin{eqnarray*}&&{{Q}_{{{D}_{1}},{{\Pi }_{{{A}_{{{k}_{1}}}}{{A}_{{{k}_{2}}}}\ldots {{A}_{{{k}_{m}}}}}}}}(\mathcal{E})\leqslant \left( 1-\frac{1}{n} \right)\left( 1-\frac{1}{{{d}_{{{A}_{{{k}_{1}}}}}}{{d}_{{{A}_{{{k}_{2}}}}}}\cdots {{d}_{{{A}_{{{k}_{m}}}}}}} \right).\end{eqnarray*}$$

Proof. Consider a projection $\bar{\Pi }={{\bar{\Pi }}_{{{A}_{{{k}_{1}}}}{{A}_{{{k}_{2}}}}\ldots {{A}_{{{k}_{m}}}}}}$ that leaves ρ invariant. Without loss of generality, assume ρ is the first state listed in the ensemble. Then

$$\begin{eqnarray}\begin{array}{rcl} {{Q}_{{{D}_{1}},\{{{\Pi }_{{{A}_{{{k}_{1}}}}{{A}_{{{k}_{2}}}}\ldots {{A}_{{{k}_{m}}}}}}\}}}(\mathcal{E}) & = & \mathop{{\rm min} }\limits_{\{\Pi ={{\Pi }_{{{A}_{{{k}_{1}}}}{{A}_{{{k}_{2}}}}\ldots {{A}_{{{k}_{m}}}}}}\}}\mathop{\sum }\limits_{i=1}^{n}{{p}_{i}}{{D}_{1}}({{\rho }_{i}},\Pi [{{\rho }_{i}}]) \\ {} & \leqslant & q{{D}_{1}}(\rho ,\bar{\Pi }[\rho ])+\mathop{\sum }\limits_{i=2}^{n}{{p}_{i}}{{D}_{1}}(\rho ,\bar{\Pi }[\rho ]) \\ {} & \leqslant & \mathop{\sum }\limits_{i=2}^{n}{{p}_{i}}(1-\frac{1}{{{d}_{{{A}_{{{k}_{1}}}}}}{{d}_{{{A}_{{{k}_{2}}}}}}\cdots {{d}_{{{A}_{{{k}_{m}}}}}}}) \\ {} & = & (1-q)(1-\frac{1}{{{d}_{{{A}_{{{k}_{1}}}}}}{{d}_{{{A}_{{{k}_{2}}}}}}\cdots {{d}_{{{A}_{{{k}_{m}}}}}}}). \\ \end{array}\end{eqnarray} \tag{ 38 }$$

The first inequality is due to the choice of a particular projection $\bar{\Pi }$. The second inequality comes from the fact that $\bar{\Pi }$ is such that $\bar{\Pi }[\rho ]=\rho$, so that ${{D}_{1}}(\rho ,\bar{\Pi }[\rho ])=0$, and from the general bound (36), applied to all the other states in the ensemble.

For the second claim, it suffices to notice that if all n states in the ensemble are classical on the individual systems ${{A}_{{{k}_{1}}}}{{A}_{{{k}_{2}}}}\ldots {{A}_{{{k}_{m}}}}$ (possibly in different local orthonormal bases), then at least one of them has the associated probability $q\geqslant 1/n$, because probabilities must sum up to 1.

It is worth recalling that, if one considers a single system, every given state is classical in its eigenbasis. So, as an application of theorem 3.8 we find the following improvement over equation (33):

$$\begin{eqnarray}&&{{Q}_{{{D}_{1}},\{\Pi \}}}(\mathcal{E})\leqslant (1-{{p}_{{\rm max} }})\left( 1-\frac{1}{d} \right)\leqslant \left( 1-\frac{1}{n} \right)\left( 1-\frac{1}{d} \right),\end{eqnarray} \tag{ 39 }$$

where d is the dimension of the system, ${{p}_{{\rm max} }}$ is the largest among all probabilities with which each state appears in the ensemble, and n is the number of elements in the ensemble. Notice that, because of the identity (18), these bounds on the disturbance of ensembles of a single system can be immediately used to bound the disturbance of correlations of $d\times n$ quantum-classical states ${{\rho }_{SX}}$.

In this section, we compute, or provide bounds for, the quantumness of ensembles, both for single systems and for correlations, for some interesting examples. These include qubit ensembles and uniform ensembles of pure states. As we will see, both kinds of examples seem to indicate that the general bound we found are quite good.

3.3.1. Qubits

We start by providing a general formula for the single-system ensemble quantumness of ensembles of qubit states.

