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Detecting entanglement of random states with an entanglement witness

The entanglement content of high-dimensional random pure states is almost maximal, nevertheless, we show that, due to the complexity of such states, the detection of their entanglement using witness operators is rather difficult. We discuss the case of unknown random states, and the case of known random states for which we can optimize the entanglement witness. Moreover, we show that coarse graining, modeled by considering mixtures of m random states instead of pure ones, leads to a decay in the entanglement detection probability exponential with m. Our results also allow to explain the emergence of classicality in coarse grained quantum chaotic dynamics.


Introduction
Random pure quantum states, i. e., states whose distribution is invariant under an arbitrary unitary transformation, are almost maximally entangled. Let us consider, for instance, a random state |ψ from N × N dimensional Hilbert space H = C N 2 . We consider a bipartition to subsystems A and B, namely |ψ as an element of a tensor product Hilbert space H A ⊗ H B where H A = C N , H B = C N are Hilbert spaces of subsystems A and B, respectively. The entanglement E of the state |ψ is then measured by the von Neumann entropy S(ρ) = − tr (ρ log 2 ρ) of the reduced density matrices ρ A = tr B |ψ ψ| or ρ B = tr A |ψ ψ|. It turns out [1,2,3] that which is close to the maximal value log 2 N attained for maximally entangled state. Since random states carry a lot of entanglement and entanglement [4,5] has no analog in classical mechanics, one can conclude that random states are highly non-classical. On the other hand, pseudo-random states with properties close to those of true random states can be efficiently generated by dynamical systems (maps) in the regime of quantum chaos [6,7,8,9,10,11]. In such chaotic maps the classical limit is recovered when N → ∞. Therefore, one can argue that for high-dimensional random states, i.e., in the limit N → ∞, the quantum expectation value of an operator with a well defined classical limit will be close to its classical microcanonical average. According to this picture random states in a way "mimic" classical microcanonical density. Expectation values are therefore close to the classical ones.
At first sight this is in striking contrast with the almost maximal entanglement of such states. How can we reconcile this apparent contradiction? In the present paper we are going to tackle this question by considering how can we detect entanglement of random states. By studying the detection of entanglement with decomposable entanglement witnesses we are going to argue that in the limit of large systems the detection of entanglement in a random state becomes increasingly difficult as it would demand the control of very finely interwoven degrees of freedom and a measurement resolution inversely proportional to the size of the Hilbert space, which seems hardly feasible experimentally. Therefore, as far as the detection of entanglement is concerned, high dimensional random states are effectively classical.
Moreover, coarse graining naturally appears. For instance, one could repeat several times the measurement of the entanglement witness for a random state and the prepared random state would be different from time to time due to unavoidable experimental imperfections. We model this problem by considering mixtures of m pure random states, namely where the |ψ i are mutually independent random pure states, but in general they are not orthogonal. We are going to show that the detection of entanglement is even more difficult for these mixed states, as it requires a number of measurements growing exponentially with m.
There are other physical contexts in which formally the same kind of coarse graining (mixing of the state) naturally appears: (i) Time averaging. For example, if a state |ψ undergoes a time evolution |ψ(t) = U(t)|ψ given in terms of some unitary dynamics U(t), then the time average of a physical observable A over an interval T is essentially determined by expectation value tr (Aρ) in the mixed state which has an effective rank m ≈ T /t corr where t corr is a dynamical correlation time of the dynamics U(t). For a quantum chaotic evolution U(t), the state |ψ(t) can be, after some time, arguably well described by a random state and the correlation time t corr is expected to be short, so ρ (3) may be considered as a mixture (2) of m uncorrelated random states. (ii) Phase space averaging. Sometimes it is useful to represent quantum states in terms of distribution functions in the classical phase space, like the Husimi function (see e.g. [7]), which can be understood as a convolution of the Wigner function or its coarse graining over a phase space volume (2π ) d in d degreees of freedom. In fact, the Husimi function of a pure state can be understood as a Wigner function of the following mixed state where T ( q, p) are unitary phase space translation operators, and α is an arbitrary squeezing parameter. A random pure state |ψ has a Wigner function with random sub-Planck structures with phase space correlaton length l corr ∼ which is semi-classically smaller than the coarse-graining width ∼ 1/2 , so ρ (4) can be again considered as a mixture (2) of m random pure states with m ∼ −1/2 . The paper is organized as follows. In Sec. 2, we review known results about entanglement witnesses and pure states entanglement. Depending on our a priori knowledge of the random state, two relevant cases can be distinguished: (i) We do not know in advance of which random state we are going to detect the entanglement, i.e., the random state is unknown. In such a case the best one can do is to choose some generic entanglement witness in advance, independently of the state we measure. Such situation can also be thought to arise in the case when we are not able to prepare an arbitrary entanglement witness but just some subset of witnesses. (ii) We know the random state in advance and we are able to prepare an arbitrary entanglement witness. In such case we can use the optimal entanglement witness for each random state separately. These two cases are discusses in Secs. 3 and 4, respectively. Finally, in Sec. 5, we provide a brief discussion of our results.

