VOLUME 72 30 MAY 1994 NUMBER 22 Statistical Distance and the Geometry of Quantum States Samuel L. Braunstein' and Carlton M. Caves Center for Advanced Studies, Department of Physics and Astronomy, University of ¹toMexico, ASuquerque, Negro Mexico 87181-1156 (Received 11 February 1994) By finding measurements that optimally resolve neighboring quantum states, we use statistical distinguishability to define a natural Riemannian metric on the space of quantum-mechanical density operators and to formulate uncertainty principles that are more general and more stringent than standard uncertainty principles. PACS numbers: 03.65.Bz, 02.50.—r, 89.70.+c One task of precision quantum measurements is to de- where p~ = r2. Using this argument, Wootters was led tect a weak signal that produces a small change in the to the distance spD, which he called statistical distance. state of some quantum system. Given an initial quantum Wootters generalized statistical distance to quantum- state, the size of the signal parametrizes a path through mechanical pure states as follows [1]. Consider neighbor- the space of quantum states. Detecting a weak signal expanded in an orthonormal basis ing pure states, I j): is thus equivalent to distinguishing neighboring quantum states along the path. We pursue this point of view by us- (2) ing the theory of parameter estimation to formulate the problem of distinguishing neighboring states. We find measurements that optimally resolve neighboring states, and we characterize their degree of distinguishability in terms of a Riemannian metric, increasing distance corre- Normalization implies that Re((/Id')) = —z(dgldg). sponding to more reliable distinguishability. These con- Measurements described by the one-dimensional projec- siderations lead directly to uncertainty principles that tors lj)(jl can distinguish I@) and lg) according to the are more general and more stringent than standard un- classical metric (1). The quantum distinguishability met- certainty principles. ric should be defined by measurements that resolve the We begin by reviewing Wootters's derivation [1] of two states optimally —i.e., that maximize Eq. (1). a distinguishability metric for probability distributions. The maximum is given by the Hilbert-space angle After drawing N samples from a probability distribution, cos t(1(pig) I), which clearly captures a notion of state distinguishability. The line one can estimate the probabilities p~ as the observed fre- corresponding element, quencies f~, The probability for the frequencies is given — —,'dsps = = 1 — I' = by a multinomial distribution, which for large N is pro- [cos '(l(414) I)]' l(@14» (A'~ I+'~) — — portional to a Gaussian exp[ (N/2)(f~ p~)2/p~] A. nearby distribution pz can be reliably distinguished from (4) ps if the Gaussian exp[ —(N/2)(p~ —ps) /p~] is small. 2"' 2 2 Thus the quadratic form —p~)2/p~ provides a nat- (p~ called the Fubini-Study metric [2], is the natural metric ural (Riemannian) distinguishability metric on the space on the manifold of Hilbert-space rays. Here of probability distributions (PD), Id@&) ld@) —1@)(@Id/) is the projection of ld@) orthogonal to d2 I@). The term in large square brackets, the variance of —= ' = = dspD ) ) p, (dlnp, ) 4) dr, , (1) the phase changes, is non-negative; an appropriate choice 2 ' 2 2 003I-9007/94/72(22)/3439(5)$06. 00 3439 l 994 The American Physical Society VOLUME 72, NUMBER 22 PH YSICAL REVl EVr' LETTERS of basis makes it zero [3]. Thus ds2ps is the maximum tistical deviation is min v N((hx}2) & ]. The v N re- value of ds&D, which means that dsps is the statistical moves the expected 1/~N improvement with the num- distance between neighboring pure states (PS). ber of measurements; the minimum means that statisti- We generalize the notion of statistical distance to im- cal distance is defined in terms of the most discriminating pure quantum states and thus obtain a natural Rieman- procedure for determining the parameter. nian geometry on the space of density operators [see We are thus led to define the distinguishability metric Eqs. (23) and (28)], where no natural inner product by guides the generalization. Our derivation, like Woot- dX' ters's, proceeds in two steps, one classical and one quan- Gs '8) min X((b'X)2)~-] tum mechanical, but it sharpens the formulation of statis- tical distance by highlighting distinct classical and quan- We take the minimum in the two steps mentioned above: tum optimization problems. For the first step, to ob- first, optimization over estimators for a given quantum tain the classical distinguishability metric, we use an ap- measurement to get the classical distinguishability met- proach based on the theory of parameter estimation [4], ric and, second, optimization over all quantum measure- This approach, which is independent of Wootters's work, ments to get the quantum distinguishability metric. maps the problem of state distinguishability onto that of The classical optimization relies on a lower bound. precision determination of a parameter. For the second called the Cramer-Rao bound [7], on the variance of any step, to obtain the quantum distinguishability metric, estimator. The proof of the Cramer-Rao bound proceeds we optimize over att quantum measurements, not just from the trivial identity measurements described by one-dimensional orthogonal projectors. 