Theorem 3.9. The $({{D}_{1}},\{\Pi \})$-quantumness of an ensemble $\mathcal{E}\;:=\;\left\{ ({{p}_{i}},{{\rho }^{(i)}}) \right\}_{i=1}^{n}$ of qubit states is given by

$$\begin{eqnarray}&&{{Q}_{{{D}_{1}},\{\Pi \}}}[\mathcal{E}]=\frac{1}{2}\mathop{{\rm min} }\limits_{\hat{v}\in {{S}^{2}}}\mathop{\sum }\limits_{i=1}^{n}{{p}_{i}}\parallel {{\vec{r}}_{i}}\parallel \left| {\rm sin} \left[ \angle (\hat{v},{{{\vec{r}}}_{i}}) \right] \right|,\end{eqnarray} \tag{ 40 }$$

where the minimization is performed over all vectors $\hat{v}$ on the Bloch sphere, ${{\vec{r}}_{i}}$ is the Bloch vector corresponding to ${{\rho }^{(i)}}$, $\parallel \cdot \parallel \equiv \parallel \cdot {{\parallel }_{2}}$ is the Euclidean norm on ${{\mathbb{R}}^{3}}$, $\angle (\hat{v},{{\vec{r}}_{i}})$ is the angle between the two vectors named.

Proof. For some qubit state $\rho =\frac{1}{2}\left(\mathbb{1} +\vec{r}\cdot \vec{\sigma } \right)$, if Π is the projective measurement along a basis corresponding to the unit vector $\hat{v}$, then

$$\begin{eqnarray}\begin{array}{rcl} \Pi [\rho ] & = & \frac{1}{{{2}^{3}}}\left(\mathbb{1} +\hat{v}\cdot \vec{\sigma } \right)\left(\mathbb{1} +\vec{r}\cdot \vec{\sigma } \right)\left(\mathbb{1} +\hat{v}\cdot \vec{\sigma } \right)+\frac{1}{{{2}^{3}}}\left(\mathbb{1} -\hat{v}\cdot \vec{\sigma } \right)\left(\mathbb{1} +\vec{r}\cdot \vec{\sigma } \right)\left(\mathbb{1} -\hat{v}\cdot \vec{\sigma } \right) \\ {} & = & \frac{1}{2}\left(\mathbb{1} +(\hat{v}\cdot \vec{r})\hat{v}\cdot \vec{\sigma } \right). \\ \end{array}\end{eqnarray} \tag{ 41 }$$

Therefore,

$$\begin{eqnarray}\begin{array}{rcl} \left\|\rho -\Pi [\rho ]\right \|{{}_{1}} & = & \left\|\frac{1}{2}\left(\mathbb{1} +\vec{r}\cdot \vec{\sigma }-\mathbb{1} -(\hat{v}\cdot \vec{r})\hat{v}\cdot \vec{\sigma } \right)\right \|{{}_{1}} \\ {} & = & \frac{1}{2}\left\|\left( \vec{r}-(\hat{v}\cdot \vec{r})\hat{v} \right)\cdot \vec{\sigma }\right \|{{}_{1}} \\ {} & = & \left\|\vec{r}-(\hat{v}\cdot \vec{r})\hat{v}\right\| \\ {} & = & \parallel \vec{r}\parallel \left| {\rm sin} \left[ (\hat{v},\vec{r}) \right] \right|. \\ \end{array}\end{eqnarray} \tag{ 42 }$$

The result follows by the definition of ${{Q}_{{{D}_{1}},\{\Pi \}}}$.

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In the case where the qubit ensemble contains two elements, we are able to provide an explicit analytical formula .

Corollary 3.10. In the case of an ensemble consisting of two one-qubit states (the case n = 2 in theorem 3.9), the minimum comes out to be

$$\begin{eqnarray}&&{{Q}_{{{D}_{1}},\{\Pi \}}}\left[ \left\{ (p,{{\rho }^{(1)}}),(1-p,{{\rho }^{(2)}}) \right\} \right]=\frac{1}{2}\left| {\rm sin} \left[ \angle ({{{\vec{r}}}_{1}},{{{\vec{r}}}_{2}}) \right] \right|{\rm min} \left[ p\parallel {{{\vec{r}}}_{1}}\parallel ,(1-p)\parallel {{{\vec{r}}}_{2}}\parallel \right].\end{eqnarray} \tag{ 43 }$$

Proof. It is clear that we should choose $\hat{v}$ to lie in the plane defined by ${{\vec{r}}_{1}}$ and ${{\vec{r}}_{2}}$. This is because, for a fixed angle between $\hat{v}$ and ${{\vec{r}}_{1}}$, the smallest angle between $\hat{v}$ and ${{\vec{r}}_{2}}$ is achieved for $\hat{v}$ lying in such a plane. Having reduced the problem to a two-dimensional one, we can prove the claim by simply considering figure 2.