Entanglement witnesses
First of all, let us introduce some known facts about entanglement witnesses and entanglement of pure states that we will need for the derivation of our results in subsequent sections. Given a pure state |ψ , its bipartite entanglement content is completely specified by the Schmidt decomposition: where the Schmidt coefficients µ i are positive real numbers satisfying i µ 2 i = 1 and |a i and |b i are orthonormal states on subspaces H A and H B . The squares of the Schmidt coefficients are equal to the eigenvalues of the reduced density matrices ρ A and ρ B . For such bipartition there are at most N nonzero Schmidt coefficients µ i . For a state having r roughly equal nonzero Schmidt coefficients (µ i ∼ 1/ √ r), the reduced von Neumann entropy is S(ρ A ) ∼ log 2 r. For instance, the simple (GHZ) state |ψ = 1 √ 2 |0 . . . 0 + 1 √ 2 |1 . . . 1 leads to two nonzero eigenvalues of ρ A (µ 1 = µ 2 = 1/ √ 2), and simple Schmidt vectors |a 1 ⊗ |b 1 = |0 . . . 0 and |a 2 ⊗ |b 2 = |1 . . . 1 , resulting in S = 1. On the other hand, a random pure state has all N eigenvalues nonzero, very complicated (random) Schmidt vectors |a i and |b i , and almost maximal entropy S ≈ log 2 N − 1/ log e 4 [1,2,3]. Note, however, that all the eigenvalues of the reduced density matrix for a bipartite random state are very small. In fact, they decrease with N, the largest being, on average, 4/N while the smallest is 1/N 3 [12]. The guiding idea of this paper is that this smallness of eigenvalues and the complexity of eigenvectors should be somehow reflected in the difficulty of detecting entanglement in such a state, despite the fact that the entanglement content of a random state is large.
Besides pure states, we will also be interested in the entanglement of a mixture of pure states (2). A quantum state ρ of a bipartite system is called separable if it can be written as where ρ Ak and ρ Bk are density matrices for the two subsystems. A state is entangled if it is not separable. To decide whether a given mixed state is entangled or not is a difficult problem [4,5]. Fortunately, there is an operational criterion which is able to detect the most useful entangled states. This is the famous positive partial transposition (PPT) criterion [13]: since separable states have positive semidefinite partial transpose ‡, all non-PPT states are entangled. Note, however, that for dimensions higher than 2 × 2 and 2 × 3 there exist PPT-entangled states, known as bound entangled states [14]. A convenient way to detect entanglement is to use the so-called entanglement witnesses [14,15]. By definition, an entanglement witness is a Hermitian operator W such that tr (W ρ sep ) ≥ 0 for all separable states ρ sep while there exists at least one state ρ ent such that tr (W ρ ent ) < 0. Therefore, the negative expectation value of W is a signature of entanglement and the state ρ ent is said to be detected as entangled by the witness W . The concept of entanglement witness is close to experimental implementations and detection of entanglement by means of entanglement witnesses has been realized in several experiments [16,17,18]. Because it is easier to measure a larger negative expectation value of an entanglement witness, one can argue that the detection of entanglement is easier the larger this negative value is. The expectation value of W also provides lower bounds to various entanglement measures [19,20]. Estimation of entanglement entropy by measurement of observables has been considered in [21]. ‡ Introducing and orthonormal basis {|iα = |i A ⊗ |α B } in the Hilbert space associated with the bipartite system, the density matrix ρ has matrix elements ρ iα,jβ = iα|ρ|jβ . The partial transpose is constructed by taking the transpose of only Latin or only Greek indexes (here Latin indexes refer to subsystem A and Greek indexes to subsystem B). For instance, the partial transpose with respect to subsystem B is given by Finally, it is interesting to note that violation of Bell inequalities can be rewritten in terms of nonoptimal entanglement witnesses [15,22]. In general, classification of entanglement witnesses is a hard problem. However, much simpler is the issue with the so-called decomposable entanglement witnesses (D-EW) [23]. D-EW is a witness which can be decomposed as that is, with positive semidefinite operators P, Q. D-EW can detect only non-PPT entangled states, i.e., those with negative eigenvalues of ρ T B . They are therefore equivalent to the PPT criterion but closer to experimental implementations, as full tomographic knowledge about the state is not needed. In the present paper we are going to limit ourselves only to D-EW. General non-decomposable entanglement witness (ND-EW) can be written in a canonical form as W = P + Q T B − ǫ½ [23,24] and can also detect entangled states with PPT. Finding an optimal ND-EW, for which violation of positivity is maximal, is in general hard [23].