0 = d(i d(z p((ilX) p((+lX) hx„, , I9) Consider now a curve p(X) on the space of density op- erators. The from neigh- problem of distinguishing p(x) where AX„& = Xe8t((i, . , (rv) —(X„r)x. Taking the boring density operators along the curve is equivalent derivative of this identity with respect to X, we obtain to the problem of determining the value of the param- eter The determination is made from the results X. of d(i d4 p((ilX) p((rv X) measurements. To be general, we must allow arbitrary generalized quantum measurements [5,6], which include ( ~. Bing((„]x)) d(x„t)x all measurements permitted by the rules of quantum me- chanics. A generalized measurement is described by a set of Applying the Schwarz inequality to Eq. (10) yields the non-negative, Hermitian operators E(E), which are com- Cramer Rao bound- in the sense that plete -' xF(x)((~x...p), & )'. d( E(E) = i = (unit operator) . ( where the Fisher information is defined by The quantity labels the "results" of the measurement; ( 2 although written here as a single continuous real variable, """'l ' d(p((lx) l l it could be discrete or multivariate. The probability den- Bx ) sity for result given the parameter is (, X, &p((lx) p((IX) = tr(E(()i(x)) p((lx} BX Consider now N such measurements, with results Converted to the form needed in the definition (8), the (i, . , (iv. One estimates the parameter X via a func- Cramer-Rao bound becomes tion — . A sensible sta- X„r, X„i,((i, , (iv). definition of 1 tistical distance is to measure the parameter increment &((bX)')x ) + &(~x)x ) F X . dX in units of the statistical deviation of the estimator away from the parameter. The appropriate measure of A nonzero value of (bx)x means that the units-corrected deviation is estimator has a systematic bias away from the parameter; x„, (6X)x is zero when the estimator is unbiased, i.e. , when (X,t)x = X locally. ld(x„t )x /dx l The Cramer-Rao bound only places a lower bound on The derivative d(xesr)x/dX removes the local difference the minimum that appears in Eq. (8). Fisher's theo- in the "units" of the estimator and the parameter. The rem [8,9], however, says that asymptotically for large X, subscript X on expectation values reminds one that they maximum-likelihood estimation is unbiased and achieves depend on the parameter. The appropriate unit of sta- the Cramer-Rao bound. Thus, for a given probabil- 3440 VOLUME 72, NUMBER 22 P H YSI CA L R EV I E VV L ETTERS 30 MAY 1994 ity distribution we arrive at the classical distin- ters metric (1) at the boundary can be removed by using p((~X), — — guishability metric ds2PD —F(X)dX2, which, given the coordinates r~, where p& —r~, which essentially remove forms (12) of F, is the Wootters metric (1) for continu- the boundary. One now shows that ous, instead of discrete alternatives. = — The second step, to optimize over quantum measure- dXp' ) dp~lj)(jl+ idX) (S~ Sl )hl~lk)(jl (21) ments, is now seen to be the problem of maximizing the Fisher information over all quantum measurements, i.e., from which it follows that symbolically —dX max . dXtr(Ap') = idX — hI d@o F(X) (14) 2) r~A~~dr~ + ) (p~ pg)A~I, ~ I&(~)) 2 j,k The subscript DO reminds one that this is a metric on = dXRe tr(pAR. '(p')) (22 density operators. The expression for F(X) involves dividing by p((~X), provided that the singular matrix elements of R: (p') so one might expect the quantum distinguishability met- are assigned any finite values consistent with Hermiticity. ric to involve "division" by p. The appropriate sense of Choosing them to vanish conveniently extends R: to this division comes from defining a superoperator the boundary 'R-(o) = -'(p -'(po+op) =) +pa)o ~lj)(kl (») 'R,='(O) =— ). + O&~ iI)(kI (23) (i,&ln, +nato} The second form is written in the orthonormal basis where = is diagonal. In the interior of the We now manipulate the Fisher information to ob- p P. p~ ~ j)(j~ (17) space of density operators —i.e., away from the bound- tain an upper bound ary, where one or more of the eigenvalues pz vanishes- Rp has a well defined inverse 'R:, with matrix elements Retr R: p' F= jE ['R: (O)]&A, = 20&1,/(p~ + pl, ) in the basis that diagonal- d( tr (E(()p) izes P.