In terms of the geometric elements present in figure 2, our objective function can be recast as

$$\begin{eqnarray*}\begin{array}{rcl} {{Q}_{{{D}_{1}},\{\Pi \}}}\left[ \left\{ (p,{{\rho }^{(1)}}),(1-p,{{\rho }^{(2)}}) \right\} \right] & = & \frac{1}{2}\mathop{{\rm min} }\limits_{\hat{v}\in {{S}^{2}}}\left( {{p}_{1}}\parallel {{{\vec{r}}}_{1}}\parallel \left| {\rm sin} \left[ \angle (\hat{v},{{{\vec{r}}}_{1}}) \right] \right| \right. \\ {} & {} & +\left. {{p}_{2}}\parallel {{{\vec{r}}}_{2}}\parallel \left| {\rm sin} \left[ \angle (\hat{v},{{{\vec{r}}}_{1}}) \right] \right| \right) \\ \end{array}\end{eqnarray*}$$

$$\begin{eqnarray*}\begin{array}{cl} {} & = & \mathop{{\rm min} }\limits_{\hat{v}\in {{S}^{2}}}\frac{1}{2}(d({{R}_{1}},{{T}_{1}})+d({{R}_{2}},{{T}_{2}})), \\ \end{array}\end{eqnarray*}$$

where we have used the notation $d(X,Y)$ to denote the Euclidean distance between two points X and Y. We now notice that $\frac{1}{2}d({{R}_{1}},{{T}_{1}})d(O,P)$ is the area of the triangle OPR₁. Similarly, $\frac{1}{2}d({{R}_{2}},{{T}_{2}})d(O,P)$ is the area of the triangle OPR₂. The sum of the two areas gives the area of the triangle $O{{R}_{1}}{{R}_{2}}$, independently of the position of P, i.e., independently of the choice of $\vec{v}$ in the plane. So

$$\begin{eqnarray*}&&\frac{1}{2}d({{R}_{1}},{{T}_{1}})d(O,P)+\frac{1}{2}d({{R}_{2}},{{T}_{2}})d(O,P)=\frac{1}{2}(d({{R}_{1}},{{T}_{1}})+d({{R}_{2}},{{T}_{2}}))d(O,P)={\rm const}.\end{eqnarray*}$$

Thus, it is clear that $\hat{v}$ should be chosen to maximize $d(O,P)$. This can be done by choosing the measurement axis $\hat{v}$ parallel to the longest ${{p}_{i}}{{\vec{r}}_{i}}$. In such a case $\frac{1}{2}(d({{R}_{1}},{{T}_{1}})+d({{R}_{2}},{{T}_{2}}))=\frac{1}{2}\left| {\rm sin} \left[ \angle ({{{\vec{r}}}_{1}},{{{\vec{r}}}_{2}}) \right] \right|{\rm min} \left[ p\parallel {{{\vec{r}}}_{1}}\parallel ,(1-p)\parallel {{{\vec{r}}}_{2}}\parallel \right]$.

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Via the relations (18), which equate the quantumness of correlations of a classical-quantum system to the quantumness of a single-system ensemble, we conclude that the classical-quantum state of two qubits that exhibits the largest one-sided quantumness of correlations, as measured by trace-distance disturbance, is, up to local unitaries, the state

$$\begin{eqnarray}&&\frac{1}{2}|0\rangle \langle 0|\otimes |0\rangle \langle 0|+\frac{1}{2}|+\rangle \langle +|\otimes |1\rangle \langle 1|,\end{eqnarray} \tag{ 44 }$$

for which ${{Q}_{{{D}_{1}},{{\Pi }_{A}}}}=1/4$. It is worth remarking that this quantumness matches the upper bound (39). The same state (44) is the classical-quantum state exhibiting the largest quantumness of correlations also according to the entropic disturbance (Gharibian et al 2011).

3.3.2. Uniform ensembles

In this section, we will consider uniform ensembles of pure states, that is, ensembles ${{\mathcal{E}}_{{\rm Haar}}}$ of pure states distributed according to the Haar measure (Bengtsson and Życzkowski 1996). We will begin with the calculation of the trace-distance single-system ensemble quantumness, ${{Q}_{{{D}_{1}},\{\Pi \}}}$, and move later to the trace-distance ensemble quantumness of correlations, both one-sided, ${{Q}_{{{D}_{1}},\{{{\Pi }_{A}}\}}}$, and two-sided, ${{Q}_{{{D}_{1}},\{{{\Pi }_{A}}\otimes {{\Pi }_{B}}\}}}$.

By symmetry considerations⁹ , it is clear that the choice of basis for the measurement is irrelevant: the average disturbance is going to be the same in all local bases. So

$$\begin{eqnarray*}&&{{Q}_{{{D}_{1}},\{\Pi \}}}({{\mathcal{E}}_{{\rm Haar}}})=\mathop{{\rm min} }\limits_{\Pi }\int {\rm d}\psi \frac{1}{2}\parallel |\psi \rangle \langle \psi |-\Pi [|\psi \rangle \langle \psi |]{{\parallel }_{1}}=\int {\rm d}\psi \frac{1}{2}\parallel |\psi \rangle \langle \psi |-\Pi [|\psi \rangle \langle \psi |]{{\parallel }_{1}},\end{eqnarray*}$$