Unknown random state
In this section we assume that the random state |ψ whose entanglement we would like to detect is unknown so that we are not able to use an optimal W for a particular |ψ . The best one can do is to choose some fixed entanglement witness W in advance, independently of the state. Since we will be interested in the average behaviour over unitary invariant ensemble of pure random states, W can be chosen to be random as well. That is, in the present section we are going to study detection of entanglement with a random entanglement witness, whose precise definition will be given later. What we want to calculate is the distribution of the expectation values ψ|W |ψ for a fixed W and an ensemble of random pure states |ψ . Averaging over random states |ψ we see that the average expectation value ψ|W |ψ is where • = dP• denotes an integration over a unique U(N 2 )-invariant distribution of pure states |ψ , and we used the fact that for a random state |ψ = i c i |i we have c i c * j = δ ij /N 2 . We fix normalization of the entanglement witness W such that tr W = 1. Therefore, the average expectation value ψ|W |ψ scales ∝ 1/N 2 . We therefore define the rescaled quantity w = N 2 ψ|W |ψ , such that w = 1, independently of the dimension N. From now on we will focus on the random variable w, and its distribution with density p(w) = dP/dw. Because operator P in D-EW (8) just shifts the expectation value towards positive values, we limit ourselves to D-EW of the form W = Q T B . Any positive semidefinite operator Q can be written in its eigenbasis in terms of positive eigenvalues d i , satisfying We will first study the case when Q is a simple rank one projector (subsection 3.1), that is W is given by W = (|φ φ|) T B . There are two reasons why this is the most important case. First, as we will see below, optimal D-EW is always of such a "projector" form. Second, the expectation value for a general Q can be written (subsection 3.2) as a sum of expectation values for individual eigenvectors of Q and therefore the probability distribution of w will be a simple convolution of distributions for the case of Q being rank one.
Most of our theoretical results are derived for one mixing component only, m = 1 (i.e. for a pure random state), since the general result for arbitrary number m of independent mixing components is obtained by simple convolutions as discussed in subsection 3.3.