where the projection on the rightmost-hand side is fixed arbitrarily. We find

$$\begin{eqnarray*}\begin{array}{rcl} \int {\rm d}\psi \frac{1}{2}\parallel |\psi \rangle \langle \psi |-\Pi [|\psi \rangle \langle \psi |]{{\parallel }_{1}} & \geqslant & \int {\rm d}\psi (1-\langle \psi |\Pi [|\psi \rangle \langle \psi |]|\psi \rangle )) \\ {} & = & 1-\int {\rm d}\psi {\rm Tr}(\Pi [|\psi \rangle {{\langle \psi |]}^{2}}) \\ {} & = & 1-\int {\rm d}\psi {\rm Tr}\left( (\Pi [|\psi \rangle \langle \psi |]\otimes \Pi [|\psi \rangle \langle \psi |])W \right) \\ \end{array}\end{eqnarray*}$$

$$\begin{eqnarray*}\begin{array}{clc} {} & = & 1-\int {\rm d}\psi {\rm Tr}\left( (|\psi \rangle \langle \psi |\otimes |\psi \rangle \langle \psi |)\left( (\Pi \otimes \Pi )[V] \right) \right) \\ {} & = & 1-{\rm Tr}\left( \left( \int {\rm d}\psi |\psi \rangle \langle \psi |\otimes |\psi \rangle \langle \psi | \right)(\Pi \otimes \Pi )[W] \right) \\ {} & = & 1-{\rm Tr}\left( \frac{\mathbb{1} +W}{d(d+1)}(\Pi \otimes \Pi )[W] \right) \\ {} & = & 1-{\rm Tr}\left( \frac{\mathbb{1} +W}{d(d+1)}\mathop{\sum }\limits_{i}|i\rangle \langle i|\otimes |i\rangle \langle i| \right) \\ {} & = & 1-\frac{2d}{d(d+1)} \\ {} & = & 1-\frac{2}{d+1}. \\ \end{array}\end{eqnarray*}$$

In the above, we have introduced the swap operator W acting on two copies of the Hilbert space according to $W|\alpha \rangle |\beta \rangle =|\beta \rangle |\alpha \rangle$. For any arbitrary choice of orthonormal basis $\{|i\rangle \}$, W admits the decomposition $W={{\sum }_{i,j}}|i\rangle \langle j|\otimes |j\rangle \langle i|$. We used two useful standard identities involving W: ${\rm T}{{{\rm r}}_{1,2}}(({{X}_{1}}\otimes {{Y}_{2}}){{W}_{1,2}})={\rm Tr}(XY)$ and $\int {\rm d}\psi |\psi \rangle \langle \psi |\otimes |\psi \rangle \langle \psi |=\frac{\mathbb{1} +W}{d(d+1)}$ ¹⁰ . On the other hand,

$$\begin{eqnarray*}\begin{array}{rcl} {{Q}_{{{D}_{1}},\{\Pi \}}}({{\mathcal{E}}_{{\rm Haar}}}) & \leqslant & \int {\rm d}\psi \sqrt{1-\langle \psi |\Pi [|\psi \rangle \langle \psi |]|\psi \rangle } \\ {} & \leqslant & \sqrt{1-\int {\rm d}\psi \langle \psi |\Pi [|\psi \rangle \langle \psi |]|\psi \rangle } \\ {} & = & \sqrt{1-\frac{2}{d+1}} \\ {} & = & 1-\frac{1}{d+1}+O({{d}^{-2}}) \\ \end{array}\end{eqnarray*}$$

In corollary 3.5, we had found that the greatest trace-distance disturbance, maximized over states, is equal to $1-1/d$ in dimension d, and used the same value to bound the (single-system) ensemble disturbance. From the calculations above we see that the ensemble disturbance over the Haar ensemble is essentially the maximal one.

We move now to the one-sided trace-distance quantumness of correlations, ${{Q}_{{{D}_{1}},\{{{\Pi }_{A}}\}}}$. We can actually follow several of the steps above to arrive to

$$\begin{eqnarray*}&&\begin{array}{l} {{Q}_{{{D}_{1}},\{{{\Pi }_{A}}\}}}({{\mathcal{E}}_{{\rm Haar}}})\quad \geqslant 1-{\rm Tr}\left( \frac{{{\mathbb{1} }_{{{A}_{1}}{{A}_{2}}{{B}_{1}}{{B}_{2}}}}+{{W}_{{{A}_{1}}{{B}_{1}}:{{A}_{2}}{{B}_{2}}}}}{{{d}_{AB}}({{d}_{AB}}+1)}({{\Pi }_{{{A}_{1}}}}\otimes {{\Pi }_{{{A}_{2}}}})[{{W}_{{{A}_{1}}{{B}_{1}}:{{A}_{2}}{{B}_{2}}}}] \right) \\ \quad =1-{\rm Tr}\left( \frac{{{\mathbb{1} }_{{{A}_{1}}{{A}_{2}}{{B}_{1}}{{B}_{2}}}}+{{W}_{{{A}_{1}}:{{A}_{2}}}}\otimes {{W}_{{{B}_{1}}:{{B}_{2}}}}}{{{d}_{AB}}({{d}_{AB}}+1)}({{\Pi }_{{{A}_{1}}}}\otimes {{\Pi }_{{{A}_{2}}}})[{{W}_{{{A}_{1}}:{{A}_{2}}}}\otimes {{W}_{{{B}_{1}}:{{B}_{2}}}}] \right) \\ \end{array}\end{eqnarray*}$$

$$\begin{eqnarray*}&&\begin{array}{l} \quad =1-{\rm Tr}\left( \frac{{{\mathbb{1} }_{{{A}_{1}}{{A}_{2}}{{B}_{1}}{{B}_{2}}}}+{{W}_{{{A}_{1}}:{{A}_{2}}}}\otimes {{W}_{{{B}_{1}}:{{B}_{2}}}}}{{{d}_{AB}}({{d}_{AB}}+1)}\left( \mathop{\sum }\limits_{i}|i\rangle \langle i{{|}_{{{A}_{1}}}}\otimes |i\rangle \langle i{{|}_{{{A}_{2}}}} \right)\otimes {{W}_{{{B}_{1}}:{{B}_{2}}}} \right) \\ \quad =1-\frac{{{d}_{A}}{{d}_{B}}+{{d}_{A}}d_{B}^{2}}{{{d}_{AB}}({{d}_{AB}}+1)} \\ \quad =1-\frac{{{d}_{B}}+1}{{{d}_{A}}{{d}_{B}}+1}. \\ \end{array}\end{eqnarray*}$$

In the derivation, we have used ${{d}_{AB}}={{d}_{A}}{{d}_{B}}$ and that the swap operator between two copies ${{A}_{1}}{{B}_{1}}$ and ${{A}_{2}}{{B}_{2}}$ of $AB$ can be written as the product of the swaps of the copies of the subsystems, i.e., ${{W}_{{{A}_{1}}{{B}_{1}}:{{A}_{2}}{{B}_{2}}}}={{W}_{{{A}_{1}}:{{A}_{2}}}}\otimes {{W}_{{{B}_{1}}:{{B}_{2}}}}$. Similarly as before, we also find the upper bound

$$\begin{eqnarray*}&&{{Q}_{{{D}_{1}},\{{{\Pi }_{A}}\}}}({{\mathcal{E}}_{{\rm Haar}}})\leqslant \sqrt{1-\frac{{{d}_{B}}+1}{{{d}_{A}}{{d}_{B}}+1}}\end{eqnarray*}$$

Notice also that in this case the average disturbance is comparable with the maximal disturbance for local projective measurements that we computed, $1-1/{{d}_{A}}$.

Finally, it should be clear from the steps in the proof above that, when we consider the two-sided ensemble quantumness of correlations, we go back to the bounds that we obtained for a single system, just with the dimension equal to the total dimension, $d={{d}_{AB}}={{d}_{A}}{{d}_{B}}$, obtaining

$$\begin{eqnarray*}&&1-\frac{2}{{{d}_{A}}{{d}_{B}}+1}\leqslant {{Q}_{{{D}_{1}},\{{{\Pi }_{A}}\otimes {{\Pi }_{B}}\}}}({{\mathcal{E}}_{{\rm Haar}}})\leqslant \sqrt{1-\frac{2}{{{d}_{A}}{{d}_{B}}+1}}.\end{eqnarray*}$$

Notice that this proves that a bound like (30) cannot hold in the case of ensembles, even if it does for single states, and that the dependence on the total dimension of the ensemble bound (35) is optimal up to constant factors, at least asymptotically, i.e., for large dimensions.

Here, we prove a theorem on the limitations of hiding using classical states. For this, we make use of the following observation.

Lemma 4.2. For any two normalized density operators ρ and σ on some Hilbert space $\mathcal{H},$

$$\begin{eqnarray}&&\frac{1}{{\rm min} \{R(\rho ),R(\sigma )\}}{{F}^{2}}(\rho ,\sigma )\leqslant {\rm Tr}(\rho \sigma )\leqslant {{F}^{2}}(\rho ,\sigma ),\end{eqnarray} \tag{ 47 }$$

where $F(\rho ,\sigma )\;:=\;{\rm Tr}\sqrt{\sqrt{\rho }\sigma \sqrt{\rho }}$ is the fidelity of the two states and $R(\rho )$ and $R(\sigma )$ are the ranks of ρ and σ, respectively.

Proof. We will use the fact that for any matrix X it holds

$$\begin{eqnarray}&&\parallel X{{\parallel }_{2}}\leqslant \parallel X{{\parallel }_{1}}\leqslant \sqrt{R(X)}\parallel X{{\parallel }_{2}},\end{eqnarray} \tag{ 48 }$$

with $\parallel X{{\parallel }_{1}}={\rm Tr}\sqrt{{{X}^{\dagger }}X}$ the 1-norm (also known as trace norm) of X and $\parallel X{{\parallel }_{2}}=\sqrt{{\rm Tr}({{X}^{\dagger }}X)}$ the 2-norm (also known as Hilbert–Schmidt norm) of X. Then the lemma is proved by simply considering $X=\sqrt{\rho }\sqrt{\sigma }$, since $F(\rho ,\sigma )=\parallel \sqrt{\rho }\sqrt{\sigma }{{\parallel }_{1}}$ and ${\rm Tr}(\rho \sigma )=\parallel \sqrt{\rho }\sqrt{\sigma }\parallel _{2}^{2}$, and by noting that $R(YZ)\leqslant {\rm min} \{R(Y),R(Z)\}$, while $R(\sqrt{\rho })=R(\rho )$ (similarly for σ).