Q is a projector
The entanglement witness is of the form W = (|φ φ|) T B , with a fixed |φ , and we would like to calculate the density of probability distribution p(w) of its normalized expectation values w for random pure states. This distribution can depend on the chosen |φ . First few moments of the density p(w) can be calculated explicitly. We have already seen that w = 1. For higher moments we can in the leading order (in Hilbert space dimension 1/N 2 ) use Gaussian averages and Wick contractions in order to approximate integrals over the unitary group [25]. Using and similarly for higher order products, we arrive, up to corrections O(N −2 ), at Here we denote by κ n the n-th cumulant. We see that the average value of w as well as the width of the distribution is 1, independently of the state |φ we use for W . While tr W = 1 is a simple normalization, second moment is 1 due to Q being of rank 1. If only the first two cumulants were nonzero our probability density p(w) would be a simple Gaussian.
Let us first see what happens if we choose for |φ a state with large Schmidt rank r (number of nonzero eigenvalues of the reduced density matrix σ A = tr B |φ φ|) of order r ∼ N. Using Schmidt decomposition (5) for the state |φ = i µ i |a i ⊗ |b i we can immediately write eigenvalues and eigenvectors of the Hermitian operator W = (|φ φ|) T B . There are N(N − 1) eigenvalues ±µ i µ j , i < j, with the corresponding eigenvectors (|a i b * j ± |a j b * i )/ √ 2 and N eigenvalues µ 2 i with the corresponding eigenvectors |a i b * i . In our notation, Using these eigenvalues one can see that the traces of powers of W are Because eigenvalues of σ A are of order ∼ 1/r, we have tr N 2 ) and so on. Note that formulas (11) are exact up to order 1/N 2 and therefore can not be used for κ n≥3 and states |φ with the full Schmidt rank. Nevertheless, we can conclude that the higher cumulants are at most ∼ 1/N 2 and therefore vanish in the limit N → ∞. Therefore, for W and |φ with an increasing rank, the probability density p(w) converges to a Gaussian in the limit N → ∞, Theoretical prediction (13) is compared with the results of numerical simulation in figure 1. Probability to measure negative w, i.e., of detecting entanglement, is Note that this entanglement detection probability is independent of the details of |φ , provided that its Schmidt rank r is large, more precisely r ∝ N.
The above Gaussian form of p(w) can be understood also as a consequence of the central limit theorem. Indeed, let us write W in its eigenbasis, then we have, writing Denoting overlaps by ψ|a i b * j = √ y ij e iϕ ij , where y ij and ϕ ij are two real numbers, we For the overlap of two random states we know that the angle ϕ is distributed uniformly, while the amplitude (scaled by N 2 ) has an exponential distribution, p(y ij ) = exp (−y ij ). Therefore, for a given set of eigenvalues λ i of σ A , and assuming y ij to be independent, we can calculate the distribution of w using the above formula. It is a convolution of distributions of individual terms. In the limit r → ∞ the central limit theorem can be used, resulting in a Gaussian distribution (13).
For |φ with a small finite Schmidt rank there will be deviations from Gaussian because higher cumulants of p(w) (11) will be in general nonzero also in the limit N → ∞. The above form for w (16) is actually very handy for an explicit calculation of p(w) for |φ with a small rank r. Let us take the extreme case of rank r = 2, where we expect strongest deviations from a Gaussian. We therefore have only two nonzero eigenvalues of σ A , λ 1 = λ and λ 2 = 1 − λ. Assuming the overlaps y ij to be independent and exponentially distributed and the angles ϕ ij to be uniform, we arrive after evaluating few convolutions at The distribution is a sum of exponentials. Probability to detect entanglement, i.e., to measure negative value of w is P(w < 0) = 1/(4 + 2/ λ(1 − λ)). As a function of λ it reaches a maximum for λ = 1/2, i.e., both eigenvalues of σ A are equal, where it is equal to 1/8 (which is less than for Gaussian distribution (13)). In figure 2 we compare results of numerical simulation for p(w) for two cases: one with λ = 1/2, for which the appropriate limit of eq. (17) gives : w > 0, 1 4 e 2w : w < 0. (18) and the case with an almost pure σ A , λ = 1/26, for which the probability of detecting entanglement is 5/72 ≈ 0.07. Note that in the limit λ → 0, i.e., of a pure product state for |φ , w will always be positive with an exponential distribution.