We are now ready to prove the theorem.

Theorem 4.3. If a classical-quantum (w.r.t. the partition $A:B$) bipartite state ${{\rho }_{AB}}={{\sum }_{i}}{{p}_{i}}|i\rangle \langle i|\otimes {{\rho }_{i}}$ is -distinguishable from another bipartite state ${{\sigma }_{AB}}$ under global operations, i.e.,

$$\begin{eqnarray}&&\frac{1}{2}\parallel {{\rho }_{AB}}-{{\sigma }_{AB}}{{\parallel }_{1}}\geqslant 1-\epsilon ,\end{eqnarray} \tag{ 49 }$$

and $R={\rm min} \{R(\rho ),R(\tilde{\sigma })\}\leqslant R(\rho )$ is the lesser of the ranks of ρ and $\tilde{\sigma }$, with $\tilde{\sigma }={{\sum }_{i}}|i\rangle \langle i{{|}_{A}}\sigma |i\rangle \langle i{{|}_{A}}$, then ρ is at least $\sqrt{2R\epsilon }$-distinguishable from ${{\sigma }_{AB}}$ under ${\rm LOCC}$ w.r.t. $A:B$, i.e.,

$$\begin{eqnarray*}&&\frac{1}{2}\parallel {{\rho }_{AB}}-{{\sigma }_{AB}}{{\parallel }_{{\rm LOCC}}}\geqslant 1-\sqrt{2R\epsilon }.\end{eqnarray*}$$

Proof. Besides lemma 4.2, we will make use of the well-known relation (Nielsen and Chuang 2000)

$$\begin{eqnarray}&&1-F(\rho ,\sigma )\leqslant D(\rho ,\sigma )\leqslant \sqrt{1-F{{(\rho ,\sigma )}^{2}}}.\end{eqnarray} \tag{ 50 }$$

The claim can be proved through the following steps:

$$\begin{eqnarray*}\begin{array}{rcl} \frac{1}{2}\parallel \rho -\sigma {{\parallel }_{{\rm LOCC}}} & \mathop{\geqslant }\limits^{(i)} & \frac{1}{2}\parallel \rho -\tilde{\sigma }{{\parallel }_{1}} \\ {} & \mathop{\geqslant }\limits^{(ii)} & 1-F(\rho ,\tilde{\sigma }) \\ {} & \mathop{\geqslant }\limits^{(iii)} & 1-\sqrt{R}\sqrt{{\rm Tr}(\rho \tilde{\sigma })} \\ {} & \mathop{=}\limits^{(iv)} & 1-\sqrt{R}\sqrt{{\rm Tr}(\rho \sigma )} \\ {} & \mathop{\geqslant }\limits^{(v)} & 1-\sqrt{R}F(\rho ,\sigma ) \\ {} & \mathop{\geqslant }\limits^{(vi)} & 1-\sqrt{R}\sqrt{1-D{{(\rho ,\sigma )}^{2}}} \\ \end{array}\end{eqnarray*}$$

$$\begin{eqnarray}\begin{array}{cl} {} & \mathop{\geqslant }\limits^{(vii)} & 1-\sqrt{2R\epsilon } \\ \end{array}\end{eqnarray} \tag{ 51 }$$

The steps are justified as follows: $(i)$ holds because a possible LOCC strategy is the one-way communication one with the first step consisting in measuring in the classical basis (for ρ) of A; $(ii)$ and $(vi)$ are due to equation (50); $(iii)$ and $(v)$ hold because of equation (47); $(iv)$ holds because of the cyclic property of the trace; $(vii)$ holds because of hypothesis equation (49).

Our result points out how, with the use of a classical state, it is impossible to have a hiding scheme with $\epsilon =0$. Indeed, our bound implies that perfect distinguishability ($\epsilon =0$) by global operations implies perfect distinguishability ($\delta =1$) also by LOCC. On the other hand, it is known that there exist good hiding schemes with perfect distinguishability; they necessarily make use of non-classical states. We will consider such a case in the example below.