Q of higher rank
So far we have discussed only the case when Q is a one-dimensional projector, Q = |φ φ|. What happens if the rank of Q is larger? If Q is of rank 2, W can be written as W = d 1 W 1 + d 2 W 2 with positive d 1,2 and W 1,2 = (|φ 1,2 φ 1,2 |) T B . If we assume W 1 and W 2 are statistically independent, so that w 1 = N 2 ψ|W 1 |ψ and w 2 = N 2 ψ|W 2 |ψ are uncorrelated, then the distribution of w = d 1 w 1 + d 2 w 2 is given by a convolution of distributions for w 1,2 . Let us calculate the second moment of w given by the above sum. Using we get We see that the width of the distribution of w is the same as in the case of convolution of two independent distributions, leading to a width d 2 1 + d 2 2 , provided the two vectors |φ 1 and |φ 2 are orthogonal. Using similar considerations for higher moments we can conclude that in the case of Q = k i d i |φ i φ i | having rank k, and because eigenvectors of Q are orthogonal, the distribution of w is Gaussian of width σ 2 = k i d 2 i , at least for sufficiently large k. Assuming for simplicity that all d i are the same, d i = 1/k, we obtain

Mixtures of random states
It is also interesting to consider the case in which Q is of rank 1 but we wish to detect the entanglement of mixed states, for instance of a mixture of m pure random states, as given in eq. (2). In such case the expectation value is w = 1 m i tr (W |ψ i ψ i |), with W = (|φ φ|) T B . Repeating the same derivation as for Q of higher rank and pure states, just replacing k by m, we see that the distribution of w will be a convolution of distributions for individual |ψ i and, provided these are statistically independent, the width of the resulting Gaussian (22) will be 1/m. Because the probability density p(w) becomes narrowly peaked about its mean w = 1 with increasing m, the probability to measure negative values decreases with m, namely for the Gaussian form of eq. (22) we have This probability decays to zero exponentially with m. Therefore, the detection of entanglement for a mixture of random states is very hard.

Known random state
In this section we assume that the random state |ψ whose entanglement we want to measure is known in advance and furthermore, that we are able to prepare an arbitrary D-EW. In addition, we have to assume that our state |ψ is neither separable, nor bound entangled, which is true with probability which converges to one exponentially in N. Therefore, for each |ψ we can prepare an optimal entanglement witness, such that its expectation value will be minimal. As far as D-EW are concerned, the optimal choice of W = W opt is to take for Q a projector to the eigenspace corresponding to the minimal (negative) eigenvalue λ min of ρ T B , W opt = (|φ min φ min |) T B . The maximal violation of positivity is therefore If we are able to measure entanglement witness with a given precision § it is the size of λ min which determines the difficulty of detecting entanglement in |ψ . Note that the optimal entanglement witness W opt depends on the state |ψ . For each state |ψ we have to pick a different W opt .

Distribution of eigenvalues of ρ T B
First, let us look at the distribution of eigenvalues after PT operation for a single random pure state. The eigenvalues can be written in terms of Schmidt coefficients µ i as λ = ±µ i µ j . Formally, we can write the distribution of λ, for λ > 0, as where distribution of µ i can be for large N obtained from Marčenko-Pastur law [26] for the distribution of τ = Nµ 2 i , namely dP/dτ = τ (4 − τ )/(2πτ ), by a simple change of variables τ → µ i . The result for the distribution of scaled eigenvalues y = Nλ reads dP dy = 1 8π 2 (16 + y 2 )K(1−y 2 /16) − 32E(1−y 2 /16) , y ∈ [−4, 4], (27) where E(x) and K(x) are elliptic integrals. Note that this distribution takes into account only the N(N − 1) "off-diagonal" eigenvalues ±µ i µ j with i < j. "Diagonal" eigenvalues µ 2 i have the same distribution as the eigenvalues of ρ A and are only N in number. The expectation value of the minimal eigenvalue equals λ min = −4/N. In fact the distribution of λ min becomes strongly peaked around −4/N with diminishing relative fluctuations as N → ∞. When we mix several independent (in general non-orthogonal) random vectors, ρ = m i |ψ i ψ i |/m, the minimal eigenvalue λ min increases and the distribution becomes increasingly sharply peaked (for m → ∞ we get ρ → ½/N 2 with all eigenvalues being equal to 1/N 2 ). We numerically verified the above theoretical prediction for dP/dy (27) in figure 3. We can also see that after mixing many random states (m ∼ N) the distribution becomes a semicircle, which is a numerical result for which we have yet no analytical explanation.

Minimal eigenvalue of ρ T B
Because λ min determines the maximal violation for an optimal witness W (24) we are going to look in more detail at the dependence of λ min on m and N. For a single pure state (m = 1) we know that the average minimal eigenvalue is λ min = −4/N. On the other hand, we also know that for large m, as we approach completely mixed state the minimal eigenvalue must scale as λ min ∼ 1/N 2 . Therefore, the scaling of λ min has to change as we increase the number of mixed states m. To confirm this expectation,  Figure 4. Dependence of λ min on m. Overlapping of curves for different dimension N signals the scaling λ min ∼ −1/N for small m in the left frame (a) and λ min ∼ −1/N 2 for large m (but smaller than m * whereλ min changes sign) in the right frame (b). We average over 1000, 100, 10, and 6 mixed random states (2), for N 2 = 2 6 , 2 8 , 2 10 and 2 12 , respectively.
Although von Neumann entropy of a random state is large all eigenvalues of ρ T B are very small and will therefore be hard to detect. If we assume that we are able to measure values of tr (W ρ) with accuracy ǫ then we can, depending on the scaling of ǫ with N, tell for which values of m the detection of entanglement is possible. If ǫ does not depend on N, i.e., precision does not increase with N, then for sufficiently large N, such that 4/N < ǫ, detection of entanglement will be impossible. Already a single random state becomes from the viewpoint of entanglement detection "classical", since measuring a negative expectation value of its optimal entanglement witness is below the detection limit. If on the other hand we are able to measure ǫ which decreases as 1/N, the critical m crit , beyond which the entanglement detection is impossible, will be independent of N, i.e., in the limit N → ∞ the ratio m crit /N → 0 (see fig. 4a). If however we are able to detect very small expectation values of order 1/N 2 , then m crit will be proportional to N 2 (see fig. 4b). Furthermore, even with arbitrary accuracy, detection of entanglement with D-EW is impossible beyond m = m * ∝ N 2 .
Worth mentioning is that for m ≥ N the mixed state ρ of eq.(2) can not be used for dense coding. Indeed a quantum state is useful for dense coding [27] only if S(ρ A ) − S(ρ) > 0 (see e.g. [28]). For mixtures of m random states S(ρ) is roughly equal to ∼ log m. On the other hand S(ρ A ) is at most log N. Therefore, S(ρ A ) will be smaller than S(ρ) for m ∼ N.

Conclusions
In this paper we have considered random states and have shown that, while their entanglement content is almost maximal, the detection of such entanglement is very difficult. This is a consequence of the complexity of a random state, which leads to a large number of small coefficients in the Schmidt decomposition of the state. Nevertheless, for random pure states, a finite success probability in the detection of entanglement exists also in the limit in which the Hilbert space dimension N 2 → ∞. This implies that quantum chaos alone is not sufficient to erase any trace of entanglement in the classical limit, provided that ideal measurements are possible. On the other hand such erasure becomes very efficient when coarse graining is taken into account, for instance when mixtures instead of pure states are considered.
We note that all our results can be straightforwardly generalized to the case of unbalanced bipartition N = dimH A = N ′ = dimH B , for instance, distribution of w = NN ′ ψ|W |ψ is Gaussian (13) provided both dimensions N, N ′ are large.
We would like to stress once more than the detection difficulties are a consequence of the complexity of random states. If instead one considers "regular states" such as the GHZ state |GHZ = 1 √ 2 (|0...0 + |1...1 ), then the optimal witness is W opt = (|φ min φ min |) T B , with |φ min = 1 √ 2 (|0...0 |1...1 − |1...1 |0...0 ) which corresponds to the minimal eigenvalue λ min = −1/2 of (|GHZ GHZ|) T B . Since the value of λ min is −1/2 instead of −4/N as for a random state, it turns out that it will be much easier to detect entanglement in a "regular" rather than in a random state. This happens in spite of the fact that the entanglement content is larger in a random than in such a regular state.