We remark that our result just puts limits on a hiding scheme that uses at least one classical state, but hiding schemes that make use of at least a classical state do exist. For example, (Hayden et al 2004) provide an example of a hiding pair in ${{\mathbb{C}}^{d}}\otimes {{\mathbb{C}}^{d}}$, where one of the states is simply the maximally mixed state, $\mathbb{1} /{{d}^{2}}$, and the other state is the result of the action of an approximately randomizing random-unitary map $\mathcal{R}$ on part of a maximally entangled, i.e., the state $\mathcal{R}\otimes {\sf id} [{{\psi }^{+}}]$, with

$$\begin{eqnarray*}&&\mathcal{R}[\rho ]=\frac{1}{n}\mathop{\sum }\limits_{i=1}^{n}{{U}_{i}}\rho U_{i}^{\dagger },\ \end{eqnarray*}$$

satisfying

$$\begin{eqnarray*}&&\left\|\rho -\frac{\mathbb{1} }{d}\right \|{{}_{\infty }}\leqslant \frac{\eta }{d}.\end{eqnarray*}$$

As proven in (Aubrun 2009), improving on (Hayden et al 2004), this can be achieved with independent Haar-random unitaries U_i and $n\geqslant Cd/{{\eta }^{2}}$, with C a constant. It is immediate to check that such a scheme achieves $\delta \leqslant \eta /2$ and $\epsilon \leqslant n/{{d}^{2}}\approx C/(d{{\eta }^{2}})$.

We have seen that the classicality of one of the hiding states does not prevent the hiding scheme from working, although it puts some 'quality' constraints on it. With the next theorem, we will see that the two states must nonetheless display a large ensemble quantumness of correlations.

Theorem 4.4. The hiding capability of a pair of bipartite states is bounded above by the $(\parallel \cdot {{\parallel }_{1}},\{{{\Pi }_{A}}\})$-quantumness of the ensemble consisting of the two states with equal weights:

$$\begin{eqnarray}&&{{\Delta }_{{\rm H}}}[{{\rho }_{AB}},{{\sigma }_{AB}}]\leqslant 2\;{{Q}_{\parallel \cdot {{\parallel }_{1}},\{{{\Pi }_{A}}\}}}\left[ \left\{ \left( \frac{1}{2},{{\rho }_{AB}} \right),\left( \frac{1}{2},{{\sigma }_{AB}} \right) \right\} \right].\end{eqnarray} \tag{ 52 }$$

Proof. By definition,

$$\begin{eqnarray*}\begin{array}{rcl} {{\Delta }_{{\rm H}}}[{{\rho }_{AB}},{{\sigma }_{AB}}] & = & \frac{1}{2}\left( \left\|{{\rho }_{AB}}-{{\sigma }_{AB}}\right \|{{}_{1}}-\left\|{{\rho }_{AB}}-{{\sigma }_{AB}}\right \|{{}_{{\rm LOCC}}} \right) \\ \end{array}\end{eqnarray*}$$

$$\begin{eqnarray}\begin{array}{clc} {} & = & \frac{1}{2}\left( \left\|{{\rho }_{AB}}-{{\sigma }_{AB}}\right \|{{}_{1}}-\mathop{{\rm max} }\limits_{\mathcal{M}\in {\rm LOCC}}\left\|\mathcal{M}[{{\rho }_{AB}}]-\mathcal{M}[{{\sigma }_{AB}}]\right \|{{}_{1}} \right) \\ {} & \leqslant & \frac{1}{2}\left( \left\|{{\rho }_{AB}}-{{\sigma }_{AB}}\right \|{{}_{1}}-\mathop{{\rm max} }\limits_{{{\Pi }_{A}}}\left\|{{\Pi }_{A}}[{{\rho }_{AB}}]-{{\Pi }_{A}}[{{\sigma }_{AB}}]\right \|{{}_{1}} \right) \\ {} & = & \frac{1}{2}\mathop{{\rm min} }\limits_{{{\Pi }_{A}}}\left( \left\|{{\rho }_{AB}}-{{\sigma }_{AB}}\right \|{{}_{1}}-\left\|{{\Pi }_{A}}[{{\rho }_{AB}}]-{{\Pi }_{A}}[{{\sigma }_{AB}}]\right \|{{}_{1}} \right) \\ {} & \leqslant & \frac{1}{2}\mathop{{\rm min} }\limits_{{{\Pi }_{A}}}\left( \left\|{{\rho }_{AB}}-{{\Pi }_{A}}[{{\rho }_{AB}}]\right \|{{}_{1}} \right. \\ {} & {} & +\left\|{{\Pi }_{A}}[{{\rho }_{AB}}]-{{\Pi }_{A}}[{{\sigma }_{AB}}]\right \|{{}_{1}}+\left\|{{\Pi }_{A}}[{{\sigma }_{AB}}]-{{\sigma }_{AB}}\right \|{{}_{1}} \\ {} & {} & -\left. \left\|{{\Pi }_{A}}[{{\rho }_{AB}}]-{{\Pi }_{A}}[{{\sigma }_{AB}}]\right \|{{}_{1}} \right) \\ {} & = & 2\;{{Q}_{\parallel \,\cdot \,{{\parallel }_{1}},\{{{\Pi }_{A}}\}}}\left[ \left\{ \left( \frac{1}{2},{{\rho }_{AB}} \right),\left( \frac{1}{2},{{\sigma }_{AB}} \right) \right\} \right]. \\ \end{array}\end{eqnarray} \tag{ 53 }$$

The first inequality above follows from the fact that one-way LOCC measurements that start with a projective measurement are a subset of all LOCC measurements. The second inequality follows from repeated application of the triangle inequality for the trace norm.

Notice that all the steps above could be adapted to the case of projective local measurements on both A and B, leading to ${{\Delta }_{{\rm H}}}[{{\rho }_{AB}},{{\sigma }_{AB}}]\leqslant {{Q}_{\parallel \,\cdot \,{{\parallel }_{1}},\{{{\Pi }_{A}}\otimes {{\Pi }_{B}}\}}}\left[ \left\{ \left( \frac{1}{2},{{\rho }_{AB}} \right),\left( \frac{1}{2},{{\sigma }_{AB}} \right) \right\} \right]$.

Example: Werner hiding pairs

A well-known hiding pair constituted by the two Werner states,

$$\begin{eqnarray}&&{{\sigma }_{\pm }}\;:=\;\frac{\mathbb{1} \otimes \mathbb{1} \pm W}{d(d\pm 1)},\end{eqnarray} \tag{ 54 }$$

with W, we recall, the swap operator. The states σ₊ and σ₋ are in fact normalized projectors onto the symmetric and antisymmetric subspaces, respectively. They form a hiding pair, with

$$\begin{eqnarray*}&&\frac{1}{2}\parallel {{\sigma }_{+}}-{{\sigma }_{-}}{{\parallel }_{1}}=1,\end{eqnarray*}$$

since they are orthogonal, and

$$\begin{eqnarray}&&\frac{1}{2}\parallel {{\sigma }_{+}}-{{\sigma }_{-}}{{\parallel }_{{\rm LOCC}}}=\frac{2}{d+1}.\end{eqnarray} \tag{ 55 }$$

That equation (55) holds comes from the fact that even positive-under-partial-transposition (PPT) measurements, more general than LOCC, do not do better [see (Matthews et al 2009, Eggeling and Werner 2002, DiVincenzo et al 2002)], and there are LOCC measurements that achieve the bound [interestingly, the bound is achieved exactly via local orthogonal projections, as easily verified¹¹ ]. Therefore, the hiding gap for this pair is exactly

$$\begin{eqnarray}&&{{\Delta }_{{\rm H}}}[{{\sigma }_{+}},{{\sigma }_{-}}]=1-\frac{2}{d+1}=\frac{d-1}{d+1}.\end{eqnarray} \tag{ 56 }$$

We calculate the ensemble quantumness of correlations to be

$$\begin{eqnarray*}&&\begin{array}{l} {{Q}_{\parallel \,\cdot \,{{\parallel }_{1}},\{{{\Pi }_{A}}\}}}\left[ \left\{ \left( \frac{1}{2},{{\sigma }_{+}} \right),\left( \frac{1}{2},{{\sigma }_{-}} \right) \right\} \right]\quad =\mathop{{\rm min} }\limits_{{{\Pi }_{A}}}\frac{1}{4}\left[ (\parallel {{\sigma }_{+}}-{{\Pi }_{A}}[{{\sigma }_{+}}]{{\parallel }_{1}}+\parallel {{\sigma }_{+}}-{{\Pi }_{A}}[{{\sigma }_{+}}]{{\parallel }_{1}}) \right. \\ \quad =\mathop{{\rm min} }\limits_{{{\Pi }_{A}}}\frac{1}{4}\left( \left\|\frac{\mathbb{1} \otimes \mathbb{1} +W}{d(d+1)}-{{\Pi }_{A}}\left[ \frac{\mathbb{1} \otimes \mathbb{1} +W}{d(d+1)} \right]\right \|{{}_{1}}+\left\|\frac{\mathbb{1} \otimes \mathbb{1} -W}{d(d-1)}-{{\Pi }_{A}}\left[ \frac{\mathbb{1} \otimes \mathbb{1} -W}{d(d-1)} \right]\right \|{{}_{1}} \right) \\ \quad =\mathop{{\rm min} }\limits_{{{\Pi }_{A}}}\frac{1}{4}\left( \frac{1}{d(d+1)}\left\|W-{{\Pi }_{A}}[W]\right \|{{}_{1}}+\frac{1}{d(d-1)}\left\|W-{{\Pi }_{A}}[W]\right \|{{}_{1}} \right) \\ \quad =\frac{1}{2({{d}^{2}}-1)}\left\|W-\mathop{\sum }\limits_{i}|i\rangle \langle i|\otimes |i\rangle \langle i|\right \|{{}_{1}} \\ \quad =\frac{1}{2({{d}^{2}}-1)}\left\|\mathbb{1} -\mathop{\sum }\limits_{i}|i\rangle \langle i|\otimes |i\rangle \langle i|\right \|{{}_{1}} \\ \quad =\frac{1}{2({{d}^{2}}-1)}({{d}^{2}}-d) \\ \ \quad =\frac{d}{2(d+1)} \\ \end{array}\end{eqnarray*}$$

Thus, our bound (52) reads ${{\Delta }_{H}}\leqslant 2\frac{d}{2(d+1)}=\frac{d}{d+1}$ and is quite tight in this case, almost matching the actual quality of the hiding scheme (56).